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NEW FIRST ORDER NECESSARY CONDITIONS FOR FREE
END-TIME PROBLEMS WITH MEASURABLY DEPENDENT DATA\ast PIERNICOLA BETTIOL\dagger AND RICHARD B. VINTER\ddagger Abstract. First order necessary conditions of optimality for free end-time optimal control problems with measurably time-dependent data include boundary conditions on the maximized Hamiltonian function evaluated along the optimal state trajectory and costate trajectory. Since pointwise
evaluation of an a.e. defined, measurable function is meaningless, boundary values of the maximized
Hamiltonian must be appropriately interpreted. In earlier papers, this difficulty was overcome by
expressing the necessary conditions in terms of elements in the essential values of the maximized
Hamiltonian. Necessary conditions involving essential values have been derived for dynamic optimization problems with dynamic constraints, either controlled differential equations or differential
inclusions. They have been extended to cover problems with pathwise state constraints. This paper
shows that, throughout this literature, available first order necessary conditions can be improved by
replacing the essential value of the maximized Hamiltonian by two, more refined, constructs, namely
the sub- and superessential values of the maximized Hamiltonian. We give an example of an optimal
control problem in which the new conditions, involving sub- and superessential values, can be used
to show that a certain admissible process is not a minimizer, while the former conditions, involving
essential values, fail to do so. The search for the appropriate definition of boundary value of the maximized Hamiltonian is intimately connected with the representation and estimation of limiting suband superdifferentials of indefinite integral functions; the paper includes a section on this topic. The
improvements to available first order necessary conditions provided by super- and subessential values
is connected with the fact that these sets capture precisely the differential properties of indefinite
integral values, whereas essential values only provide estimates, which may not be exact.
Key words. optimal control, free end-time problems, nonsmooth analysis
MSC codes. 49K15, 49K21, 49J52
DOI. 10.1137/24M1649265
1. Introduction. This paper concerns first order necessary conditions of optimality for free end-time dynamic optimization problems, that is, problems in which
the left and right end-times are included among the choice variables. Minimum time
problems, in which the aim is to drive the state from an initial state to a target set in
the state space in minimum time, are important examples of such problems. We shall
consider several formulations of the free end-time problem but, in this introduction,
we restrict attention to
\left\{ Minimize g(S, x(S), T, x(T ))
over intervals [S, T ], arcs x \in W 1,1 ([S, T ]; \BbbR n )
and measurable functions u : [S, T ] \rightarrow \BbbR m satisfying
x(t)
\. = f (t, x(t), u(t)) a.e. t \in [S, T ],
(F T )
u(t) \in U (t) a.e. t \in [S, T ],
and
(S, x(S), T, x(T )) \in C .
\ast Received by the editors March 25, 2024; accepted for publication (in revised form) March 7,
2025; published electronically July 18, 2025.
https://doi.org/10.1137/24M1649265
Funding: This research was partially funded by the French Agence Nationale de la Recherche
(ANR), under grant ANR-22-CE40-0010-01 (project COSS).
\dagger Univ Brest, UMR CNRS, 6205 Laboratoire de Math\'
ematiques de Bretagne Atlantique, F-29200
Brest, France (piernicola.bettiol@univ-brest.fr).
\ddagger Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road,
London, SW7 2BT, UK (r.vinter@imperial.ac.uk).
2547
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2548
PIERNICOLA BETTIOL AND RICHARD B. VINTER
The data for this problem comprise functions g : \BbbR 1+n+1+n \rightarrow \BbbR and f : \BbbR \times \BbbR n \times \BbbR m \rightarrow \BbbR n , a nonempty multifunction U : \BbbR ; \BbbR m , and a closed set C \subset \BbbR 1+n+1+n . This is
called a ``free end-time"" problem since the end-times S and T of the state trajectories
involved are included among the choice variables. The domain of this problem is the
set of elements ([S, T ], x, u) comprising a time interval [S, T ], a control u : [S, T ] \rightarrow \BbbR m ,
and a corresponding solution of the differential equation. In this paper attention
focuses on free end-time problems in which the time-dependence of the data is assumed
to be merely measurable. Let H denote the (maximized) Hamiltonian function for
(FT):
H(t, x, p) := sup p \cdot f (t, x, u).
u\in U (t)
When we seek to generalize first order necessary conditions for fixed time problems
to free end-time problems, we must find supplementary first order conditions, to
take account of the extra degrees of freedom in the problem formulation. These
traditionally take the form of boundary conditions on the maximized Hamiltonian t \rightarrow H(t, x(t),
\=
p(t)) , evaluated along the optimal state trajectory x
\= and the corresponding
costate trajectory p, at the optimal end-times S\= and T\=, namely
\= x(
\= p(S)),
\= p(S),
\= H(T\=, x(
( - H(S,
\= S),
\= T\=), p(T\=)), - p(T\=))
\= x(
\= T\=, x(
\= x(
\= T\=, x(
\in \lambda \partial g(S,
\= S),
\= T\=)) + NC (S,
\= S),
\= T\=)) .
Since the data are assumed measurably time dependent, it is natural to assume that
the maximized Hamiltonian is only almost everywhere defined, w.r.t. the time variable. But then the boundary conditions on t \rightarrow H(t, x(t),
\=
p(t)) have no meaning,
in a classical sense and require interpretation. The approach of Clarke and Vinter [7] was to express the boundary conditions in terms of the essential values of
h(t) := H(t, x(t),
\=
p(t)) at the optimal end-times. The essential value of this function
(at time t) is a multifunction defined according to
\Bigl[ \Bigl( \Bigr) \Bigl( \Bigr) \Bigr] \prime ess
h(t
)
:=
lim
ess
inf
h(t)
,
lim
ess
sup
h(t)
.
\prime t \rightarrow t
\epsilon \downarrow 0
\= ,t+\epsilon \= ]
t\in [t - \epsilon \epsilon \downarrow 0
\= ,t+\epsilon \= ]
t\in [t - \epsilon (The concept had precursors in the earlier Russian literature [9] and also in classical
constructions of the ``essential range"" of mappings on measure spaces [13].) The
essential value is a plausible candidate for interpreting the boundary values, because
it is invariant under changes on a null-set to the original function h and, in the case
that h is continuous, the essential value is point valued and coincides with point
evaluation of h.
There is a literature on first order necessary conditions for free end-time problems with measurably time-dependent data. Such conditions have been derived for
dynamic optimization problems formulated in terms of a collection of differential inclusions (``multiprocess"" systems) [7, 8], differential inclusions [6], systems with state
constraints ([10] and [12]), and problems involving controlled differential equations [11]
and [7], [8]. The free time component of the transversality condition in all these papers
is expressed in terms of the essential value of the maximized Hamiltonian evaluated
along the extremal of interest, at the optimal end-times.
The main contributions of this paper are to introduce new concepts of ``suband superessential values"" and then to derive necessary conditions in which the ``free
time"" transversality conditions are expressed in terms of these constructs. (Individual sections cover different formulations of the optimal control problems, namely
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FREE END-TIME PROBLEMS
2549
those involving controlled differential equations, differential inclusions, and state constraints.) The new necessary conditions improve on those of [7], [8], [6], [10], [12], and
[11] because, in each case, they incorporate a stronger transversality condition. The
proof techniques employed are based on Clarke's perturbation methods, according to
which we derive optimality conditions for a succession of optimal control problems,
of increasing complexity, ending in (FT). Such techniques have been used in earlier
work cited above. The difference is that, for the first and simplest problem, we derive
necessary conditions expressed in terms of sub- and superessential values, in place
of essential values, and we find that these conditions are retained in all the subsequent problems, including (FT). A summary version of some material in this paper
is included in the monograph [2].
We give a simple example of an optimal control problem, of a type encountered in
resource economics, which illustrates the benefits of the new necessary conditions. In
this example, there is an admissible process that satisfies earlier necessary conditions
(involving essential values) and therefore, in view of previous theory, might be considered a candidate for being a minimizer. But we can use the new necessary conditions
to eliminate this candidate, by showing that it violates the new, stronger necessary
conditions (involving sub- and superessential values).
The derivation of first order necessary conditions for free end-time problems is
based on the analysis of variations to the end-times. It turns out that obtaining
first order information about changes to the cost under such variations is intimately
connected with the representation and estimation of the limiting sub- and superdif\= of the indefinite integral function \psi ferentials, at a point t,
\int t
\psi (t) =
h(t\prime )dt\prime ,
S \prime in which h is a given integrable function and S \prime \in \BbbR . The advantages of expressing the
necessary conditions in terms of limiting sub- and superessential values is connected
with the fact that these constructs provide precise representations of the limiting sub\= whereas essential values offer only estimates which,
and superdifferentials of \psi at t,
in certain cases, are inexact. It is for these reasons that the paper begins with an
analysis of indefinite integral functions.
After this introduction, the remaining sections of the paper are organized as
follows. Section 2 introduces sub- and superessential values; these concepts underpin
all the new necessary conditions of this paper. Sections 3 and 4 provide a statement
and proof of our necessary conditions featuring these new constructs, in the case the
dynamic constraint is formulated as a controlled differential equation. Section 3 also
includes an example highlighting the differences with earlier necessary conditions and
illustrating how the new necessary conditions can be used. In section 5 we derive new
necessary conditions, also involving sub- and superessential values, covering problems
for which the dynamics are formulated as a differential inclusion. Finally, in section 6,
we indicate how the preceding necessary conditions can be generalized to take account
of pathwise state constraints.
Notation. Given an interval [a, b], L1 ( [a, b]; \BbbR n ) is the space of integrable nvector valued functions on [a, b]. We simply write L1 (a, b) for L1 ( [a, b]; \BbbR ). W 1,1 ( [a, b];
\BbbR n ) is the space of absolutely
x : [a, b] \rightarrow \BbbR n , equipped with the
\int b continuous functions
\oplus norm | | x| | W 1,1 := | x(a)| + a | x(t)| dt.
\.
We write C (a, b) for the set of elements in the
topological dual of the space of continuous functions C([a, b]; \BbbR ) with supremum norm,
taking nonnegative values on nonnegative valued functions in C([a, b]; \BbbR ). \| \mu \| T.V.
denotes the total variation of \mu \in C \oplus (a, b). By N BV ([a, b]; \BbbR n ) we denote the space
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2550
PIERNICOLA BETTIOL AND RICHARD B. VINTER
of \BbbR n -valued functions of bounded variation defined on [a, b] which are normalized in
the sense that they are right-continuous on (a, b). The Euclidean norm of a vector
x \in \BbbR n is written | x| . The closed unit ball in Euclidean space is written \BbbB with interior
\circ \BbbB . Given numbers a and b, a\wedge b and a\vee b mean, respectively, min\{ a, b\} and max\{ a, b\} .
co A denotes the convex hull of the set A, and dA (x) is the Euclidean distance of a
point x from A. The graph of a multifunction \Gamma : D ; \BbbR k (D \subset \BbbR n ) is the set
Gr \Gamma := \{ (x, \xi ) \in D \times \BbbR k : \xi \in \Gamma (x)\} .
We make use of some constructs from nonsmooth analysis, described in detail, for
example, in [5] and [11]: given a closed set C \subset \BbbR n and x \in C, the proximal normal
cone of C at x is
NCP (x):=\{ \zeta \in \BbbR n : \exists M > 0 s.t. \zeta \cdot (y - x) \leq M | x - y| 2 for all y \in C\} .
The (limiting) normal cone of C at x is
\Bigl\{ \Bigr\} NC (x) := lim \zeta i : \zeta i \in NCP (xi ) and xi \in C for all i, and xi \rightarrow x .
i\rightarrow \infty Given a lower semicontinuous function f : \BbbR n \rightarrow \BbbR \cup \{ +\infty \} and a point x such that
f (x) < +\infty , the proximal subdifferential of f at x is
\partial P f (x) := \{ \zeta \in \BbbR n : \exists M > 0, \epsilon > 0 such that
f (y) - f (x) \geq \zeta \cdot (y - x) - M | y - x| 2 for all y \in x + \epsilon \BbbB \} .
\^ (x), is the set
The strict subdifferential of f at x, \partial f
\Biggr\} \Biggl\{ \zeta \cdot (y
- x)
- (f
(y)
- f
(x))
\^ (x) := \zeta \in \BbbR n : lim sup
\leq 0 .
\partial f
| y - x| y\rightarrow x
The limiting subdifferential of f at x, \partial f (x), is the set
\Bigl\{ \Bigr\} \partial f (x) :=
lim \zeta i : \zeta i \in \partial P f (xi ), xi \rightarrow x, f (xi ) \rightarrow f (x) .
i\rightarrow \infty When the function f depends on two variables, we write \partial x f (\=
x, y)
\= to denote the
(limiting) subdifferential of x \rightarrow f (x, y)
\= at x.
\=
2. Differential properties of indefinite integral functions. We have observed in the introduction that there is a connection between first order necessary
conditions for dynamic optimization problems and differential properties of indefinite
integral functions. The purpose of this section is to assemble the analytical tools for
representing and estimating these differentials.
Take an integrable function h : [S, T ] \rightarrow \BbbR and t\= \in (S, T ). Fix any S \prime \in [S, T ] and
define the absolutely continuous function \psi : [S, T ] \rightarrow \BbbR to be
\int t
(2.1)
\psi (t) :=
h(s)ds .
S \prime (The value of the indefinite integral function at a point t < S \prime has its usual interpre\int S \prime tation, namely - t h(s)ds.) The following multipart definition introduces concepts
which are useful in characterizing and estimating limiting sub- and supergradients of
this indefinite integral function.
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FREE END-TIME PROBLEMS
2551
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Definition 2.1. Take h \in L1 (S, T ) and t\= \in (S, T )
(A1 ): (Subessential value)
sub-ess h(t)
t\rightarrow t\=
\Bigl\{ := \zeta \in \BbbR : \exists ti \rightarrow t\= and \zeta i \rightarrow \zeta s.t.
\int ti
\int ti +\epsilon lim sup \epsilon - 1
h(s)ds \leq \zeta i \leq lim inf \epsilon - 1
h(s)ds,
\epsilon \downarrow 0
\epsilon \downarrow 0
ti - \epsilon ti
\Bigr\} for each i ;
(A2 ): (Superessential value)
super-ess h(t)
t\rightarrow t\=
\Bigl\{ := \zeta \in \BbbR : \exists ti \rightarrow t\= and \zeta i \rightarrow \zeta s.t.
\int ti +\epsilon \int ti
lim sup \epsilon - 1
h(s)ds \leq \zeta i \leq lim inf \epsilon - 1
h(s)ds,
\epsilon \downarrow 0
\epsilon \downarrow 0
ti
ti - \epsilon \Bigr\} for each i ;
(B): (Essential value)
\Bigr] \Bigl[ ess h(t) := lim ess inf h(t) , lim ess sup h(t) ;
t\rightarrow t\=
\epsilon \downarrow 0 t - \epsilon \= \leq t\leq t+\epsilon \=
\epsilon \downarrow 0 t - \epsilon \= \leq t\leq t+\epsilon \=
(C): (Lebesgue essential value)
\scrL -ess h(t)
t\rightarrow t\=
\Bigl\{ \Bigr\} \= \leq \epsilon \} > 0 for all \epsilon > 0\} .
:= \zeta \in \BbbR : \scrL -meas\{ s \in [S, T ] : | h(s) - \zeta | \leq \epsilon , | s - t| Given a function of two variables h(t, x), esst\rightarrow t\= h(t, x) indicates the essential value of
t \rightarrow h(t, x), for fixed x, etc.
Comments.
(i): Sub- and superessential values are new concepts, whose role will be precisely
to capture limiting sub- and super differentials of indefinite integral functions.
Observe that since the subessential value of h and the superessential value of
h at a point t\= \in (S, T ) do not depend on S \prime \in [S, T ], the limiting sub- and
superdifferentials of the indefinite integral function \psi at t\= do not depend on
the choice of the point S \prime \in [S, T ] used to define \psi , either.
(ii): Essential values, which provide an outer approximation to limiting sub- and
superdifferentials, have been used to formulate first order free time necessary
conditions since they were introduced into the Western literature in 1989
[7, 8]. As we shall see, this approximation can fail to be tight. In this
earlier work, they were given an equivalent definition, for essentially bounded
functions h, as the convex hull of Lebesgue essential values. Subsequently
(see, e.g., [6], [11]) the above definition of essential value above has become
standard.
(iii) The Lebesgue essential value is a classical concept [13], aimed at identifying
``significant"" points in the range of a function, whose domain is a measure
space (for example, Lebesgue measure on [S, T ]).
We begin by stating, without proof, some elementary properties of essential values,
sub- and superessential values, and Lebesgue essential values. (For a proof of the
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2552
PIERNICOLA BETTIOL AND RICHARD B. VINTER
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properties of the essential values, see [7], [8], and [11, section 8.3]. The analysis can
also be easily adapted to cover other cases.)
Proposition 2.2. Take h : [S, T ] \rightarrow \BbbR to be an integrable function. Let A :
(S, T ) ; \BbbR be either (i) the subessential value, or (ii) the superessential value, or (iii)
the essential value, or (iv) the Lebesgue essential value of h at t for t \in (S, T ). Then,
in each case,
(a): A is invariant under changes to the values of h on a set of Lebesgue measure
zero,
(b): if h is continuous at t \in (S, T ), then A(t) = \{ h(t)\} ,
(c): A(t) is closed, for each t \in (S, T ),
(d): lim supt\prime \rightarrow t A(t\prime ) \subset A(t), for each t \in (S, T ).
The next proposition provides the information that sub- and superessential values
precisely capture the sub- and superdifferentials of the indefinite integral function.
Proposition 2.3. Take an integrable function h : [S, T ] \rightarrow \BbbR and t\= \in (S, T ).
Define the absolutely continuous function \psi : [S, T ] \rightarrow \BbbR according to (2.1). Then
\= and super-ess h(t) = - \partial ( - \psi )(t)
\= .
sub-ess h(t) = \partial \psi (t)
t\rightarrow t\=
t\rightarrow t\=
Furthermore
sub-ess h(t) \subset ess h(t), super-ess h(t) \subset ess h(t), and \scrL -ess h(t) \subset ess h(t) .
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
\= if and only if there exists \zeta i \rightarrow \zeta and
Proof. Since \psi is continuous, \zeta \in \partial \psi (t)
\^ i ) for each i. (We have used the characterization of limiting
ti \rightarrow t\= such that \zeta i \in \partial \psi (t
\^ i )"" can be equivasubdifferentials as limits of strict subdifferentials.) But ``\zeta i \in \partial \psi (t
lently stated
lim sup \epsilon - 1 (\psi (ti ) - \psi (ti - \epsilon )) \leq \zeta i \leq lim inf \epsilon - 1 (\psi (ti + \epsilon ) - \psi (ti ))
\epsilon \downarrow 0
\epsilon \downarrow 0
or
(2.2)
lim sup \epsilon \epsilon \downarrow 0
- 1
\int ti
h(s)ds \leq \zeta i \leq lim inf \epsilon - 1
\int ti +\epsilon \epsilon \downarrow 0
ti - \epsilon h(s)ds .
ti
An analogous analysis can be carried out for points in the limiting superdifferential
\= We have proved the first assertion. Now take any \epsilon \= > 0. Then, for i
- \partial ( - \psi )(t).
sufficiently large, (2.2) implies
ess inf
s\in [ti - \=
\epsilon ,ti +\=
\epsilon ]
h(s) \leq \zeta i \leq ess sup h(s) .
s\in [ti - \=
\epsilon ,ti +\=
\epsilon ]
(We allow the possibility that either of these bounds is infinite.) Passing to the limit
as i \rightarrow \infty , we see that \zeta \in [ess infs\in [t - \epsilon \= ,t+\epsilon \= ] h(s), ess sups\in [t - \epsilon \= ,t+\epsilon \= ] h(s)]. Now letting
\epsilon \= \downarrow 0, we conclude that \zeta \in esst\rightarrow t\= h(t).
Now take \zeta \in \scrL -esst\rightarrow t\= h(t). It follows directly from the definition that lim\epsilon \downarrow 0
ess inft - \epsilon h(t) \leq \zeta \leq lim\epsilon \downarrow 0 ess supt - \epsilon h(t). We have shown \scrL -esst\rightarrow t\= h(t) \subset \= \leq t\leq t+\epsilon \=
\= \leq t\leq t+\epsilon \=
esst\rightarrow t\= h(t).
The following proposition concerns graph closure properties of sub- and superessential values concerning a family of functions t \rightarrow h(t, x), parameterized by an
additional variable x.
Proposition 2.4. Take an interval [S, T ] \subset \BbbR , a set A \subset \BbbR n , and a function
h : [S, T ] \times A \rightarrow \BbbR . Assume that
(a): t \rightarrow h(t, x) is integrable on [S, T ] for each x \in A;
(b): x \rightarrow h(t, x) is continuous on A, uniformly over t \in [S, T ].
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2553
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FREE END-TIME PROBLEMS
\= x) \in (S, T ) \times A,
Then, for each (t,
A
\= xi \rightarrow (i): for any sequences ti \rightarrow t,
x, and \zeta i \rightarrow \zeta such that \zeta i \in sub-esst\rightarrow ti h(ti , xi )
for each i, we have
\zeta \in sub-ess h(t, x) ;
t\rightarrow t\=
A
\= xi \rightarrow x, and \zeta i \rightarrow \zeta such that \zeta i \in super-esst\rightarrow t h(ti , xi )
(ii): for any sequences ti \rightarrow t,
i
for each i, we have
\zeta \in super-ess h(t, x) .
t\rightarrow t\=
A
\= xi \rightarrow Proof. We only prove (i). The proof of (ii) is similar. Take ti \rightarrow t,
x,
and \zeta i \rightarrow \zeta such that \zeta i \in sub-esst\rightarrow ti h(ti , xi ) for each i. Then, by Proposition 2.4,
\zeta i \in \partial t \psi (ti , xi ), for each i. Now
\Bigl( \Bigr) \zeta i \in \partial t \psi (ti , x) + (\psi (ti , xi ) - \psi (ti , x) .
Under the ``uniform continuity"" hypothesis, t \rightarrow (\psi (t, xi ) - \psi (t, x) is Lipschitz continuous, with Lipschitz constant ki such that ki \rightarrow 0 as i \rightarrow \infty . But then, by the sum
rule for limiting subdifferentials
\zeta i \in \partial \psi (ti , x) + ki \BbbB .
\= we conclude that
Since \zeta i \rightarrow \zeta , ki \rightarrow 0 and t \rightarrow \partial t \psi (t, x) has closed graph near t,
\= x) = sub-ess h(t, x) .
\zeta = lim \zeta i \in \partial \psi (t,
t\rightarrow t\=
i
More can be said about the relation between limiting sub- and superessential
values under an essential boundedness condition.
Proposition 2.5. If, in Proposition 2.3, h is essentially bounded on a neighbor\= then
hood of t,
\biggl\{ \biggr\} \biggl\{ \biggr\} \biggl\{ \biggr\} co sub-ess h(t) = co super-ess h(t) = ess h(t) and co \scrL -ess h(t) = ess h(t) .
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
Comment. The assertions of Proposition 2.5 are false without the ``essential
boundedness"" hypothesis. To see this consider the absolutely continuous function
h1 : [S, T ]\rightarrow \BbbR defined a.e. by
\biggl\{ 1
| 1/2 - t| - 2
if t \in [0, 1/2),
h1 (t) :=
1
- | t - 1/2| - 2 if t \in (1/2, 1] .
Take t\= := 1/2. An easy calculation reveals
sub-ess h1 (t) = \emptyset and super-ess h1 (t) = ess h1 (t) = ( - \infty , \infty ) .
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
It follows that
\biggl\{ \emptyset = co
\biggr\} sub-ess h1 (t) =
\not ess h1 (t) = ( - \infty , \infty ).
t\rightarrow t\=
t\rightarrow t\=
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2554
PIERNICOLA BETTIOL AND RICHARD B. VINTER
- Proof of Proposition 2.5. Write \zeta + := lim\epsilon \downarrow 0 (ess sups\in [t - \epsilon :=
\= ,t+\epsilon \= ] h(s)) and \zeta lim\epsilon \downarrow 0 (ess sups\in [t - \epsilon h(s)).
Denote
by
\scrS the
Lebesgue
points
of
h
:
[S,
T
]
\rightarrow \BbbR .
\= ,t+\epsilon \= ]
Since \scrS has full Lebesgue measure, there exists a sequence of points ti \rightarrow t\= such that
ti \in \scrS for each i and h(ti ) \rightarrow \zeta + as i \rightarrow \infty .
\int t +\epsilon For each i, by the Lebesgue point property, h(ti ) = lim\epsilon \downarrow 0 \epsilon - 1 tii h(s)ds =
\int t
lim\epsilon \downarrow 0 \epsilon - 1 tii - \epsilon h(s)ds. But then, for each i,
\int ti
\int ti +\epsilon lim sup \epsilon - 1
h(s)ds \leq h(ti ) \leq lim inf \epsilon - 1
h(s)ds.
\epsilon \downarrow 0
\epsilon \downarrow 0
ti - \epsilon ti
It follows that \zeta = limi\rightarrow \infty h(ti ) \in sub-esst\rightarrow t\= h(t). Likewise we show that \zeta - \in sub-esst\rightarrow t\= h(t).
But then esst\rightarrow t\= h(t) = [\zeta - , \zeta + ] \subset co(sub-esst\rightarrow t\= h(t)). It follows from Proposition 2.3 that esst\rightarrow t\= h(t) = co(sub-esst\rightarrow t\= h(t)). The proof that esst\rightarrow t\= h(t) =
co(super-esst\rightarrow t\= h(t)) is similar.
It follows directly from the definitions that \zeta \in \scrL -esst\rightarrow t\= h(t) for \zeta = \zeta + or \zeta = \zeta - .
+
(\zeta and \zeta - were defined above). Then, esst\rightarrow t\= h(t) = [\zeta - , \zeta + ] \subset co \scrL -esst\rightarrow t\= h(t). This
relation combines with Proposition 2.3 to give esst\rightarrow t\= h(t) = co \scrL -esst\rightarrow t\= h(t).
+
We know from Proposition 2.3 that the subessential value captures the limiting
subdifferential of the indefinite integral function \psi . We now provide some counterexamples, illustrating that the essential value and the Lebesgue essential value may
fail precisely to characterize the limiting subdifferential of \psi . Consider the bounded
function h1 : [ - 1, +1] \rightarrow \BbbR ,
\biggl\{ - 1 if t \in [ - 1, 0],
h1 (t) :=
1
if t \in (0, 1],
\int t
and set t\= := 0. Write \psi 1 (t) := - 1 h1 (s)ds. It is a straightforward exercise to calculate
sub-ess h1 (t) = [ - 1, 1], \scrL -ess h1 (t) = \{ - 1\} \cup \{ 1\} , ess h1 (t) = [ - 1, 1].
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
\= we conclude
Since sub-esst\rightarrow t\= h1 (t) = \partial \psi 1 (t),
strict
\= .
\scrL -ess h1 (t) \subset \partial \psi 1 (t)
t\rightarrow t\=
Now define h2 (t) := - h1 (t) and again set t\= = 0. Write \psi 2 (t) :=
calculate
\int t
- 1
h2 (s)ds. We
sub-ess h2 (t) = \{ - 1\} \cup \{ 1\} , \scrL -ess h2 (t) = \{ - 1\} \cup \{ 1\} , ess h2 (t) = [ - 1, 1].
t\rightarrow t\=
t\rightarrow t\=
t\rightarrow t\=
We conclude
strict
\= \subset ess h2 (t) .
\partial \psi 2 (t)
t\rightarrow t\=
3. A maximum principle for free end-time problems. We reproduce the
free end-time problem (F T ) of the introduction,
\left\{ Minimize g(S, x(S), T, x(T ))
over intervals [S, T ], arcs x \in W 1,1 ([S, T ]; \BbbR n )
and measurable functions u : [S, T ] \rightarrow \BbbR m satisfying
x(t)
\. = f (t, x(t), u(t)) a.e. t \in [S, T ],
(F T )
u(t) \in U (t) a.e. t \in [S, T ],
and
(S, x(S), T, x(T )) \in C .
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2555
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FREE END-TIME PROBLEMS
A process is taken to be a triple ([S, T ], x, u) in which [S, T ] is an interval, x (the
state trajectory) is an element in W 1,1 ([S, T ]; \BbbR n ), and u (the control function) is a
measurable \BbbR m valued function on [S, T ] satisfying, for a.e. t \in [S, T ],
\biggl\{ x(t)
\. = f (t, x(t), u(t)),
u(t) \in U (t).
\= T\=], x,
A process ([S,
\= u)
\= which satisfies the constraints of (F T ) is said to be a W 1,1
local minimizer if there exists \delta > 0 such that
\= x(
\= T\=, x(
g(S, x(S), T, x(T )) \geq g(S,
\= S),
\= T\=))
for every process ([S, T ], x, u) which satisfies the constraints of (FT) and also
\= T\=], x))
d(([S, T ], x), ([S,
\= \leq \delta .
Here, d is the metric
d(([S, T ], x), ([S \prime , T \prime ], x\prime )) := | S - S \prime | + | T - T \prime | (3.1)
+ | x(S) - x\prime (S \prime )| +
\int T \vee T \prime S\wedge S \prime | x\. e (s) - x\. \prime e (s)| ds,
in which, we recall, S \wedge S \prime := min\{ S, S \prime \} and T \vee T \prime := max\{ T, T \prime \} . Here, xe denotes
the extension to ( - \infty , +\infty ) of the function x : [S, T ] \rightarrow \BbbR n on the finite interval [S, T ],
defined as follows:
\left\{ x(S) if t < S,
x(t) if t \in [S, T ],
xe (t) :=
x(T ) if t > T .
Notice that, according to this definition, x\. e (t) = 0 for a.e. t \in / [S, T ].
Let \scrH denote the unmaximized Hamiltonian function for (FT):
\scrH (t, x, p, u) := p \cdot f (t, x, u).
Then the maximized Hamiltonian H has the representation H(t, x, p) := supu\in U (t)
\scrH (t, x, p, u).
\= T\=], x,
Theorem 3.1 (free end-time maximum principle). Let ([S,
\= u)
\= be a W 1,1
local minimizer for (F T ) such that T\= - S\= > 0. Assume that there exist \epsilon > 0 and
\=
\sigma \in (0, (T\= - S)/2)
such that
\= x(
\= T\=, x(
(H1): g is Lipschitz continuous on a neighborhood of (S,
\= S),
\= T\=)) and C is a
closed set,
(H2): Gr U is an \scrL \times \scrB m measurable set,
(H3): for each x \in \BbbR n , f (., x, .) is \scrL \times \scrB m measurable; there exists a function kf :
\BbbR \times \BbbR m \rightarrow \BbbR such that t \rightarrow kf (t, u(t))
\=
is integrable on [S\= - \sigma , T\= + \sigma ] and
| f (t, x\prime , u) - f (t\prime , x\prime \prime , u)| \leq kf (t, u)| x\prime - x\prime \prime | for all x\prime , x\prime \prime \in x
\=e (t) + \epsilon \BbbB and u \in U (t), a.e. t \in [S\= - \sigma , T\= + \sigma ],
(H4): there exists c\= \geq 0 such that, for a.e. t \in [S\= - \sigma , S\= + \sigma ] \cup [T\= - \sigma , T\= + \sigma ],
| f (t, x, u)| \leq c\=
for all x \in x
\=e (t) + \epsilon \BbbB and u \in U (t).
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2556
PIERNICOLA BETTIOL AND RICHARD B. VINTER
\= T\=]; \BbbR n ) and a real number \lambda \geq 0 such that
Then there exist p \in W 1,1 ([S,
(i): (p, \lambda ) \not = (0, 0),
\= T\=],
(ii): - p(t)
\. \in co\partial x \scrH (t, x(t),
\=
p(t), u(t))
\=
a.e. t \in [S,
\= p(S))
\= and \xi 1 \in super-esst\rightarrow T\= H(t, x(
(iii): there exist \xi 0 \in sub-esst\rightarrow S\= H(t, x(
\= S),
\= T\=),
p(T\=)) such that
\= \xi 1 , - p(T\=)) \in \lambda \partial g(S,
\= x(
\= T\=, x(
\= x(
\= T\=, x(
( - \xi 0 , p(S),
\= S),
\= T\=)) + NC (S,
\= S),
\= T\=)),
\= T\=].
(iv): \scrH (t, x(t),
\=
p(t), u(t))
\=
= maxu\in U (t) \scrH (t, x(t),
\=
p(t), u) a.e. t \in [S,
\= x0 , T, x1 ) : (x0 , T, x1 ) \in C\} ,
\~ for some closed
If the initial time S\= is fixed, i.e., C = \{ (S,
set C\~ \subset \BbbR n \times \BbbR \times \BbbR n , and g(S, x0 , T, x1 ) = g(x
\~ 0 , T, x1 ) for some g\~ : \BbbR n \times \BbbR \times \BbbR n \rightarrow \BbbR ,
then the above conditions are satisfied when (iii) is replaced by: there exists \xi 1 \in super-esst\rightarrow T\= H(t, x(
\= T\=), p(T\=)) such that
\= T\=x(
\= \xi 1 , - p(T\=)) \in \lambda \partial g(\=
\= T\=, x(
x(S),
\= T\=)),
(iii)\prime : (p(S),
\~ x(S),
\= T\=)) + NC\~ (\=
and when, in hypothesis (H4), the stated conditions are required to hold only for
t \in [T\= - \sigma , T\= + \sigma ]. The assertions of the theorem can be similarly modified when the
final time T\= is fixed.
The proof is given in section 4 below.
Example. The optimal control problem below is a very simple example, chosen
to illustrate the distinction between the new necessary conditions of this paper and
former ones, of optimal control problems in resource economics (see, e.g., [4]), in
which effort is expended to generate a profitable asset. In this example, energy is
purchased continuously to build up assets that generate a continuous financial return;
we must determine a purchasing strategy (and the time period over which the strategy
is deployed) to maximize net profits, taking account of both the cost of the energy
and of the sale of the assets at the end of the time period.
\left\{ \int T \Bigl( \Bigr) Minimize - r(x(t)) - c(t)u(t) dt - g(x(T ))
0
(F T )
over T \in [0, Tmax ], meas. u : [0, T ] \rightarrow \BbbR and x \in W 1,1 (0, T ) s.t.
x(t)
\. = f (u(t)), a.e. t \in [0, T ],
u(t) \in [0, umax ], a.e. t \in [0, T ],
x(0) = x0 .
Here, x(t) denotes accumulated assets and u(t) is the rate of energy comsumption at
time t. (The problem has been converted to a minimization problem by taking the
negative of the cost.)
In this problem, r : \BbbR \rightarrow \BbbR , c : \BbbR \rightarrow \BbbR , f : \BbbR \rightarrow \BbbR , and g : \BbbR \rightarrow \BbbR are given
functions, x0 \geq 0, Tmax > 0, and umax > 0. (r(x) is the profit rate (for assets x).
f (u) is the rate of financial return when the rate of energy consumption is u. c(t)u is
the price per unit time at time t when the energy consumption rate is u. g(x) is the
sale value of the acquired assets. x0 is prior assets, and umax > 0 and Tmax > 0 are
bounds on the rate of energy consumption and time horizon, respectively.
Take f (u) = u, x0 = 0, umax = 1, Tmax = 3, and, for each x, r(x) = x - 2,
g(x) = 3x, and
\left\{ 5
for - \infty < t \leq 1,
0.25 for 1 < t < 2,
c(t) :=
2
for 2 \leq t < \infty .
This problem has a minimizer, which (after state augmentation to eliminate the integral cost) satisfies the hypotheses for validity of the necessary conditions of Theorem 3.1. Since the right end-point of the state variable is free, we can normalize the
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FREE END-TIME PROBLEMS
2557
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cost multiplier, thus \lambda = 1. Let (T\=, x,
\= u,
\= p) be an extremal (with unity cost multiplier).
The Hamiltonian is
\scrH (t, x, p, u) = (p - c(t))u - 2 + x
and the maximized Hamiltonian is H(t, x, p) = (p - c(t))+ - 2 + x.
The costate trajectory is the solution to the differential equation p\. = - 1 on [0, T\=]
with right boundary condition p(T\=) = 3, namely p(t) = (T\= - t) + 3, for t \in [0, T\=].
We deduce from this formula for p and the Weierstrass condition that if T\= \in (0, 2],
then
\biggl\{ (0, 0)
for t \in [0, 1) \cap [0, T\=],
(3.2)
(\=
u(t), x(t))
\=
=
(1, t - 1) for t \in [1, 2) \cap [0, T\=] .
Moreover, when T\= \in (2, 3], we have
\biggl\{ 1 for t \in [0, T\= - 2] \cup [2, 3],
u(t)
\= =
(3.3)
0 for t \in (T\= - 2, 1]
and therefore
\left\{ (3.4)
x(t)
\= =
t
for t \in [0, T\= - 2],
\=
T - 2
for t \in (T\= - 2, 1],
\=
T - 2 + t - 1 for t \in [1, 3].
To compete verification that (T\=, x,
\= u,
\= p) is an extremal, for an appropriate value of
T\=, in remains to examine the ``free time"" transversality condition at time T\=. Write
h(T\=, t) := H(t, x(
\= T\=), p(T\=)). We can calculate
\left\{ \{ - 2\} if T\= \in (0, 1),
\{ - 2\} \cup \{ 0.75\} if T\= = 1,
\=
\{ T\= - 0.25\} if T\= \in (1, 2),
super-ess h(T , t) =
t\rightarrow T\=
[ - 1, 1.75]
if T\= = 2,
\=
\{ T - 1\} if T\= \in (2, 3].
The transversality condition, relating to the optimal end-time, is 0 \in super-esst\rightarrow T\=
h(T\=, t). The unique extremal (T\=, x,
\= u,
\= p), with 0 < T\=\leq Tmax , is given by (3.2)--(3.4)
with T\= = 2.
Notice that if we applied the earlier free time necessary conditions with transversality condition 0 \in esst\rightarrow T\= h(T\=, t), the processes (T\=, x,
\= u)
\= given by (3.2)--(3.4) would
be extremals for T\= taking values either T\= = 1 or T\= = 2. We see that the earlier condition introduces an additional extremal, which is not a minimizer. This is because,
when T\= = 1,
0 \in / super-ess h(t) = \{ - 2\} \cap \{ 0.75\} but 0 \in ess h(t) = co (\{ - 2\} \cup \{ 0.75\} = [ - 2, 0.75].
t\rightarrow T\=
t\rightarrow T\=
The example suggests that for free end-time dynamic optimization problems of resource economics and related areas, in which there are abrupt changes in interest
rates or other charges (to encourage or suppress activity), optimality conditions involving super- and subessential values can provide more precise information about
minimizers than earlier conditions expressed in terms of essential values.
We point out that real-world optimization dynamic problems in mathematical
economics or engineering are almost always solved not by direct application of optimality conditions such as those derived in this paper, but by means of numerical
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2558
PIERNICOLA BETTIOL AND RICHARD B. VINTER
methods. Improved necessary conditions of optimality can have a role, nonetheless,
in the solution of such problems. This is because an algorithm for computational
dynamic optimization typically generates, in the limit, a feasible process and its associated costate trajectory [1]. It is then common practice [3] to test the optimality of
this process by checking whether the process and costate trajectory satisfy first order
necessary conditions of optimality. The better the necessary conditions, the better
the test.
4. Proof of Theorem 3.1. Step 1 (a free end-point problem). We first
derive necessary conditions of optimality for the following special case of problem
(FT):
\left\{ (F ET )
Minimize g(S, x(S), T, x(T ))
over intervals [S, T ] \subset \BbbR , and arcs x \in W 1,1 ([S, T ]; \BbbR n )
and measurable functions u : [S, T ] \rightarrow \BbbR m satisfying
x(t)
\. = f (t, x(t), u(t)) a.e. t \in [S, T ],
u(t) \in U (t) a.e. t \in [S, T ] ,
| T - T \prime | , | S - S \prime | \leq \sigma \prime .
Here, [S \prime , T \prime ] is a given interval and \sigma \prime > 0 is a given constant such that \sigma \prime <
(T - S \prime )/2.
\prime Proposition 4.1. Let ([S \prime , T \prime ], x\prime , u\prime ) be a global minimizer for (F ET ). Assume
that, for some \epsilon \prime > 0 and \sigma \prime \in (0, (T \prime - S \prime )/2),
(FET1): g is Lipschitz continuous;
(FET2): Gr U is an \scrL \times \scrB m measurable set;
(FET3): for each x \in \BbbR n , f (., x, .) is \scrL \times \scrB m measurable; there exists kf \in L1 (S \prime - \sigma \prime , T \prime + \sigma \prime ) such that
| f (t, x, u) - f (t, x1 , u)| \leq kf (t)| x - x1 | for all x, x1 \in x\prime e (t) + \epsilon \prime \BbbB and u \in U (t), a.e. t \in [S \prime - \sigma \prime , T \prime + \sigma \prime ];
(FET4): there exists c\= \geq 0 such that, for a.e. t \in [S \prime - \sigma \prime , S \prime + \sigma \prime ] \cup [T \prime - \sigma \prime , T \prime + \sigma \prime ],
| f (t, x, u)| \leq c\=
for all x \in x\prime e (t) + \epsilon \prime \BbbB and u \in U (t).
Then there exists p \in W 1,1 ([S \prime , T \prime ]; \BbbR n ) such that
(i): - p(t)
\. \in co\partial x \scrH (t, x\prime (t), p(t), u\prime (t)) a.e. t \in [S \prime , T \prime ],
(ii): there exist \xi 0 \in sub-esst\rightarrow S \prime H(t, x\prime (S \prime ), p(S \prime )) and \xi 1 \in super-esst\rightarrow T \prime H(t, x\prime (T \prime ), p(T \prime )) such that
( - \xi 0 , p(S \prime ), \xi 1 , - p(T \prime )) \in \partial g(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )),
(iii): \scrH (t, x\prime (t), p(t), u\prime (t)) = maxu\in U (t) \scrH (t, x\prime (t), p(t), u) a.e. t \in [S \prime , T \prime ].
The proof is based on perturbational methods, in which we employ the following distance function on admissible processes for (F ET ): given admissible processes
([S, T ], x, u) and ([S \prime , T \prime ], x\prime , u\prime ) for (F ET ) we define
(4.1) dcontrol (([S, T ], x, u), ([S \prime , T \prime ], x\prime , u\prime )) := | x(S) - x\prime (S \prime )| + meas \{ t \in [S, T ] \cap [S \prime , T \prime ] : u(t) \not = u\prime (t)\} + | S - S \prime | + | T - T \prime | .
The required properties of this distance function are summarized in the following
lemma, whose straightforward proof we omit.
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Lemma 4.2. Assume that, for some interval [S \prime , T \prime ] and \sigma \prime \in (0, (T \prime - S \prime )/2),
hypotheses (FET2)--(FET4) of Proposition 4.1 are satisfied. Let
\scrM = \{ admissible processes ([S, T ], x, u) for (F ET ) \} .
Then (\scrM , dcontrol ) is a complete metric space.
Proof of Proposition 4.1. Observe that, without loss of generality, we can reduce
attention to the case when (FET3) is replaced by the stronger hypothesis (FET)\ast , in
which k\= is a constant:
(FET3)\ast : There exists a constant k\= > 0 such that, for a.e. t \in [S \prime - \sigma \prime , T \prime + \sigma \prime ],
\= - x1 | | f (t, x, u) - f (t, x1 , u)| \leq k| x
for all x, x1 \in x\prime e (t) + \epsilon \prime \BbbB .
Indeed, it is not restrictive to assume that kf (t) \geq 1 for a.e. t \in [S \prime - \sigma \prime , T \prime + \sigma \prime ]
(otherwise we can always replace kf by kf \vee 1). Consider the change of the independent
variable s = \sigma (t),
\^
where \sigma \^ : \BbbR \rightarrow \BbbR is defined by
\left\{ \int t
\prime - S \prime +2\sigma \prime k(\tau )d\tau for all t \in [S \prime - \sigma \prime , T \prime + \sigma \prime ],
S \prime - \sigma \prime + T \| k
f \| L1
\sigma (t)
\^ :=
\prime \prime S - \sigma t otherwise.
(Here, \| kf \| L1 = \| kf \| L1 (S \prime - \sigma \prime ,T \prime +\sigma \prime ) .) Notice that \sigma \^ is a strictly increasing function
such that \sigma (S
\^ \prime - \sigma \prime ) = S \prime - \sigma \prime and \sigma (T
\^ \prime + \sigma \prime ) = T \prime + \sigma \prime . Then, we can consider a
modified version of problem (F ET ) in which the function f and the multifunction U
are respectively replaced by
f\^(s, x, u) :=
\| kf \| L1
\prime \prime (T - S + 2\sigma \prime ) \times (k
\^
f \circ \sigma - 1 )(s)
f (\^
\sigma - 1 (s), x, u)
for all (s, x, u) \in \BbbR \times \BbbR n \times \BbbR m
and
\^ (s) := U (\^
U
\sigma - 1 (s)) for all s \in \BbbR .
Observe that the process ([S\^ := \sigma \^ - 1 (S \prime ), T\^ := \sigma \^ - 1 (T \prime )], x
\^ := x\prime \circ \sigma - 1 , u
\^ := u\prime \circ \sigma - 1 )
\^
is a global minimizer for this new problem, and that f satisfies (FEP3)\ast with k\= =
\| kf \| L1
\^ is an absolutely continuous function on [S \prime - T \prime - S \prime +2\sigma \prime . Notice also that, since \sigma \^ is an \scrL \times \scrB m measurable set.
\sigma \prime , T \prime + \sigma \prime ], Gr U
It is then easy to derive the necessary conditions for the original problem, with
reference to the minimizer ([S \prime , T \prime ], x\prime , u\prime ), via the inverse change of independent variable s = \sigma (t),
\^
where the Lagrange multiplier p\^ obtained in the new problem is replaced
by p = p\^ \circ \sigma \^ in the original one.
Consider first the case of (F ET ), in which g has the structure
(4.2)
g(S, x0 , T, x1 ) = g(S,
\~ x0 , T, x1 ) + e(S, x0 , T, x1 ) ,
for some twice differentiable function g\~ and some Lipschitz continuous function e.
(This situation will be obtained later in the proof after applying the argument based
on the inf convolution of g.) When g has the structure in (4.2) we shall prove a less
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precise version of the necessary conditions in the proposition statement, in which (ii)
is replaced by
\surd 1
( - \xi 0 , p(S \prime ), \xi 1 , - p(T \prime )) \in \partial g(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + 2 2ke (1 + c\=2 ) 2 \BbbB ,
where ke is a Lipschitz constant for e,
(x\prime , u\prime ) is a minimizer for a version of problem (F ET ), in which the underlying
time interval is fixed at [S \prime , T \prime ]. The hypotheses of the Clarke nonsmooth maximum
principle are satisfied. In consequence of the fact that the problem has no state endpoint constraints, the cost multiplier can be set to 1. It follows that there exists
p \in W 1,1 (S \prime , T \prime ) satisfying conditions (i) and (iii) and the following tranversality
condition:
(p(S \prime ), - p(T \prime )) \in \nabla x0 ,x1 g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + ke \BbbB .
(4.3)
For s \in (0, \sigma \prime ], ([S \prime , T \prime - s], x\prime , u\prime ) is admissible and
g(S
\~ \prime , x\prime (S \prime ), T \prime - s, x\prime (T \prime - s)) + e(S \prime , x\prime (S \prime ), T \prime - s, x\prime (T \prime - s))
\geq g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + e(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) .
Since g\~ is a C 2 function, there exists r1 > 0, that does not depend on s, such that
for all s \in (0, \sigma \prime ]
1
0 \geq g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) - g(S
\~ \prime , x\prime (S \prime ), T \prime - s, x\prime (T \prime - s)) - ke (1 + c\=2 ) 2 s
\int T \prime \Bigl( \Bigr) 1
=
\nabla T g(S
\~ \prime , x\prime (S \prime ), t, x\prime (t)) + \nabla x1 g(S
\~ \prime , x\prime (S \prime ), t, x\prime (t)) \cdot x\. \prime (t) dt - ke (1 + c\=2 ) 2 s
\prime T - s
\int T \prime 1
\prime \prime \prime \prime \prime \prime \geq \nabla T g(S
\~ , x (S ), T , x (T ))s - H(t, x\prime (T \prime ), p(T \prime )) dt - r1 s2 - ke (\=
c + (1 + c\=2 ) 2 )s.
T \prime - s
To derive the final inequality in the above relation, we have used the facts that
- p(T \prime ) \in \nabla x1 g\~ (S \prime , x(S \prime ), T \prime , x(T \prime )) + ke \BbbB and that (from (FET3)\ast and (FET4)) we
have
\=cs, a.e. t \in [T \prime - s, T \prime ].
p(T \prime ) \cdot x\. \prime (t) \leq H(t, x\prime (T \prime ), p(T \prime )) + | p(T \prime )| k\=
We have shown that
\left\{ T \prime is a local minimum over [S \prime , T \prime ] of
\int T \prime (4.4)
T \rightarrow - \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime ))(T \prime - T ) - T ( - H)(t, x\prime (T \prime ), p(T \prime )) dt
1
+r1 (T - T \prime )2 + ke (\=
c + (1 + c\=2 ) 2 )| T - T \prime | .
\BbbR m
Next, for each integer k \geq 1 we can select a measurable function uk : [T \prime , T \prime +\sigma \prime ] \rightarrow such that uk (t) \in U (t) for a.e. t \in [T \prime , T \prime + \sigma \prime ] and
1
p(T \prime ) \cdot f (t, x\prime (T \prime ), uk (t)) \geq sup p(T \prime ) \cdot f (t, x\prime (T \prime ), u) - ,
k
u\in U (t)
a.e. t \in [T \prime , T \prime + \sigma \prime ].
By Filippov's existence theorem (cf. [11, Theorem 2.4.3]), for each k, there exists a
process (yk , uk ) on [T \prime , T \prime + \sigma \prime ] such that yk (T \prime ) = x\prime (T \prime ); from (FET3)\ast and (FET4)
we also have
\int T \prime +s
\=cs2 for all s \in [0, \sigma \prime ].
(4.5)
| f (t, yk (t), uk (t)) - f (t, x\prime (T \prime ), uk (t))| dt \leq k\=
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We now construct a sequence of admissible processes ([S \prime , T \prime + \sigma \prime ], x\prime k , u\prime k ) by
concatenating ([S \prime , T \prime ], x\prime , u\prime ) and ([T \prime , T \prime + \sigma \prime ], yk , uk ). By optimality,
0 \leq (\~
g + e)(S \prime , x\prime k (S \prime ), T \prime + s, x\prime k (T \prime + s)) - (\~
g + e)(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime ))
1
\leq g(S
\~ \prime , x\prime (S \prime ), T \prime + s, yk (T \prime + s)) - g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + ke (1 + c\=2 ) 2 s
\int T \prime +s\Bigl( \Bigr) 1
=
\nabla T g(S
\~ \prime , x\prime (S \prime ), t, yk (t)) + \nabla x1 g(S
\~ \prime , x\prime (S \prime ), t, yk (t)) \cdot y\. k (t) dt + ke (1 + c\=2 ) 2 s
T \prime for all s \in [0, \sigma \prime ]. Since g\~ is twice continuously differentiable, there exists r1 > 0 (which
can be assumed to be same constant appearing in (4.4)) independent of s, such that
0 \leq \int T \prime +s \Bigl( T \prime \prime \Bigr) 1
\nabla T g(S
\~ \prime , x\prime (S \prime ), t, yk (t)) + \nabla x1 g(S
\~ \prime , x\prime (S \prime ), t, yk (t)) \cdot y\. k (t) dt + ke (1 + c\=2 ) 2 s
\prime \prime \prime \prime \prime \int T \prime +s
\leq \nabla T g(S
\~ , x (S ), T , x (T ))s +
T \prime \nabla x1 g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) \cdot f (t, x\prime (T \prime ), uk (t)) dt
1
+ r1 s2 + ke (1 + c\=2 ) 2 s .
Recalling that - p(T \prime ) \in \nabla x1 g\~ (S \prime , x(S \prime ), T \prime , x(T \prime )) + ke \BbbB , from (4.5) we deduce
that for all k
\nabla x1 g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) \cdot f (t, x\prime (T \prime ), uk (t))
1
\leq ( - H)(t, x\prime (T \prime ), p(T \prime )) + + ke c\= a.e. t \in [T \prime , T \prime + s].
k
Therefore, combining the preceding two relations and taking the limit as k \rightarrow +\infty ,
we arrive at
0 \leq \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime ))s
\int T \prime +s
1
+
( - H)(t, x\prime (T \prime ), p(T \prime )) dt + r1 s2 + ke (\=
c + (1 + c\=2 ) 2 )s .
T \prime We have shown that
\left\{ \prime T is a local minimum over [T \prime , T \prime + \sigma \prime ] of
\int T
(4.6)
T \rightarrow \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime ))(T - T \prime ) + T \prime ( - H)(t, x\prime (T \prime ), p(T \prime )) dt
1
+r1 | T - T \prime | 2 + ke (\=
c + (1 + c\=2 ) 2 )| T - T \prime | .
\int T \prime \int T
Interpreting, as usual, the integral T \prime ( - H)(t, x\prime (T \prime ), p(T \prime ))dt as - T ( - H)(t, x\prime (T \prime ),
p(T \prime ))dt for T < T \prime , we deduce from (4.4) and (4.6) that
\left\{ \prime T is a local minimum over [S \prime , T \prime + \sigma \prime ] of
\int T
T \rightarrow \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime ))(T - T \prime ) + T \prime ( - H)(t, x\prime (T \prime ), p(T \prime )) dt
1
+r1 (T - T \prime )2 + ke (\=
c + (1 + c\=2 ) 2 )| T - T \prime | .
Applying
\int t Proposition 2.3, in which we take the indefinite integral function \psi to be
\psi (t) := T \prime ( - H)(s, x\prime (T \prime ), p(T \prime ))ds, t \in [S \prime , T \prime + \sigma \prime ], noting that 0 is a limiting subgradient of a function at a minimizing point, and also applying the sum rule for limiting
subdifferentials of Lipschitz functions, we arrive at
1
- \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) \in sub-ess
( - H)(t, x\prime (T \prime ), p(T \prime )) + ke (\=
c + (1 + c\=2 ) 2 )\BbbB .
\prime t\rightarrow T
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Then, using Proposition 2.3, we obtain
1
\nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) \in super-ess H(t, x\prime (T \prime ), p(T \prime )) + ke (\=
c + (1 + c\=2 ) 2 )\BbbB .
t\rightarrow T \prime A similar analysis, in which we vary the left end-time, yields
1
- \nabla S g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) \in sub-ess
H(t, x\prime (S \prime ), p(S \prime )) + ke (\=
c + (1 + c\=2 ) 2 )\BbbB .
\prime t\rightarrow S
In view of (4.3), we have confirmed that there exist elements \xi 0 \in sub-esst\rightarrow S \prime H(t,
x\prime (S \prime ), p(S \prime )) and \xi 1 \in super-esst\rightarrow T \prime H(t, x\prime (T \prime ), p(T \prime )) such that
\surd 1
( - \xi 0 , p(S \prime ), \xi 1 , - p(T \prime )) \in \nabla \~
g(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + 2 2ke (1 + c\=2 ) 2 \BbbB .
We have proved a special case of the proposition (with weakened transversality condition) when g has a special structure. We now assume that g is an arbitrary Lipschitz
continuous function.
For i = 1, 2, . . . let gi be the ith quadratic inf convolution of g [5]. Consider, for
each i, the following variant of (F ET ), in which we replace g by gi :
\left\{ Minimize gi (S, x(S), T, x(T ))
over intervals [S, T ], arcs x \in W 1,1 ([S, T ]; \BbbR n )
and measurable functions u : [S, T ] \rightarrow \BbbR m satisfying
x(t)
\. = f (t, x(t), u(t)) a.e. t \in [S, T ],
u(t) \in U (t) a.e. t \in [S, T ],
| T - T \prime | , | S - S \prime | \leq \sigma \prime .
\surd Write \alpha i := kg / i, where kg is a Lipschitz constant for g. According to the properties
of quadratic inf convolutions (cf. [5]), ([S \prime , T \prime ], x\prime , u\prime ) is an \alpha i2 minimizer for this
problem.
We can express the preceding problem as
Minimize \{ Ji ([S, T ], x, u):= gi (S, x(S), T, x(T )) : ([S, T ], x, u) \in \scrM \} .
(\scrM was defined in the statement of Lemma 4.2.) According to this lemma, (\scrM ,
dcontrol ) is a complete metric space. Ji is continuous on (\scrM , dcontrol ). It follows then
from Ekeland's theorem that, for each i, there exists an element ([Si , Ti ], xi , ui ) \in \scrM that minimizes Ji ([S, T ], x, u) + \alpha i dcontrol (([S, T ], x, u), ([Si , Ti ], xi , ui ) over \scrM and
dcontrol (([Si , Ti ], xi , ui ), ([S \prime , T \prime ], x\prime , u\prime )) \leq \alpha i .
This implies that xi \rightarrow x\prime uniformly, Si \rightarrow S \prime , and Ti \rightarrow T \prime .
Fix a measurable selection u
\~ : [S \prime - \sigma \prime , T \prime + \sigma \prime ] \rightarrow \BbbR m such that u(t)
\~ \in U (t) for a.e.
\prime \prime \prime \prime \prime t \in [S - \sigma , T +\sigma ]. We write ui the measurable extension of the control ui defined on
[Si , Ti ] \subset [S \prime - \sigma \prime , T \prime + \sigma \prime ] to the interval [S \prime - \sigma \prime , T \prime + \sigma \prime ] obtained by concatenating
ui with u:
\~
u\prime i (t) :=
\Bigl\{ ui (t)
u(t)
\~
if t \in [Si , Ti ],
if t \in [S \prime - \sigma \prime , T \prime + \sigma \prime ] \setminus [Si , Ti ].
Noting that, for each interval [S, T ] \subset [S \prime - \sigma \prime , T \prime + \sigma \prime ],
(4.7)
meas \{ t \in [S, T ] \cap [Si , Ti ] : u(t) \not = ui (t)\} \leq meas \{ t \in [S, T ] : u(t) \not = u\prime i (t)\} Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
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FREE END-TIME PROBLEMS
2563
(with equality if [S, T ] = [Si , Ti ]), we see that ([Si , Ti ], xi , ui ) is a minimizer for
\left\{ \Biggl( \int T
Minimize gi (S, x(S), T, x(T ))) + \alpha i
mi (t, u(t))dt + | S - Si | + | T - Ti | S
\bigm| \bigm| \bigm| +| x(S) - xi (Si )\bigm| ) over [S, T ] \subset \BbbR , x \in W 1,1 ([S, T ]; \BbbR n )
\bigm| and measurable functions u : [S, T ] \rightarrow \BbbR m satisfying
x(t)
\. \in f (t, x(t), u(t)) and u(t) \in U (t), a.e. t \in [S, T ] ,
| T - T \prime | , | S - S \prime | \leq \sigma \prime ,
(F ET )i
in which
\biggl\{ mi (t, u) :=
0
1
if u = u\prime i (t),
if u \not = u\prime i (t).
By properties of the quadratic inf convolution, there exists a quadratic function
g\~i such that
\nabla \~
gi (Si , xi (Si ), Ti , xi (Ti )) \in \partial g(zi )
for some zi \in (Si , xi (Si ), Ti , xi (Ti )) + (kg /i)\BbbB , and
\biggl\{ g\~i (z) \geq gi (z) for all z \in \BbbR \times \BbbR n \times \BbbR \times \BbbR n ,
g\~i (z) = gi (z) when z = (S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) .
It follows that ([Si , Ti ], xi , ui ) is a minimizer for a variant of (F ET )i , in which
the quadratic function g\~i replaces gi , namely
\left\{ \Biggl( \int T
Minimize g\~i (S, x(S), T, x(T ))) + \alpha i
\Biggr) mi (t, u(t))dt + | x(S) - xi (Si )| S
+| S - Si | + | T - Ti | over [S, T ] \subset \BbbR , x \in W 1,1 ([S, T ], \BbbR n )
and measurable functions u : [S, T ] \rightarrow \BbbR m satisfying
x(t)
\. \in f (t, x(t), u(t)) and u(t) \in U (t), a.e. t \in [S, T ] ,
| T - T \prime | , | S - S \prime | \leq \sigma \prime .
Since Si \rightarrow S \prime and Ti \rightarrow T \prime , we can arrange (by eliminating initial sequence terms)
that | Si - S \prime | , | Ti - T \prime | \leq \sigma \prime /2. But then ([Si , Ti ], xi , ui ) remains a minimizer when we
replace the end-time constraint by | T - Ti | , | S - Si | \leq \sigma \prime /2. Absorbing the integral
term in the cost into the dynamic constraint via state augmentation, we arrive at an
example of the special case of (F ET ), for which analysis employed in the first part
of the proof provides necessary conditions of optimality. The relevant hypotheses
are satisfied when \sigma \prime /2 replaces the parameter \sigma \prime . It follows that there exists pi \in W 1,1 ([Si , Ti ]; \BbbR n ) such that
(A)\prime : - p\.i (t) \in co \partial x [pi (t) \cdot f (t, xi (t), ui (t))] a.e. t \in [Si , Ti ],
(B)\prime : pi (t) \cdot f (t, xi (t), ui (t)) \geq pi (t) \cdot f (t, xi (t), u) - \alpha i for all u \in U (t) a.e. t \in [Si , Ti ],
(C)\prime : there exist \xi 0i \in sub-esst\rightarrow Si H(t, xi (Si ), p(Si )) and \xi 1i \in super-ess
t\rightarrow Ti H(t,
\surd 1
xi (Ti ), p(Ti )) such that ( - \xi 0i , pi (Si ), \xi 1i , - pi (Ti ) \in \partial g(ei ) + (2 2)\alpha i (1 + c\=2 ) 2 \BbbB .
We now use our ``constant extrapolation"" convention to extend the domains
\prime \prime of the pi 's to the entire real line. We deduce from (A) and (C) that the pi 's
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2564
PIERNICOLA BETTIOL AND RICHARD B. VINTER
are uniformly bounded with uniformly integrably bounded derivatives. We can arrange then, by subsequence extraction, that pi \rightarrow p uniformly and p\.i \rightarrow p\. weakly
in L1 for some p \in W 1,1 . The sequences \{ \xi 0i \} and \{ \xi 1i \} are bounded. A further
subsequence extraction ensures that they have limits \xi 0 and \xi 1 , respectively. We
know, however, that ([Si , Ti ], xi , ui ) \rightarrow ([S \prime , T \prime ], x\prime , u\prime ) w.r.t. the dcontrol metric and
ei \rightarrow (S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )). A standard convergence analysis permits us to pass to
\prime \prime the limit as i \rightarrow \infty in conditions (A) --(C) and arrive at the relations asserted in
the proposition statement. Observe that we are making use of the stability, under
limit taking, properties of the limiting process, expressed in terms of the sub- and
superessential values.
Step 2 (completion of the proof ). Making use of standard techniques (cf.
[11, Chapter 6]) it is possible to show that the assertions of the theorem are true in
general if we can demonstrate their validity under the additional hypothesis:
(A): There exist integrable functions k0 : [S\= - \sigma , T\= + \sigma ] \rightarrow \BbbR , c0 : [S\= - \sigma , T\= + \sigma ] \rightarrow \BbbR ,
and a constant c\= \geq 0 such that
(i): | f (t, x, u) - f (t, x\prime , u)| \leq k0 (t)| x - x\prime | for all x, x\prime \in x(t)
\= + \epsilon \BbbB , u \in U (t), a.e.
t \in [S\= - \sigma , T\= + \sigma ],
(ii): | f (t, x, u)| \leq c0 (t) for all x \in x(t)
\= + \epsilon \BbbB , u \in U (t), a.e. t \in [S\= - \sigma , T\= + \sigma ],
(iii): | f (t, x, u)| \leq c\= for all x \in x(t)
\= + \epsilon \BbbB , u \in U (t), a.e. t \in [S\= - \sigma , S\= + \sigma ] \cup [T\= - \sigma , T\= + \sigma ].
\= T\=], x,
We can also assume, without loss of generality, that ([S,
\= u)
\= is a (global)
\= T\=], x,
minimizer for (FT). To see this, assume that ([S,
\= u)
\= is a minimizer merely w.r.t.
\= T\=], x,
candidate admissible processes ([S, T ], x, u) satisfying d(([S, T ], x, u), ([S,
\= u))
\= \leq \beta \= T\=], x,
for some \beta > 0. Then ([S,
\= u)
\= is a (global) minimizer for (FT) when we add the
constraint
\int T
\=
| f (t, x(t), u(t)) - f (t, x(t),
\=
u(t))| dt
\=
+ | x(S) - x(
\= S)| S
\= \leq \beta .
+ (1 + 2\=
c)(| T - T\=| + | S - S| )
\= T\=], x,
By applying the special case of the theorem in which ([S,
\= u)
\= is a minimizer
to this modified problem, following absorption of the integral term in the added
constraint into the dynamics via state augmentation, and noting the added constraint
is inactive, we validate the assertions of the theorem for the original problem, when
\= T\=], x,
([S,
\= u)
\= is merely a W 1,1 local minimizer.
Take \alpha i \downarrow 0 and, for each i, define
\= x(
\= T\=, x(
\ell i (S, x0 , T, x1 ) := max\{ g(S, x0 , T, x1 ) - g(S,
\= S),
\= T\=)) + \alpha i2 , dC (S, x0 , T, x1 )\} .
Now consider the problem
\left\{ (P1i )
Minimize J1i (([S, T ], x, u)) subject to
x(t)
\. = f (t, x(t), u(t)), a.e. t \in [S, T ],
u(t) \in U (t), a.e. t \in [S, T ],
\= | T - T\=| \leq \sigma ,
| S - S| ,
\= T\=], x,
in which J1i (([S, T ], x, u)) := \ell i (S, x(S), T, x(T )). Since J1i (([S,
\= u))
\= = \alpha i2 , and J1i
2
\=
\=
is nonnegative valued, ([S, T ], x,
\= u)
\= is an \alpha i -minimizer.
Let \scrM denote the set of admissible processes ([S, T ], x, u) for (P1i ). Observe that
the data characterizing the admissible processes for (P1i ) satisfy the assumptions of the
data which characterize the admissible processes for (F ET ), and so from Lemma 4.2
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we know that (\scrM , dcontrol ) is a complete metric space. Notice also that the function
J1i : \scrM \rightarrow \BbbR is continuous w.r.t. the metric dcontrol . By Ekeland's theorem, we
are assured of the existence of a minimizer ([Si , Ti ], xi , ui ) for a modification of the
above optimization problem, in which J1i ([S, T ], x, u) is replaced by J1i ([S, T ], x, u) +
dcontrol (([S, T ], x, u), ([Si , Ti ], xi , ui )). Furthermore
\= T\=], x,
dcontrol (([Si , Ti ], xi , ui ), ([S,
\= u))
\= \leq \alpha i .
(4.8)
Similarly as before, we fix a measurable selection u
\~ : [S\= - \sigma , T\= +\sigma ] \rightarrow \BbbR m such that
u(t)
\~ \in U (t) for a.e. t \in [S\= - \sigma , T\= + \sigma ], and we write u
\~i the measurable extension to
[S\= - \sigma , T\= + \sigma ] of the control ui (defined on [Si , Ti ] \subset [S\= - \sigma , T\= + \sigma ]), which is obtained
as follows:
\biggl\{ ui (t) if t \in [Si , Ti ],
u
\~i (t) :=
u(t)
\~
if t \in [S\= - \sigma , T\= + \sigma ] \setminus [Si , Ti ] .
Taking account of inequality (4.7), we conclude that ([Si , Ti ], xi , ui ) is a minimizer for
\left\{ \Biggl( \int T
Minimize \ell i (S, x(S), T, x(T )) + \alpha i
S
(P\~1i )
mi (t, u(t))dt
\Biggr) +| x(S) - xi (Si )| + | S - Si | + | T - Ti | over [S, T ] \subset \BbbR , x \in W 1,1 ([S, T ]; \BbbR n )
and measurable functions u : [S, T ] \rightarrow \BbbR m satisfying
x(t)=f
\.
(t, x(t), u(t)) and u(t) \in U (t), a.e. t \in [S, T ] ,
\= | T - T\=| \leq \sigma ,
| S - S| ,
in which
\biggl\{ mi (t, u) :=
0 if u = u
\~i (t),
1 if u \not = u
\~i (t).
It can be deduced from (4.8) that | | xi - x| | \= L\infty \rightarrow 0 as i \rightarrow \infty (remember our
\= | Ti - T\=| \rightarrow 0. It
``extension"" convention for interpreting such relations) and | Si - S| ,
follows that, for i sufficiently large, (xi , ui ) is a minimizer for (P\~1i ), when the end\= | Ti - T\=| \leq \sigma /2."" Now absorb the integral
time constraint is replaced by ``| Si - S| ,
cost term into the dynamics by state augmentation. The data for the problem we
thereby obtain satisfies the hypotheses of Proposition 4.1. We deduce the existence
of a costate trajectory pi such that
\prime (b) : - p\.i (t) \in co \partial x [ pi (t) \cdot f (t, xi (t), ui (t)) ], a.e. t \in [Si , Ti ].
\prime (c) : pi (t)\cdot f (t, xi (t), ui (t)) \geq pi (t)\cdot f (t, xi (t), u) - \alpha i for all u \in U (t) a.e. t \in [Si , Ti ],
\prime \prime (d) : there exist \xi 0i \in sub-esst\rightarrow Si H(t, xi (Si ), pi (Si )) and \xi 1i \in super-esst\rightarrow Ti H(t,
xi (Ti ), pi (Ti )) such that
\surd ( - \xi 0i , pi (Si ), \xi 1i , - pi (Ti )) \in \partial \ell i (Si , xi (Si ), Ti , xi (Ti )) + 3\alpha i \BbbB .
\prime \prime Let us examine the implications of the transversality condition (d) . We observe
that, for i sufficiently large,
(4.9)
\= x(
\= T\=, x(
max\{ g(Si , xi (Si ), Ti , xi (Ti )) - g(S,
\= S),
\= T\=)) + \alpha i2 , dC (Si , xi (Si ), Ti , xi (Ti ))\} > 0 .
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2566
PIERNICOLA BETTIOL AND RICHARD B. VINTER
\= x(
\= T\=,
Indeed if this were not the case, we would have g(Si , xi (Si ), Ti , xi (Ti )) - g(S,
\= S),
x(
\= T\=)) \leq - \alpha i2 and dC (Si , xi (Si ), Ti , xi (Ti )) = 0. This contradicts the L\infty local opti\= T\=], x,
mality of ([S,
\= u)
\= for problem (F T ). In consequence of the max rule for limiting
subdifferentials, there exists \lambda i \in [0, 1] such that
\partial \ell i (Si , xi (Si ), Ti , xi (Ti )) \subset \lambda i \partial g(Si , xi (Si ), Ti , xi (Ti ))
+ (1 - \lambda i )\partial dC (Si , xi (Si ), Ti , xi (Ti )).
A familiar argument, based on the max rule for limiting subdifferentials and limiting
subgradient properties of the distance function (cf. [11, Chapters 4 and 5]), allows us
to deduce from (4.9) that, in the preceding relation, \partial dC can be replaced by \partial dC \cap \partial \BbbB ,
thus
\prime (d) : ( - \xi 0i , p\Bigl( i (Si ), \xi 1i , - pi (Ti )) \in \lambda i \partial g(Si , xi (S
Ti , xi (Ti ))
\Bigr) i ), \surd +(1 - \lambda i ) \partial dC (Si , xi (Si ), Ti , xi (Ti )) \cap \partial \BbbB + 3\alpha i \BbbB .
Since | \xi 0i | , | \xi 1i | \leq c| | p
\= i | | \surd L\infty we deduce from the preceding relation that 2(1 +
c)| | p
\= i | | L\infty \geq (1 - \lambda i ) - \lambda i kg - 3\alpha i . Here kg is a Lipschitz constant for g and c\= is the
constant of hypothesis (H4) in the theorem statement. It follows that
\surd \prime (a) : 2(1 + c)| | p
\= i | | L\infty + (1 + kg )\lambda i \geq 1 - 3\alpha i .
Under the strengthened hypotheses on f (t, x, u) introduced at the beginning of
this step of the proof (``uniform Lipschitz continuity w.r.t. x with integrable Lip\prime \prime schitz bound""), we can deduce from (b) and (d) , with the help of Gronwall's lemma,
that the pi 's (extended to all of ( - \infty , \infty ) by constant extrapolation) are uniformly
bounded, with uniformly integrably bounded derivatives. But then, along a subsequence, pi \rightarrow p uniformly and p\.i \rightarrow pi , weakly in L1 , for some absolutely continuous
function p. We can also arrange, by further subsequence extraction, that \lambda i \rightarrow \lambda ,
\xi 0i \rightarrow \xi 0 , and \xi 1i \rightarrow \xi 1 .
\prime \prime We will recognize in conditions (a) --(d) perturbed versions of the desired conditions. A standard convergence analysis permits us to pass to the limit as i \rightarrow \infty . We
thereby recover the conditions asserted in the theorem statement.
5. Differential inclusions. The preceding sections show how previously employed analytical techniques can be modified, to provide a new free end-time maximum principle for dynamic optimization problems involving controlled differential
equations. Similar modifications can be made to earlier techniques, to derive improved first order necessary conditions, for problems involving differential inclusions.
Consider
\left\{ Minimize g(S, x(S), T, x(T ))
over intervals [S, T ] and arcs x \in W 1,1 ([S, T ], \BbbR n )
\prime satisfying
(F T )
x(t)
\. \in F (t, x(t)) a.e. t \in [S, T ],
(S, x(S), T, x(T )) \in C.
Here g : \BbbR \times \BbbR n \times \BbbR \times \BbbR n \rightarrow \BbbR is a given function, F : \BbbR \times \BbbR n ; \BbbR n is a given
multifunction, and C \subset \BbbR \times \BbbR n \times \BbbR \times \BbbR n is a given set.
Now, an arc x \in W 1,1 ([S, T ]; \BbbR n ) will usually be denoted ([S, T ], x), to emphasize
the underlying time interval [S, T ]. In the present context, an arc ([S, T ], x), where x is
an F trajectory on [S, T ] that satisfies the end-point constraint (S, x(S), T, x(T )) \in C,
is referred to as an admissible F trajectory. We shall consider a notion of minimizers
which is very general, indeed it is considered relative to a given multifunction B in
the following sense. Given a multifunction B : ( - \infty , +\infty ) ; \BbbR n , we say that an
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FREE END-TIME PROBLEMS
2567
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\= T\=], x) is a W 1,1 local minimizer for (F T ) relative to B if
admissible F trajectory ([S,
there exists some \beta > 0 such that
\= x(
\= T\=, x(
g(S, x(S), T, x(T )) \geq g(S,
\= S),
\= T\=))
for all admissible F trajectories ([S, T ], x) satisfying the following properties: x(t)
\.
\in \= T\=] \cap [S, T ] and
x(t)
\=\. + B(t) for a.e. t \in [S,
\= T\=], x))
d(([S, T ], x), ([S,
\= \leq \beta .
(Here, d is the distance function defined in equation (3.1)).
The necessary conditions in the following theorem bring together versions of the
Euler--Lagrange inclusion, the Weierstrass condition, and the transversality condition
for free end-time problems, where the dynamic constraint takes the form of a differential inclusion. Here, the formulation of the Weierstrass condition involves, for a.e.
\= T\=], the regular velocity set \Omega 0 (t) for the differential inclusion, relative to the
t \in [S,
nominal F trajectory x:
\=
(5.1)
\= T\=]
\Omega 0 (t) := \{ e \in F (t, x(t))
\=
: F (t, .) is pseudo-Lipschitz near (\=
x(t), e)\} for t \in [S,
and the transversality condition is expressed in terms of essential values of the Hamiltonian at the optimal end-times. We recall that \Gamma is pseudo-Lipschitz continuous near
(\=
x, v)
\= (with parameters \epsilon , R, and k) if
\circ \Gamma (x\prime ) \cap (\=
v + R\BbbB ) \subset \Gamma (x) + k| x\prime - x| \BbbB for all x\prime , x \in x
\= + \epsilon \BbbB .
Now the Hamiltonian H(t, x, p) is taken to be
H(t, x, p) :=
sup p \cdot v.
v\in F (t,x)
Theorem 5.1 (free end-time generalized Euler--Lagrange condition). Take a
measurable multifunction B : \BbbR ; \BbbR n such that B(t) is open for a.e. t \in \BbbR . Let
x
\= be a W 1,1 local minimizer for (F T )\prime relative to B. Assume that, for some \epsilon > 0 and
\=
\sigma \in (0, (T\= - S)/2),
the following hypotheses are satisfied:
\= x(
\= T\=, x(
(G1): g is Lipschitz continuous on a neighborhood of (S,
\= S),
\= T\=)) and C is a
closed set,
(G2): F (t, x) is nonempty for all (t, x) \in \BbbR \times \BbbR n , Gr F (t, .) is closed for each t \in \BbbR ,
and F is \scrL \times \scrB n measurable,
(G3): There exists a measurable function R : [S\= - \sigma , T\= + \sigma ] \rightarrow (0, \infty ) \cup \{ +\infty \} (a
\circ \= T\=] and the following
``radius function""), such that R(t) \BbbB \subset B(t) a.e. t \in [S,
conditions are satisfied:
(a): (Pseudo-Lipschitz continuity) There exists k \in L1 (S\= - \sigma , T\= + \sigma ) such that
\circ F (t, x\prime ) \cap (x(t)
\=\. + R(t)\BbbB ) \subset F (t, x) + k(t)| x\prime - x| \BbbB , for all x, x\prime \in x(t)
\=
+ \epsilon \BbbB , a.e. t \in [S\= - \sigma , T\= + \sigma ],
(b): (Tempered growth) There exist r \in L1 (S\= - \sigma , T\= + \sigma ), r0 > 0, and \gamma \in (0, 1)
such that r0 \leq r(t), \gamma - 1 r(t) \leq R(t), a.e. t \in [S\= - \sigma , T\= + \sigma ], and
F (t, x) \cap (x(t)
\=\. + r(t)\BbbB ) \not = \emptyset for all x \in x(t)
\= + \epsilon \BbbB , a.e. t \in [S\= - \sigma .T\= + \sigma ],
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2568
PIERNICOLA BETTIOL AND RICHARD B. VINTER
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(G4): There exists cF \geq 0 such that, for a.e. t \in [S\= - \sigma , S\= + \sigma ] \cup [T\= - \sigma , T\= + \sigma ],
(5.2)
F (t, x) \subset cF \BbbB for all x \in x(t)
\= + \epsilon \BbbB .
\= T\=]; \BbbR n ) and \lambda \geq 0 such that
Then there exist p \in W 1,1 ([S,
(i): (p, \lambda ) \not = (0, 0),
\= T\=],
(ii): p(t)
\. \in co\{ \zeta : (\zeta , p(t)) \in NGrF (t,\cdot ) (\=
x(t), x(t))\} ,
\=\.
a.e. t \in [S,
\=
\=
(iii): there exist \xi 0 \in sub-esst\rightarrow S\= H(t, x(
\= S), p(S)) and \xi 1 \in super-esst\rightarrow T\= H(t, x(
\= T\=),
\=
p(T )) such that
\= \xi 1 , - p(T\=)) \in \lambda \partial g(S,
\= x(
\= T\=, x(
\= x(
\= T\=, x(
( - \xi 0 , p(S),
\= S),
\= T\=)) + NC (S,
\= S),
\= T\=)) ,
\= T\=].
\=\. + B(t)) , a.e. t \in [S,
(iv): p(t) \cdot x(t)
\=\. \geq p(t) \cdot v for all v \in co \Omega 0 (t) \cap (x(t)
\= x0 , T, x1 ) : (x0 , T, x1 ) \in C\} ,
\~ for some
If the initial time S\= is fixed, i.e., C = \{ (S,
closed set C\~ \subset \BbbR n \times \BbbR \times \BbbR n , and g(S, x0 , T, x1 ) = g(x
\~ 0 , T, x1 ) for some g\~ : \BbbR n \times \BbbR \times n
\BbbR \rightarrow \BbbR , then the above conditions are satisfied when (iii) is replaced by: there exists
\xi 1 \in super-esst\rightarrow T\= H(t, x(
\= T\=), p(T\=)) such that
\prime \= \xi 1 , - p(T\=)) \in \lambda \partial g(\=
\= T\=, x(
\= T\=, x(
(iii) : (p(S),
\~ x(S),
\= T\=)) + NC\~ (\=
x(S),
\= T\=)),
and when condition (5.2) in (G4) is satisfied only for a.e. t \in [T\= - \sigma , T\= + \sigma ]. The
assertions of the theorem can be similarly modified when the final time T\= is fixed.
It is possible to prove alternative versions of Theorem 5.1 in which the ``pseudoLipschitz continuity"" with ``tempered growth"" hypothesis (G3) is replaced by alternative, stronger hypotheses. We record here only one corollary of Theorem 5.1, where
(G3) is replaced by the requirement that F is pseudo-Lipschitz continuous for arbitrary constant radius functions R and that the associated integrable Lipschitz bounds
have polynomial growth w.r.t. R.
Corollary 5.2. The assertions of Theorem 5.1 remain valid when hypothesis
(G3) is replaced by the following:
(G3)*: There exist numbers \alpha > 0 and \epsilon > 0, and nonnegative measurable functions k and \beta such that k and t \rightarrow \beta (t)k \alpha (t) are integrable and, for each
N \geq 0,
F (t, x\prime \prime ) \cap (x(t)
\=\. + N \BbbB ) \subset F (t, x\prime ) + kN (t)| x\prime - x\prime \prime | \BbbB ,
\= T\=],
for all x\prime , x\prime \prime \in x(t)+\epsilon \=
\BbbB and for a.e. t \in [S,
where kN (t) := k(t) + \beta (t)N \alpha .
Proof of Theorem 5.1 (sketch). The proof is accomplished in several steps. We
provide here the first step of the proof in which we derive necessary optimality conditions for a problem where there are no constraints on the end-times and end-states,
and the differential inclusion F (t, .) is ``globally"" Lipschitz continuous, with integrable
Lipschitz constant, in the sense that the pseudo-Lipschitz condition is satisfied with
radius function R \equiv +\infty . Starting from this point, earlier analytic techniques can be
employed to complete the proof, which now involves sub- and superessential values
and takes advantage of the robustness properties of sub- and superessential values
under limit taking.
Take a function g : \BbbR \times \BbbR n \times \BbbR \times \BbbR n \rightarrow \BbbR and a multifunction F : \BbbR \times \BbbR n ; \BbbR n .
Consider the following free end-point dynamic optimization problem:
\left\{ Minimize g(S, x(S), T, x(T ))
over [S, T ] \subset \BbbR and x \in W 1,1 ([S, T ]; \BbbR n ) satisfying
(F EP )
x(t)
\. \in F (t, x(t)) a.e. t \in [S, T ] .
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2569
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FREE END-TIME PROBLEMS
Proposition 5.3. Let ([S \prime , T \prime ], x\prime ) be a W 1,1 local minimizer for (FEP). Assume
that, for some \epsilon \prime > 0 and \sigma \prime \in (0, (T \prime - S \prime )/2),
(FEP1): g is Lipschitz continuous,
(FEP2): F (t, x) is a nonempty, closed set for all (t, x) \in \BbbR \times \BbbR n , Gr F (t, .) is closed
for each t \in \BbbR , and F is \scrL \times \scrB n measurable,
(FEP3): there exists a constant kF > 0 such that, for a.e. t \in [S \prime - \sigma \prime , T \prime + \sigma \prime ],
F (t, x) \subset F (t, y) + kF | x - y| \BbbB for all x, y \in x\prime (t) + \epsilon \prime \BbbB ,
(FEP4): there exists cF \geq 0 such that, for a.e. t \in [S \prime - \sigma \prime , S \prime +\sigma \prime ]\cup [T \prime - \sigma \prime , T \prime +\sigma \prime ],
F (t, x) \subset cF \BbbB for all x \in x\prime (t) + \epsilon \prime \BbbB .
Then there exists p \in W 1,1 ([S \prime , T \prime ]; \BbbR n ) such that
(A): p(t)
\. \in co\{ \zeta : (\zeta , p(t)) \in NGr F (t,\cdot ) (x\prime (t), x\. \prime (t))\} , a.e. t \in [S \prime , T \prime ],
(B): p(t) \cdot x\. \prime (t) \geq p(t) \cdot v for all v \in F (t, x\prime (t)), a.e. t \in [S \prime , T \prime ],
(C): there exist \xi 0 \in sub-esst\rightarrow S \prime H(t, x\prime (S \prime ), p(S \prime )) and \xi 1 \in super-esst\rightarrow T \prime (t, x\prime (T \prime ),
p(T \prime )) such that
( - \xi 0 , p(S \prime ), \xi 1 , - p(T \prime )) \in \partial g(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )).
Proof. Assume that \beta > 0 is a parameter such that the W 1,1 local minimizer
([S , T \prime ], x\prime ) is minimizing w.r.t. admissible F trajectories satisfying
\prime d(([S, T ], x), ([S \prime , T \prime ], x\prime )) \leq \beta .
(Here d is the distance function (3.1).)
Consider first the special case of (FEP) in which g has the structure
g(S, x0 , T, x1 ) = g(S,
\~ x0 , T, x1 ) + e(S, x0 , T, x1 )
for some twice differentiable function g\~ and some Lipschitz continuous function e, with
Lipschitz constant ke . We shall prove a coarser version of the necessary conditions in
the proposition statement, in which condition (C) is replaced by
\surd 1
( - \xi 0 , p(S \prime ), \xi 1 , - p(T \prime )) \in \nabla \~
g(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + 2 2ke (1 + c2F ) 2 \BbbB .
Note that x\prime is a W 1,1 local minimizer for a version of problem (FEP) in which
the underlying time interval is fixed at [S \prime , T \prime ]. We can then apply well-known
Euler--Lagrange conditions (cf. [11, Theorem 7.4.1]) and deduce the existence of
p \in W 1,1 ([S \prime , T \prime ]; \BbbR n ) satisfying conditions (A) and (B) and the following transversality condition:
(5.3)
(p(S \prime ), - p(T \prime )) \in \nabla x0 ,x1 g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + ke \BbbB .
It is immediate to see that d(([S \prime , T \prime - s], x\prime ), ([S \prime , T \prime ], x\prime )) = s. But then, for s \in [0, \beta ],
by W 1,1 local optimality of ([S \prime , T \prime ], x\prime ), ([S \prime , T \prime - s], x\prime ) must have cost not less than
([S \prime , T \prime ], x\prime ). It follows that
g(S
\~ \prime , x\prime (S \prime ), T \prime - s, x\prime (T \prime - s)) + e(S \prime , x\prime (S \prime ), T \prime - s, x\prime (T \prime - s))
\geq g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + e(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) .
Since g\~ is a C 2 function, there exists r1 > 0 such that for all s sufficiently small
1
0 \geq g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) - g(S
\~ \prime , x\prime (S \prime ), T \prime - s, x\prime (T \prime - s)) - ke (1 + c2F ) 2 s
\int T \prime \Bigl( \Bigr) 1
\nabla T g(S
\~ \prime , x\prime (S \prime ), t, x\prime (t)) + \nabla x1 g(S
\~ \prime , x\prime (S \prime ), t, x\prime (t)) \cdot x\. \prime (t) dt - ke (1 + c2F ) 2 s
=
T \prime - s
\int T \prime 1
\prime \prime \prime \prime \prime \prime \geq \nabla T g(S
\~ , x (S ), T , x (T ))s - H(t, x\prime (T \prime ), p(T \prime )) dt - r1 s2 - ke (cF +(1+c2F ) 2 )s.
T \prime - s
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2570
PIERNICOLA BETTIOL AND RICHARD B. VINTER
To derive the final inequality in the above relation, we have used the following facts:
- p(T \prime ) \in \nabla x1 g\~ (S \prime , x(S \prime ), T \prime , x(T \prime ))+ke \BbbB ; moreover, by (FEP3) and (FEP4), we have,
for a.e. t \in [T \prime - s, T \prime ],
x\. \prime (t) \in F (t, x\prime (T \prime )) + kF | x\prime (t) - x\prime (T \prime )| \BbbB \subset F (t, x\prime (T \prime )) + kF cF s\BbbB ,
and so, for a.e. t \in [T \prime - s, T \prime ],
p(T \prime ) \cdot x\. \prime (t) \leq max\{ p(T \prime ) \cdot v : v \in F (t, x\prime (T \prime ))\} + kF cF | p(T \prime )| s
= H(t, x\prime (T \prime ), p(T \prime )) + kF cF | p(T \prime )| s .
We have shown that
\left\{ T \prime is a local minimum over [S \prime , T \prime ] of
\int T \prime (5.4)
T \rightarrow - \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime ))(T \prime - T ) - T ( - H)(t, x\prime (T \prime ), p(T \prime )) dt
1
+r1 (T - T \prime )2 + ke (cF + (1 + c2F ) 2 )| T - T \prime | .
Next, select a measurable function \xi : [T \prime , T \prime + \sigma \prime ] \rightarrow \BbbR n such that \xi (t) \in F (t, x\prime (T \prime )) a.e. and
p(T \prime ) \cdot \xi (t) =
max
v\in F (t,x\prime (T \prime ))
p(T \prime ) \cdot v,
a.e. t \in [T \prime , T \prime + s\prime ].
We deduce from Filippov's existence theorem that there exist s\prime \in (0, \sigma \prime ) and an F
trajectory y on [T \prime , T \prime + s\prime ] such that y(T \prime ) = x\prime (T \prime ) and
\int T \prime +s
(5.5)
| y(t)
\. - \xi (t)| dt \leq (1/2)ekF s kF cF s2 for all s \in [0, s\prime ].
T \prime We now construct an F trajectory ([S \prime , T \prime + s\prime ], x) by concatenating ([S \prime , T \prime ], x\prime )
and ([T \prime , T \prime + s\prime ], y). For s\prime \in (0, \sigma \prime ) sufficiently small, we have that d(([S \prime , T \prime +
s], x), ([S \prime , T \prime ], x\prime )) \leq \beta for all s \in [0, s\prime ]. In view of the W 1,1 local optimality of
([S \prime , T \prime ], x\prime ), for all s \in [0, s\prime ] the cost of ([S \prime , T \prime + s], x) exceeds that of ([S \prime , T \prime ], x\prime ). It
follows that
0 \leq (\~
g + e)(S \prime , x\prime (S \prime ), T \prime + s, y(T \prime + s)) - (\~
g + e)(S \prime , x\prime (S \prime ), T \prime , y(T \prime ))
\int T \prime +s \Bigl( \Bigr) 1
\leq \nabla T g(S
\~ \prime , x\prime (S \prime ), t, x(t)) + \nabla x1 g(S
\~ \prime , x\prime (S \prime ), t, x(t)) \cdot x(t)
\.
dt + ke (1 + c2F ) 2 s
T \prime for all s \in [0, s\prime ].
Arguing as in the analysis preceding (5.4), we can show that, for some constant
r2 \geq r1 and for all s \in [0, s\prime ],
\int T \prime +s \Bigl( \Bigr) \nabla T g(S
\~ \prime , x\prime (S \prime ), t, x(t)) + \nabla x1 g(S
\~ \prime , x\prime (S \prime ), t, x(t)) \cdot x(t)
\.
dt
T \prime \prime \prime \prime \prime \prime \int T \prime +s
\leq (\nabla T g(S
\~ , x (S ), T , x(T ))s +
( - H)(t, x\prime (T \prime ), p(T \prime )) dt
T \prime \prime + (1 + (1/2)ekF s kF cF )r2 s2 + ke cF s .
We have shown, for some s\prime \in (0, \sigma \prime ),
\left\{ \prime T is a local minimum over [T \prime , T \prime + s\prime ] of
\int T
(5.6)
T \rightarrow \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime ))(T - T \prime ) +
( - H)(t, x\prime (T \prime ), p(T \prime )) dt
\prime T \prime 1
+(1 + (1/2)ekF s kF cF )r2 | T - T \prime | 2 + ke (cF + (1 + c2F ) 2 )| T - T \prime | .
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2571
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FREE END-TIME PROBLEMS
\int T
\int T \prime Interpreting, as usual, the integral T \prime ( - H)(t, x\prime (T \prime ), p(T \prime ))dt as - T ( - H)(t, x\prime (T \prime ),
p(T \prime ))dt for T < T \prime , we deduce from (5.4) and (5.6) that
\left\{ \prime T is a local minimum over [S \prime , T \prime + s\prime ] of
\int T
T \rightarrow \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime ))(T - T \prime ) + T \prime ( - H)(t, x\prime (T \prime ), p(T \prime )) dt
\prime 1
+(1 + (1/2)ekF s kF cF )r2 (T - T \prime )2 + ke (cF + (1 + c2F ) 2 )| T - T \prime | .
Applying
Proposition 2.3, we take the indefinite integral function \psi to be \psi (t) :=
\int t
( - H)(s,
x\prime (T \prime ), p(T \prime ))ds, t \in [S \prime , T \prime + s\prime ], noting that 0 is a limiting subgradient
T \prime of a function at a minimizing point, and also applying the sum rule for limiting
subdifferentials of Lipschitz functions, we deduce that
1
- \nabla T g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) \in sub-ess
( - H)(t, x(T
\= \prime ), p(T \prime )) + ke (cF + (1 + c2F ) 2 )\BbbB .
\prime t\rightarrow T
A similar analysis, in which we take the left end-time, yields
1
- \nabla S g(S
\~ \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) \in sub-esst\rightarrow S \prime H(t, x\prime (S \prime ), p(S \prime )) + ke (cF + (1 + c2F ) 2 )\BbbB .
In view of (5.3) and Proposition 2.3, we have confirmed that there exist elements
\xi 0 \in sub-esst\rightarrow S \prime H(t, x\prime (S \prime ), p(S \prime )) and \xi 1 \in super-esst\rightarrow T \prime H(t, x\prime (T \prime ), p(T \prime )) =
- sub-esst\rightarrow T \prime ( - H)(t, x\prime (T \prime ), p(T \prime )) such that
\surd 1
( - \xi 0 , p(S \prime ), \xi 1 , - p(T \prime )) \in \nabla \~
g(S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) + 2 2ke (1 + c2F ) 2 \BbbB .
Recall that, up to this point in the proof, we have assumed the cost function g
has a special structure. We now drop this assumption; henceforth we assume that g
is an arbitrary Lipschitz continuous function.
For i = 1, 2, . . . let gi be the ith quadratic inf convolution of g. Since ([S \prime , T \prime ], x\prime )
is a W 1,1 local minimizer for (FEP), there exists \beta > 0 such that ([S \prime , T \prime ], x\prime ) is a
minimizer for (FEP), when we append the constraint d(([S, T ], x), ([S \prime , T \prime ], x\prime )) \leq \beta .
Consider, for each i, the following variant on (FEP), in which we replace g by gi and
add a distance constraint:
\left\{ Minimize gi (S, x(S), T, x(T ))
over [S, T ] \subset \BbbR and x \in W 1,1 ([S, T ], \BbbR n ) satisfying
(F EP )i
x(t)
\. \in F (t, x(t)) a.e.
d(([S, T ], x), ([S \prime , T \prime ], x\prime )) \leq \beta .
\surd Write \alpha i := kg / i, where kg is a Lipschitz constant for g. According to the properties
of quadratic inf convolutions, ([S \prime , T \prime ], x\prime ) is an \alpha i2 minimizer for this problem.
We can express the preceding problem as
Minimize \{ Ji ([S, T ], x) : ([S, T ], x) \in \scrM \} ,
where \scrM denotes the space of admissible arcs ([S, T ], x) for problem (FEP)i and
Ji ([S, T ], x) := gi (S, x(S), T, x(S)). Ji is continuous on the closed set \scrM , equipped
with the d metric topology. It follows then from Ekeland's theorem that there exists
an element ([Si , Ti ], xi ) in \scrM that minimizes Ji ([S, T ], x) + \alpha i d([S, T ], x), ([Si , Ti ], xi )
over \scrM and
(5.7)
d(([Si , Ti ], xi ), ([S \prime , T \prime ], x\prime )) \leq \alpha i .
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2572
PIERNICOLA BETTIOL AND RICHARD B. VINTER
For i sufficiently large, \alpha i < \beta /2. Then, by the triangle inequality applied to the
metric d, ([Si , Ti ], xi ) remains a minimizer for (FEP)i , when the distance constraint
in the definition of \scrM is replaced by d(([S, T ], x), ([Si , Ti ], xi )) \leq \beta /2. It now follows
that ([Si , Ti ], xi ) is a minimizer for
\left\{ \Biggl( \int T
Minimize gi (S, x(S), T, x(T )) + \alpha i S | x(t)
\. - x\. i (t)| dt + | x(S) - xi (Si )| \Biggr) +(1 + 2cF )(| S - Si | + | T - Ti | ) ,
over [S, T ] \subset \BbbR and x \in W 1,1 ([S, T ], \BbbR n ) satisfying
x(t)
\. \in F (t, x(t)), a.e. t \in [S, T ],
d(([S, T ], x), ([Si , Ti ], xi )) \leq \beta /2.
We have shown that ([Si , Ti ], xi , yi \equiv 0) is a W 1,1 local minimizer for
\left\{ Minimize gi (S, x(S), T, x(T ))
+\alpha i (y(T ) - y(S) + | x(S) - xi (Si )| + (1 + 2cF )(| S - Si | + | T - Ti | ))
\prime (F EP )i
over [S, T ] \subset \BbbR and (x, y) \in W 1,1 ([S, T ], \BbbR n+1 ) satisfying
(x(t),
\.
y(t))
\.
\in F\~ (t, x(t)) a.e.,
in which F\~ : \BbbR \times \BbbR n ; \BbbR n+1 is the multifunction
F\~ (t, x) := \{ (v, | v - x\. i (t)| ) : v \in F (t, x)\} for (t, x) \in \BbbR \times \BbbR n .
By properties of quadratic inf convolutions, there exists a quadratic function g\~i such
that
\nabla \~
gi (Si , xi (Si ), Ti , xi (Ti )) \in \partial P g(zi )
for some zi \in (Si , xi (Si ), Ti , xi (Ti )) + (kg /i)\BbbB , and
\biggl\{ g\~i (z) \geq gi (z) for all z \in \BbbR \times \BbbR n \times \BbbR \times \BbbR n ,
g\~i (z) = gi (z) when z = (S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) .
\prime It follows that ([Si , Ti ], xi , yi \equiv 0) is also a W 1,1 local minimizer for a variant of (FEP)i ,
in which the quadratic function g\~i replaces gi , namely
\left\{ Minimize g\~i (S, x(S), T, x(T )))
+\alpha i (y(T ) - y(S) + | x(S) - xi (Si )| + (1 + 2cF )(| S - Si | + | T - Ti | ))
over [S, T ] \subset \BbbR and (x, y) \in W 1,1 ([S, T ]; \BbbR n+1 ) satisfying
(x(t),
\.
y(t))
\.
\in F\~ (t, x(t)) a.e.
This problem is an example of the special case of (FEP), for which the analysis
of Step 1 provides necessary conditions of optimality. The relevant hypotheses are
satisfied. To interpret these conditions, we note that, given a proximal normal vector
P
(\xi , (\eta , \eta 0 )) \in NGr
(\~
x, (\~
v, | \~
v - x\. i (t)| )), there exists M > 0 such that (\~
x, v)
\~ is a local
F\~ (t,.)
minimizer of the function
(x, v) \rightarrow - \xi \cdot x - \eta \cdot v - \eta 0 | v - x\. i (t)| + M (| x - x| \~ 2 + | v - v| \~ 2
+ (| v - x\. i (t)| - | \~
v - x\. i (t)| )2 )
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FREE END-TIME PROBLEMS
2573
over (x, v) \in Gr F (t, .). But then, by the exact penalization principle (cf. [11, Theorem 3.2.1]), there exists a constant K > 0 such that (\~
x, v)
\~ is a local minimizer of the
function
(x, v) \rightarrow - \xi \cdot x - \eta \cdot v - \eta 0 | v - x\. i (t)| + KdGr F (t,.) (x, v) + M (| x - x| \~ 2 + | v - v| \~ 2
+ (| v - x\. i (t)| - | \~
v - x\. i (t)| )2 ).
Since NGr F (t,.) = \{ \lambda \partial dGr F (t,.) : \lambda \geq 0\} , it follows that
(5.8)
(\xi , \eta ) \in NGr F (t,.) (\~
x, v)
\~ + | \eta 0 | \BbbB .
Consideration of limits of proximal normal vectors to Gr F\~ yields the information that
(5.9)
(\xi , \eta , \eta 0 ) \in NGr F\~ (t,.) (x, v, | v - x\. i (t)| ) =\Rightarrow (\xi , \eta ) \in NGr F (t,.) (x, v) + | \eta 0 | \BbbB .
Now apply the earlier derived necessary conditions of Proposition 5.3 when g has the
stated special structure. Notice that, in consequence of the Euler--Lagrange inclusion
and the transversality condition, the costate arc component associated with the y
state component is a constant and takes value - \alpha i . We deduce that there exists
pi \in W 1,1 ([S \prime , T \prime ]; \BbbR n ) such that
\prime (A) : p\.i (t) \in co\{ \zeta : (\zeta , pi (t)) \in NGr F (t,\cdot ) (xi (t), x\. i (t))\} + \alpha i \BbbB , a.e. t \in [Si , Ti ],
\prime (B) : pi (t) \cdot x\.i (t) + \alpha i | v - x\. i (t)| \geq pi (t) \cdot v for all v \in F (t, xi (t)), a.e. \in [Si , Ti ],
\prime (C) : there exist \xi 0i \in sub-esst\rightarrow Si H(t, xi (Si ), p(Si )) and \xi 1i \in super-esst\rightarrow Ti H(t,
xi (Ti ), p(Ti )) such that
\surd ( - \xi 0i , pi (Si ), \xi 1i , - pi (Ti )) \in \partial g(zi ) + (2 2)\alpha i (1 + 2cF )2 \BbbB .
We use our ``constant extrapolation"" convention to extend the domains of the
\prime \prime pi 's to the entire real line. We deduce from (A) and (C) that the pi 's are uniformly
bounded with uniformly integrably bounded derivatives. We can arrange then, by
subsequence extraction, that pi \rightarrow p uniformly and p\.i \rightarrow p\. weakly in L1 for some
p \in W 1,1 . The sequences \{ \xi 0i \} and \{ \xi ii \} are bounded. A further subsequence extraction ensures that they have limits \xi 0 and \xi 1 , respectively. We know, however,
that ([Si , Ti ], xi ) \rightarrow ([S \prime , T \prime ], x\prime ) w.r.t. the d metric and zi \rightarrow (S \prime , x\prime (S \prime ), T \prime , x\prime (T \prime )) as
i \rightarrow \infty . A by now familiar convergence analysis permits us to pass to the limit as
\prime \prime i \rightarrow \infty in conditions (A) --(C) and arrive at the relations asserted in the proposition
statement. The novel feature of the analysis, as compared with treatment of fixed
time problems, is that we now make use of the stability, under limit taking, of relations expressed in terms of the sub- and superessential values. We note specifically
that under the state hypotheses, the conditions \xi 0i \in sub-esst\rightarrow Si H(t, xi (Si ), p(Si ))
and \xi 1i \in super-esst\rightarrow Ti H(t, xi (Ti ), p(Ti )), for each i, Proposition 2.4 implies that
\xi 0 \in sub-esst\rightarrow S \prime H(t, x\prime (S \prime ), p(S \prime )) and \xi 1 \in super-esst\rightarrow T \prime H(t, x\prime (T \prime ), p(T \prime )).
6. Pathwise state constraints problems. We provide in this section extensions of state constrained free end-time problems. As in our earlier treatment of free
end-time free problems with free end-times (where state constraints were absent), the
extra relations corresponding to the free end-time take the form of sub- and superessential values of the maximized Hamiltonian.
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2574
PIERNICOLA BETTIOL AND RICHARD B. VINTER
Consider the following state constrained problem with free end-times:
\left\{ Minimize g(S, x(S), T, x(T ))
over [S, T ] \subset \BbbR , x \in W 1,1 ([S, T ]; \BbbR n ) and measurable functions u satisfying
x(t)
\.
= f (t, x(t), u(t)) a.e. t \in [S, T ],
(F T C)
u(t) \in U (t) a.e. t \in [S, T ],
h(t, x(t)) \leq 0 for all t \in [S, T ],
(S, x(S), T, x(T )) \in C,
the data for which comprise functions g : \BbbR \times \BbbR n \times \BbbR \times \BbbR n \rightarrow \BbbR , f : \BbbR \times \BbbR n \times \BbbR m \rightarrow \BbbR n , and h : \BbbR \times \BbbR n \rightarrow \BbbR , a multifunction U : \BbbR ; \BbbR m , and a set C \subset \BbbR \times \BbbR n \times \BbbR \times \BbbR n . Now, an admissible process is a triple ([S, T ], x, u) whose elements satisfy
all the constraints of problem (F T C), which include the pathwise state constraint
`h(t, x(t)) \leq 0 for all t \in [S, T ]'.
\= T\=],
Theorem 6.1 (state constrained free end-time maximum principle). Let ([S,
x,
\= u)
\= be a W 1,1 local minimizer for (F T C) such that T\= - S\= > 0. Assume that there
\=
exist \epsilon > 0 and \sigma \in (0, (T\= - S)/2)
such that hypotheses (H1)--(H4) of Theorem 3.1 are
satisfied. Suppose, in addition, that
(H5): h is upper semicontinuous and there exists a constant kh such that
| h(t, x) - h(t, x\prime )| \leq kh | x - x\prime | for all x, x\prime \in x
\=e (t) + \epsilon \BbbB , for all t \in [S\= - \sigma , T\= + \sigma ]; furthermore
(6.1)
\= x(
\= < 0
h(S,
\= S))
and
h(T\=, x(
\= T\=)) < 0.
\= T\=]; \BbbR n ), \lambda \geq 0, a measure \mu \in C \oplus (S,
\= T\=), and a Borel
Then there exist p \in W 1,1 ([S,
n
\=
\=
measurable function \gamma : [S, T ] \rightarrow \BbbR , such that the following conditions, in which
\= T\=]; \BbbR n ) is the function
q \in N BV ([S,
\biggl\{ \=
p(t) \int if t = S,
(6.2)
q(t) :=
\= \=
p(t) + [S,t]
\= \gamma (s)d\mu (s) if t \in (S, T ] ,
are satisfied:
(i) (p, \mu , \lambda ) \not = (0, 0, 0),
\= T\=],
(ii) - p(t)
\.
\in co \partial x q(t) \cdot f (t, x(t),
\=
u(t))
\=
a.e. t \in [S,
\= \xi 1 , - q(T\=)) \in \lambda \partial g(S,
\= x(
\= T\=, x(
\= x(
\= T\=, x(
(iii) ( - \xi 0 , q(S),
\= S),
\= T\=))+NC (S,
\= S),
\= T\=)) for some
\= p(S))
\= and \xi 1 \in super-esst\rightarrow T\= H(t, x(
\xi 0 \in sub-esst\rightarrow S\= H(t, x(
\= S),
\= T\=), p(T\=)),
\= T\=],
(iv) q(t) \cdot f (t, x(t),
\=
u(t))
\=
\geq q(t) \cdot f (t, x(t),
\=
u) for all u \in U (t), a.e. t \in [S,
\= T\=].
(v) \gamma (t) \in \partial x> h(t, x(t)),
\=
\mu -a.e. t \in [S,
\= x0 , T, x1 ) : (x0 , T, x1 ) \in C\} ,
\~ for some closed
If the initial time S\= is fixed, i.e., C = \{ (S,
set C\~ \subset \BbbR n \times \BbbR \times \BbbR n , and g(S, x0 , T, x1 ) = g(x
\~ 0 , T, x1 ) for some g\~ : \BbbR n \times \BbbR \times \BbbR n \rightarrow \BbbR ,
then the above conditions are satisfied when (iii) is replaced by: there exists \xi 1 \in super-esst\rightarrow T\= H(t, x(
\= T\=), p(T\=)) such that
\= \xi 1 , - p(T\=)) \in \lambda \partial g(\=
\= T\=, x(
\= T\=, x(
\~ x(S),
\= T\=)) + NC\~ (\=
x(S),
\= T\=)),
(iii)\prime : (p(S),
and when, in hypothesis (H4), the stated conditions are required to hold only for
t \in (T\= - \sigma , T\= + \sigma ), and (6.1) in (H5) is replaced by ``h(T\=, x(
\= T\=)) < 0."" The assertions
of the theorem can be similarly modified when the final time T\= is fixed.
Here,
\partial x> h(t, x(t))
\=
:= co lim sup \{ \partial x h(ti , xi ) : ti \rightarrow t, xi \rightarrow x(t)
\= and h(ti , xi ) > 0 for each i\} ,
where the lim sup is taken in the Kuratowski sense.
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FREE END-TIME PROBLEMS
2575
We also provide necessary conditions that are ``state constrained"" analogues of
the free end-time necessary conditions of section 5. Consider the following state
constrained problem, involving a differential inclusion, with free end-times:
\left\{ Minimize g(S, x(S), T, x(T ))
over intervals [S, T ] and arcs x \in W 1,1 ([S, T ], \BbbR n ) satisfying
\prime x(t)
\. \in F (t, x(t)) a.e. t \in [S, T ],
(F T C)
(S, x(S), T, x(T )) \in C,
h(t, x(t)) \leq 0 for all t \in [S, T ].
Accordingly, an F trajectory ([S, T ], x) is said to be admissible if it satisfies all the
constraints of (F T C)\prime .
Theorem 6.2 (state constrained free end-time problems). Take a measurable
\= T\=], x)
multifunction B : \BbbR ; \BbbR n such that B(t) is open for a.e. t \in \BbbR . Let ([S,
\= be a
W 1,1 local minimizer for (F T C)\prime relative to B such that T\= - S\= > 0. Assume that,
for some \epsilon > 0 and \sigma > 0, the data satisfy hypotheses (G1)--(G4) of Theorem 5.1.
Suppose, in addition, that
(G5): h is upper semicontinuous and there exists a constant kh such that
| h(t, x) - h(t, x\prime )| \leq kh | x - x\prime | for all x, x\prime \in x
\=e (t) + \epsilon \BbbB , for all t \in [S\= - \sigma , T\= + \sigma ]; furthermore
(6.3)
\= x(
\= < 0
h(S,
\= S))
and
h(T\=, x(
\= T\=)) < 0.
\= T\=]; \BbbR n ), real numbers \lambda \geq Then there exist an absolutely continuous arc p \in W 1,1 ([S,
\oplus \= \=
\= T\=] \rightarrow \BbbR n ,
0, \xi 0 , \xi 1 , a measure \mu \in C (S, T ), and a Borel measurable function \gamma : [S,
n
\= T\=]; \BbbR ) is the function
such that the following conditions hold, in which q \in N BV ([S,
\biggl\{ \=
\=
p(S)
if t = S,
\int q(t) :=
\= T\=] .
p(t) + [S,t] \gamma (s)d\mu (s) if t \in (S,
(i) \lambda + \| p\| L\infty + \| \mu \| T.V. = 1,
\= T\=],
(ii) p(t)
\.
\in co\{ \zeta : (\zeta , q(t)) \in NGrF (t,\cdot ) (\=
x(t), x(t))\} ,
\=\.
a.e. t \in [S,
\=
\=
\=
\=
\=
\=
\=
\= T\=, x(
(iii) ( - \xi 0 , q(S), \xi 1 , - q(T )) \in \lambda \partial g(S, x(
\= S), T , x(
\= T )) + NC (S, x(
\= S),
\= T\=)),
\= T\=],
\=\. + B(t)) a.e. t \in [S,
(iv) q(t) \cdot x(t)
\=\. \geq q(t) \cdot v for all v \in co \Omega 0 (t) \cap (x(t)
\= T\=],
\=
\mu - a.e. t \in [S,
(v) \gamma (t) \in \partial x> h(t, x(t))
\= q(S)),
\=
(vi) \xi 0 \in sub-esst\rightarrow S\= H(t, x(
\= S),
(vii) \xi 1 \in super-esst\rightarrow T\= H(t, x(
\= T\=), q(T\=)).
(In condition (i), \Omega 0 (t) is as defined by (5.1).)
\= x0 , T, x1 ) : (x0 , T, x1 ) \in C\} \~ for some set C)
\~
If the initial time is fixed (i.e., C = \{ (S,
and g(S, x0 , T, x1 ) = g(x
\~ 0 , T, x1 ), for some function g\~ : \BbbR n \times \BbbR \times \BbbR n \rightarrow \BbbR , then the
above assertions (except (vi)) remain true when condition (5.2) in (G4) is satisfied
only for a.e. t \in [T\= - \sigma , T\= + \sigma ] and (6.3) in (G5) is replaced by ``h(T\=, x(
\= T\=)) < 0""; in
this case, (iii) can be written
\= \xi 1 , - q(T\=)) \in \lambda \partial g(\=
\= T\=, x(
\= T\=, x(
(q(S),
\~ x(S),
\= T\=)) + NC\~ (\=
x(S),
\= T\=)).
Hypotheses (G4) and (G5) and the assertions of the theorem are modified similarly,
when the right end-time is fixed.
The proofs of Theorems 6.1 and 6.2, which we omit, follow the pattern of their
fixed end-time counterparts and can be constructed by formulating a sequence of state
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2576
PIERNICOLA BETTIOL AND RICHARD B. VINTER
constraint-free problems that approximate the reference state constrained problem.
It is then possible to apply simpler (or well-known) necessary optimality conditions
(such as Clarke's nonsmooth maximum principle, when the dynamic constraint is a
controlled differential equation). The difference now is that, in place of fixed endtime necessary conditions, we make use of free end-time necessary conditions; in
particular, we can employ the free end-time maximum principle Theorem 3.1 for
problems involving controlled differential equations. Once again the extra information
expressed in terms of sub- and superessential values of the maximized Hamiltonian is
preserved under limit taking and manifests itself as the free end-time transversality
condition (iii) above.
We also observe that Theorem 6.2 extends earlier results on free end-time problems involving differential inclusions, such as those in [12], in other respects as well.
Indeed, the notion of local minimizer here is more general, and condition (G3) covers
assumptions previously considered in the literature, such as (G3)* of Corollary 5.2
(which pertains to the case discussed in [12]).
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