Proceedings of the Royal Society of Edinburgh, 124A, 211-251,1994
Necessary and sufficient conditions for optimal controls in
viscous flow problems
H. O. Fattorini* and S. S. Sritharan|
Department of Mathematics, University of California-Los Angeles,
Los Angeles, California 90024-1555, U.S.A.
(MS received 10 February 1992. Revised MS received 30 November 1992)
A class of optimal control problems in viscous flow is studied. Main results are
the Pontryagin maximum principle and the verification theorem for the
Hamilton-Jacobi-Bellman equation characterising the feedback problem. The
maximum principle is established by two quite different methods.
1. Introduction
Optimal control theory of viscous flow has several applications in engineering science.
In [20], a fundamental optimal control problem in exterior hydrodynamics was
studied. In that paper, the task of accelerating an obstacle from rest to a given speed
in a given time, minimising the energy expenditure, was considered. In [13] a unified
formulation of optimal control problems in viscous hydrodynamics, covering wind
tunnel flow, flow inside containers and exterior hydrodynamics, was considered. Both
of these papers were concerned with proving existence theorems for optimal control.
The present paper is a sequel to these papers. In this paper we establish the following
two fundamental steps in the optimal control of viscous incompressible flow:
(i) the Pontryagin maximum principle to obtain the necessary conditions;
(ii) an analysis of the feedback problem using the infinite dimensional HamiltonJacobi-Bellman equations.
In Section 2, we consider a nonlinear evolution equation in a Hilbert space with
a certain type of cost functional. The form of this system represents several control
problems in fluid mechanics. The major theorems of this paper are stated in Section 2
and proved in later sections. This section also contains the hypotheses on various
operators and, as shown in [13], these are in fact satisfied for the specific flow
control problems.
In Section 3 we consider the task of computing the optimal control. The Pontryagin
maximum principle we prove in this section provides the necessary conditions for
such computations, in the form of an adjoint backward linear evolution problem
and a variational inequality which states that a certain Hamiltonian takes its maximum value at the optimal control. The maximum principle proved in this section is
* Supported by the NSF grant DMS-90011793.
t Mailing address: Code 574, NCCOSC, San Diego, CA 92152-5000, U.S.A.; supported by the
ONR-URI Grant No-N00014-91-J-4037.
212
H. Fattorini and S. S. Sritharan
powerful, in the sense that it accommodates a very general target condition. Ekeland's
variational principle plays a key role in this proof.
In Sections 4, 5 and 6 we elaborate the dynamic programming concept introduced
to fluid mechanics in [19].
In Section 4 we consider the feedback problem. The value function is defined as
the minimum value of the cost. It is shown that the value function is locally Lipschitz.
An important result proved in this context is that the value function is a viscosity
solution (in the sense of Crandall and Lions [5, 6]) of the Hamilton-Jacobi-Bellman
equation associated with our control problem.
In Section 5 we use the method in [3, 4] to provide another proof for the
Pontryagin maximum principle. In this proof, however, no target set is included.
Finally, we establish the verification theorem for the Hamilton-Jacobi-Bellman
equation in Section 6. This theorem provides the mathematical resolution of the
feedback control problem for the Navier-Stokes equations.
Some of the results of this paper were announced in [21].
2. Unified mathematical formulation and main theorems
As in [13], we consider the general control system:
te(T,T),
(2.1)
(2.2)
in a Hilbert space H. The operator si satisfies the following hypothesis:
HYPOTHESIS 2.1. si is self-adjoint and positive:
(siy,y)H^0,
VyeD(si).
The origin belongs to the resolvent set of si.
For a ^ 0, we denote by Ha the space D(si"). This space is a Hilbert space equipped
with its natural inner product (y, z)a = (s/"y, si*z)H. The inner product (-,-)« corresponds to the norm ||j>||a= I I ^ ^ H H - For oc^O, Ha is the closure of H under the
norm ||-|| a .
The nonlinear term ^T(-) is unbounded and satisfies the following hypothesis:
2.2. There exists p, 0 ^ /? < 1/2 such that the map JT(-):Hi-+H-f is
continuous and locally bounded and has a Frechet derivative [DyjV](y) which is
continuous and locally bounded (as an S£(H^,H.^-valued function).
HYPOTHESIS
The control U(t) takes values in a Hilbert space F. The linear operator 3$ satisfies:
HYPOTHESIS 2.3. J 1 e Z£(F; H).
We shall assume that controls {/(•) belong to L 2 (T, T;F). By definition, solutions
(or trajectories) of the initial value problem (2.1)-(2.2) in an interval x rg t ^ T are
H±-valued functions y(t) continuous in the norm of H± and satisfying
S(t-r)®U(r)dr,
(2.3)
Optimal controls in viscous flow problems
213
The following result is useful in the justification of (2.3) and other integral equations
of this paper.
LEMMA 2.4 ([13,16,18]). Let si be self-adjoint and non-negative definite, and let
S{t) = exp {—tsi) be the analytic semigroup generated by — si. Then:
(I) for any CeH, si^S{-% e L2(0, oo; H) and
ll^(o,=o:fl) = - ^ l l f l U .
(2-4)
Let g(-) e L2(a, b; H) be given. If we define the function
y(t)=
then:
(II)
S(t-r)g(r)dr,
a^t^b,
(2.5)
y(-)eC{<ia,by,D{j**))and
(III) The function
y(-) e L2(a, b; D(si)) and
WyWWcaa.bWA1'2))^-/^ \\g(-)\\L2(a,b;H)(2-6)
V2
y(-) has a derivative 8ty(t) in L2(a, b; H); moreover,
\L^MH),
(2-7)
\\L2ia,b,H),
(2.8)
and
dty(t) + s/y(t)=g(t),
for t a.e. in [a,
ft].
(2.9)
In certain of our results, we shall consider controls whose values are restricted to lie
in some subset U (the control set) off. In these cases, we denote by Ji{Q, T; U) the
space of all strongly measurable functions [/(•) such that U(t)eU a.e. in 0 ^ t ^ T.
The space Jfad(0, T; U) of admissible controls is
J?ad(0, T;U) = Jt(0,T,U),
when U is bounded, and
J?ad(0, T; U) = JK{0,T, U)nL2(0, T; U)
when f/is unbounded. Solvability of (2.3) was studied in [13]. The solutions of (2.3)
will be denoted by y(t) = y(t, T; C; U) to indicate dependence on the parameters.
The cost functional of the problem is of the form
tf(r, T, f, U) = <po(y(T, T; f; I/)) + j
{&(t,y(t, r, f; U)) + Y(t, U(t))} dt.
(2.10)
The task is to find the optimal control U e Jiai{§, T; U) that minimises the cost
functional <^(T, x, £ C7) subject to the target condition
y(T,r,£U)eY<=H
(2.11)
to be satisfied by the trajectories, where the closed set Y is called the target set. Here
<p0, 0 and Y are real-valued functions satisfying the following properties:
214
H. Fattorini and S. S. Sritharan
HYPOTHESIS 2.5. <po(-):H->R + and has a continuous Frechet derivative [D<po~](y).
HYPOTHESIS 2.6. 0(t, j ) : [ 0 , T] x H+.-tR is continuous locally bounded and the
Frechet derivative [2)^0] (£, y) continuous and locally bounded (as an / / _ ^-valued
function) in [0, T] x Hi.
For the feedback problem, the following stronger hypothesis on 0 ( - , •) will be used:
HYPOTHESIS 2.7. 0 ( - , •):[(), T] x H^-^R is continuous locally bounded and the
Frechet derivative [Dy0~](t,y) continuous and locally bounded (as an / / _ ^valued
function) in [0, T] x H±. Moreover,
^CAyWX,
VyeHt,
Vte[0,T]
and
)llff_ 1/2 ^C 2 ||.>i i ,
HYPOTHESIS
VyeHit
Vf6[0,T].
2.8. Y(-, •): [0, T] x F->(— oo, + oo] is continuous. There exists Co such
that
|Y(t,C7)|^ Coll 1/HI, O^t^T,
Vl/eC/
and for every R>0 there exists C(R) such that
r-T
T
Y(t, U(t)) dt^R
implies
\\ U(t) \\2F dt ^ C(R).
We assume that the cost functional <^(T, T, £ U) (or rather, the part that depends
directly on U) is weakly lower semicontinuous:
2.9. For every sequence {[/"}, UneJtai(x, T; U) such that [/"(•)-• £/(•)
weakly in L 2 (T, T; F), we have
T
rr
Y(t, U(tj) dt ^ lim inf
Y(t, U"(t)) dt.
HYPOTHESIS
For the feedback problem we again use a stronger hypothesis on Y(-, •):
HYPOTHESIS 2.10. Y(t, -):F-*(— oo, +oo] is convex and the conjugate function
Y*(t,-):F*-+R:
Y*(-, P) = sup {<F, U}F*XF - Y(-, U); U e U}
is Gateaux differ entiable with locally Lipschitz Gateaux derivative [VpY*](t, P(t)).
The value function V: [0, T] x H ->/? + is denned as
r(x, C) = min {<£(T, x; f; U); U e L2(0, T, U)}.
We associate to the above control problem the Hamilton-Jacobi-Bellman
equation:
,dlr)
= o,
ts[x,T],
(2.12)
Here, @* e y(H; F*) is the adjoint of & and J f ( • , - , - ) is the true Hamiltonian
Optimal controls in viscous flow problems
215
defined as
Let us now state the central theorems of this paper. The first theorem provides
the necessary conditions for optimal control and two proofs will be given. The proof
in Section 3 uses Hypotheses 2.1, 2.2, 2.3, 2.5, 2.6, 2.8 and 2.9. The proof in Section 5
uses Hypotheses 2.1-2.3 and 2.5-2.10 as well as certain additional properties of the
nonlinear operator Jf which are specific to the Navier-Stokes control problem.
2.11 (Pontryagin maximum principle for hydrodynamics). Let
U e L 2 (T, T; U) be the optimal control and y(t, T; £; U) be the optimal trajectory corresponding to the initial datum CeD(s/) at t = x. Then, under Hypotheses 2.1, 2.2, 2.3,
2.5, 2.6, 2.8 and 2.9 3pe C([T, I ] ; f l ) n I 2 ( T , T; H^) such that dtp e L2(t, T; H.^) and,
THEOREM
-d,p + dp + [_DyjVY{y)p = [Dy9-](t, y)
(2.13)
and
p(T)-lD<pol(y(T))eNY(y(T)),
where NY(y(T)) is the Clarke normal cone to Y at y(T) (see Definition 3.4). Moreover,
Y*(t, -31*p{t)) = ~<®*P(t), U(t)yF*XF-Y(t,
iJ(t)),
t a.e. in [T, T]
and if in addition Hypothesis 2.10 is satisfied, then
Here [_DyJ/"\*(y) is the adjoint of the Frechet derivative of jV(y(t)) at y.
In Sections 4-6 we prove, for the case of Y = H (constraint free final state), the
following theorem under Hypotheses 2.1-2.3 and 2.5-2.10 and certain additional
properties (see [13, Theorem 3]) of the nonlinearity Jf.
2.12 (Verification Theorem). Let the target set be Y = Hand let Hypotheses
2.1-2.3 and 2.5-2.10 be satisfied. Then the value function f " e C ( [ 0 , T\ x H). For
each te[%,T], r(t,-)
is Lipschitz in £ e / 7 and for each £e£>(«s/), f (-,£) is
absolutely continuous in tE[%,T~\. The super differential d^i^(t,C) is nonempty
V (t, 0 e [T, T] x D(st) and, for each C e
THEOREM
dt-T - 3V(t, C,p) = 0,
t a.e. in [T, T]
and, for some p e d^ i^{t, £),
Moreover, the optimal control U is given by the feedback relation:
U(-) = VY*(-, -3i*p(-)),
for somep(-) e d+ r{- J{-, r, £ U)).
(2.14)
216
H. Fattorini and S. S. Sritharan
Here the super differential 3C+ Y is defined as follows (t is suppressed) [6, 5]:
8? Tito) = \p e H; lim sup (
I
C^Co
\
°
I I S " toll H
^Jl)
S 0>.
/
(2.15)
)
3. Pontryagin maximum principle via the Ekeland variational principle
In this section we prove the maximum principle when the final state y(t, T; £; U) is
subjected to the target condition (2.11). We first formulate our control problem as
a nonlinear programming problem and derive a necessary condition in terms of the
set of variations of the state and the cost functional. We then analyse the linearisation
of the system (2.1). These results are used in the following two subsections, and also
later in the analysis of the feedback problem. We also compute the variations and
discuss their convergence properties. Finally we use the results of the first three
subsections to establish the maximum principle.
Infinite-dimensional programming and the maximum principle
We shall use Ekeland's approach to the maximum principle [7] as generalised to
infinite dimensional systems in [8,9,10,11]. In the last two works optimal control
problems are treated as particular cases of the following problem:
3.1 (Infinite-dimensional nonlinear programming problem). Let J be a
metric space, H a Hilbert space, f:J?->H,fc:J?^>Rbe
given maps and Y a subset
of H. Find U e Ji such as to:
PROBLEM
minimise fc(U),
UeJt,
subject to/([/) e y.
(3.1)
(3.2)
In the application we have in mind, f(U) is defined as the final value of the state:
f(U)=y(T,z;C,U)
and fc(U) is taken as the cost functional:
Let us first recall certain basic tools of nonsmooth analysis.
DEFINITION 3.2 (Contingent cone). Let Y be an arbitrary subset of a Hilbert space
H. Given y e H, the contingent cone to Y at y is the set KY(y) defined as follows:
w e KY(y) if and only if there exists a sequence {Xk} of positive numbers with Xk-+§
and a sequence {yk} c F such that
Equivalently, w e KY(y) if and only if there exists a sequence {Xk} as above and there
exists a sequence {wk} <=H such that wk -> w and y + kkwk e Y.
Ky{y) is a closed and in general nonconvex cone.
DEFINITION 3.3 (Clarke tangent cone). Let Y be an arbitrary subset of a Hilbert space
H. Given y e H, the Clarke tangent cone to Y at y is the set CY(y) which consists of
all w € H such that, for every sequence {kk} of positive numbers and every sequence
Optimal controls in viscous flow problems
217
{jj} c y such that Ak->0, yk->y, there exists a sequence {yk} <= Y such that
^f 1 O'*-.F*)- > M'
as/c->oo.
Equivalently, w> e Cy(j>) for every sequence {Xk} and {j 4 } <= Y as above there exists
a sequence {wk} c f f such that wk->w and yk + Xkwk e F.
It can be shown that CY(y) is convex and closed (see [1, p. 407]). It is evident from
the definitions that CY(y) <= KY(y). These cones are in general different, although if
Y is a convex set we have ([1, p. 407])
Let X be a subset of a Hilbert space H. The negative polar (cone) X
by
of A' is denned
Obviously, X~ is a closed convex cone (that is, X~ is closed and, if x, z e .Y" and
A ^ 0, then x + z, Xz e X~).
DEFINITION
3.4 (Clarke normal cone). The negative polar cone of the Clarke tangent
cone
is called the Clarke normal cone to Y at y.
As a trivial example of the above concepts, we consider the exterior hydrodynamics
control problem in [13, Section 3.2], where H = H{Q) x R and Y = H(C1) + {lT}. In
this case, we have for any y e Y,
{lT}
and
NY(y)={0}xR.
3.5 (Variation). Let J b e a metric space, E a linear topological space,
and let g: Jt —• E be an arbitrary mapping, defined in a subset D(g) of Ji. Given a
point U £ D(g), a vector S e E is called a variation of g at U if and only if there
exists a sequence {Xk} of positive numbers with Xk->0 and a sequence {C/k}
such that
DEFINITION
l / ) ^
(3.3)
and
jimXkl(g(Uk)-g(U)) = E.
(3.4)
The set of variations of g at U will be denoted by dg(U).
A special kind of variation is a directional derivative: S e £ is a directional derivative
if and only if there exists S > 0 and a function U(-): [0, (5] ->D(g) with
d(l/(A), 17) ^ A
(3.5)
and
lim X-1(g(U(X))-g(U))
= E.
(3.6)
218
H. Fattorini and S. S. Sritharan
The set of directional derivatives of g at U will be denoted by dog(U).
Finally, if {Zn} is a sequence of subsets of a metric space Ji, then Kuratowski 's
lim inf is denned as the set
lim inf Z-~ \z e Ji;3 {z.}, zn e Zn such that z = lim zn>.
n->
QO
I
n—* oo
I
Our proof of the maximum principle will be based on the following theorem of
Kuhn-Tucker type. The assumptions are the following, where (£m denotes the minimum of (3.1) subject to (3.2).
(a) The metric space Ji is complete.
(b) The function f:Ji^H
(respectively fc:Ji->R)
is denned in D{f)<=,Ji
(respectively D(/ c ) s Ji) with D(f)nD(fc) # 0 and the real-valued functions
&(U,v)=
V
and
fmax (0, fc(U) -6)
l+oo
if U e D(fc),
iIUtD(fe),
are lower semicontinuous for any y e Y and 6 ^ ^ m — e for some e > 0.
(c) The target set Y is closed.
3.6. Let U be a solution of (3.1)—(3.2). Then there exists a sequence
{&„} c / ? + , <>„—>0, a sequence {Un} cz Ji, a sequence {y") c Y such that
THEOREM
?>n
(3.7)
and a sequence {(fin, zn)} c R x H satisfying
Hn^O,
||(^,zn)||i{XH=l,
such that, for every (S", 3") e conv do(fc,f)(U")
(3.8)
and every w" e KY(y") we have:
(3.9)
where conv denotes closed convex hull. Moroever, for every weak limit point (/z, z) of
the sequence {(/xn, zn)} a R x H, we have
H^O,
and for every (Ec, S) e lim inf^^ conv
zeNY(f(U))
(3.10)
do(fc,f)(U")
LiEc + (z,Z)H^O.
(3.11)
Finally, assume that there exists p > 0 and a compact set Qc H such that
0 {[n(conv3(/ c , /)(£/"))] -KY(y«)^B(0,p)+
Q}
(3.12)
1 = 1
contains an interior point in H, where Yl denotes the canonical projection from R x H
into H. Then (n,z)^ 0.
Optimal controls in viscous flow problems
219
Proof. Given an arbitrary sequence {en} of positive numbers tending to zero, we
define for each n the real-valued function fSn by
9H(U, y) = {Pe(U, <$m - enf + &{U,yf^
(3.13)
in the space M x Y endowed with the product distance
,y),(U',y'))
=
d(U,U')+\\y-y'\\H,
which is a complete metric space. The function rSn is lower semicontinuous and
positive in Ji x Y and
%n(U,f(U)) = snSM<#n
+ sn .
(3.14)
Thus, by the Ekeland variational principle [ 7 ] there exists (U",y")e J? x Y such
that
en,
(3.15)
et
(3-16)
n
and V (V,y) ^ (U , y") in J/ x Y,
t V , U")+\\y-y»\\B}.
(3.17)
Let w" e KY(y"), so that there exists a sequence {Xnk}cR + with Xnk->0 and a
sequence {wnk} in H with wnk->w" andy" + lnkwnk e Y(see Definition 3.2). In (3.17),
we set y = y" + lnkwnk and V = \J\, where {[/£} is the sequence used in the Definition
(3.3)-(3.4) of variation. Thus, using the expression (3.13) for ^ ( - , •), we obtain
{[max (0,fe(Ul) ~%m + e j ] 2 + \\f(U"k)~yn
\\f(Un)~yn\\2H}i-et{d(U"k,U")
n
-
+ lnk\\Wnk\\H}.
n
n
Note that from (3.3) we have d(U k, U ) ^ Xnk. Now let (E c, a") e
Dividing by Xnk and letting k-> oo, we obtain (3.9) with
(3.18)
do(fc,f)(Un).
(^,zn) = (yn,xn)/\\(yn,xn)\\RxH,
(3.19)
(yn, xn) = {max (0,/ c (£/") - ^m + en),f(U»)-y"}.
(3.20)
Obviously (3.9) extends to conv do(fc,f)(U"). Inequality (3.11) is obtained from (3.9)
setting w" = 0 and taking limits as n —• oo. We note here that in this limiting argument
we have used the fact that zn—>z weakly and H"—>3 strongly implies
(zn, E")H->(z, E)H. The first condition (3.10) is obvious, thus it only remains to prove
the second, namely that z eNY(f(U)). Let we CY(f{U)) and let {Xn}cR+ be such
that
lj\\(7n, xJW^O asn^oo.
(3.21)
Since \\(yn, xn)\\ = ^n(U",y") ^ e n ->0, we must have Xn^>-0 as well, thus by Definition
3.3 of C y (/((/)) we can pick a sequence {wn} in H such that w"->w strongly and
y" + lnw" e Y. We then have from (3.17) that
220
H. Fattorini and S. S. Sritharan
That is,
{||(yn, xn) ||2 - 2Xn(f(U") - y \ w»)H + A2 || w" | | 2 } *
= II (y n ,*JII{l - 2 ( ^ / | | (yn,xn)\\)(zn, "")„ + % II w" 117II (?„,*,,) II2}*
= I (y». x») I {1 - a . / I (?», *„) I )(z«. W")H + o(XJ
^ ^ ( l / - , /•) - e* {XJ ||fa,,x j ||} || (yn, xn) || || w" ||.
Subtracting || (?„,*„) II (which is equal to <gn(U",y")) from both sides, dividing by
an
I.*II)II
d letting n->co, we deduce that
Since w e C y (/(t/)) is arbitrary, we deduce from Definition 3.4 that z e Afy(/(l/)). As
before, in this limiting argument we have used the fact that zn->z weakly and w" -»w
strongly implies (zn, w")H->(z, w)H.
It only remains to show that (3.12) guarantees that (/x, z ) ^ 0 . This will be a
consequence of the following result:
3.7. Let E be a Hilbert space. Let {An} be a sequence of sets in E and P cz E
a compact set such that the set
LEMMA
A= f\{com(An)
+ P}
(3.22)
contains an interior point. Let {zn} be a sequence such that
l|z.ll£ = l
(3-23)
and
(zn,y)E^en^0,
VjeAn.
(3.24)
Then every weakly convergent subsequence of {zn} has a nonzero limit.
Proof. Let {zn} be a subsequence (denoted by the same symbol) convergent to zero.
Let x be an interior point of A and let B(x, S), S > 0 be a ball contained in A. For
each n there exists yn e conv (AJ and pneP with
x+
8zn=yn+pn.
By compactness, we may assume that {/>„} is strongly convergent. Since (3.24)
extends from An to conv (An), we have
(zn,x)E = (zn,yn)E-3
||z n ||1 + (zn,pn)E^sn-S
+ (zn,pn)E.
Since zn is weakly convergent to zero and pn is strongly convergent, we should have
il m J ( z n./'n)£ = 0We thus conclude that
\im(zn,x)E^
-5.
This contradicts the fact that {zn} is weakly convergent to zero.
•
Completion of the proof of Theorem 3.6. Assume that (3.12) contains a ball B(x, 3).
Optimal controls in viscous flow problems
221
Set £ = R x H, P = [0, 1] x Q, and
If (3.12) contains an interior point in H, it is then evident that the intersection (3.22)
will contain an interior point in E as well. We apply Lemma 3.7 to the sequence
{ — (nn,zn)}. The second condition in (3.8) implies (3.23); as for (3.24) it is (3.9).
Remark 3.8. We note here that in the above arguments concerning the condition
(3.12), the compact set Q does not seem to play any role. In fact this set can be
omitted in condition (3.12) if the remaining set
AR =
£ \ { t n ( c o n v dW<> /)(U"M - KY(y")nB(0, p)}
contains an interior point. However, there are special cases such as the exterior
hydrodynamics problem described in [13] where the target set is in the form F =
X x {lT} with X<= Ha hyperplane (subspace of codimension one) and {lT} is a point
in the one-dimensional linear submanifold. In such cases, the contingent cone of Y
cannot contain an interior point due to the fact that its intersection with this linear
submanifold is a point. It is precisely this type of case where AR can be compensated
by adding a compact set Q so that the sum will contain an interior point. Here the
intersection of Q with such a finite dimensional linear manifold can simply be a ball.
The linearised equation
We study in this section the linear initial value problem
z^t^T,
Z(T) = Z 0
(3.25)
(3.26)
under Hypothesis 2.1 on «$/ and the following assumption on L(t):
HYPOTHESIS 3.9. For each t,x^t^T,
L(t) is a linear bounded operator from Hx into
H-0. The function t —>L(t) e S^(Ha; H*_ p) is continuous (in the uniform operator topology of operators).
Hypothesis 3.9 is equivalent to requiring that for each t e [T, T], stf ~pL(t)srf~* be a
linear bounded operator in H and that the function t^>stf~l3L(t)3?~a e J5?(/7; H) be
continuous in the uniform operator topology. The results on (3.25)-(3.26) will be
applied to the operator L(t) = [_DyjV](y(t, x; £; C/)), where y(t, z; (; U) is a trajectory
of the nonlinear system (2.1); here we use Hypothesis 2.2 and take a = \. We shall
also treat the adjoint 'final value problem'
dsz{s)-(^ + L(s)*)z(s) = h(s), O^s^T,
z(T) = z0.
x
l)
(3.27)
(3.28)
ll
Hypothesis 3.9 implies that stf~ L(s)*stf~ = (s/~ L(s)£/~°")* is a continuous function in the uniform topology of operators, so that L(s)* also satisfies Hypothesis 3.9
with a and )3 switched. We only treat the case a == \ below, since the case a. < \ is
simpler; the necessary modifications will be pointed out later.
In the computation of the variations, we only need to consider z0 = 0 in
(3.25)-(3.26); however, in the adjoint initial value problem we have no information
222
H. Fattorini and S. S. Sritharan
on the final condition z 0 , thus we have to consider the general case £ e H. To unify
the two cases, we shall only assume that z 0 e H in (3.26).
3.10. Let A(-) e L 2 (T, T; H) be given. Then the system (3.25)-(3.26), or, rather,
the integral equation
LEMMA
ft
z(t) = S{t-x)z0-
rt
s/l)S(t-a)sf-l)L(<j)z((T)d(T+
S(t - a)h(a) da
Jz
(3.29)
JZ
possesses a unique solution z(-) in x S t t== T. This solution has the following properties:
(I) If the initial data z0 e Ha, then z(-) e C([T, T]; Ha) with
^},
T<t^T.
(3.30)
( I I ) Let z 0 e H, then:
(a) z(-) is an Hx-valued
continuous
function
in x <t^T
satisfying
||z(t)L^C 2 {(t-T)-||z 0 || H +||A|| L 2 ( l i T ; H ) }, x<t^T.
(3.31)
Moreover,
<? (~* / II
(c)
II
_i_ II A II
\
^-" t <*" T
(*X 'Xr)\
z(-)eL2(x,T;Hx),
Proof. Proof of case (I) is similar to that for the nonlinear problem. We use the
analogue of the integral equation ([13, (23)]) with w(t) = s/"z(t),
fx + l!S(t — a)stf~"L(a)stf~ V(«r) da
S(t-a)h(a)da.
(3.34)
Existence of local solutions of (3.34) is proved in exactly the same way as for
[13,(23)]. To show that the solutions actually exist (and are unique) in the whole
interval x ^ t ^ T, we note, using Holder's inequality if a <\ and Lemma 2.4 (II) if
a = 5 in the integral equation (3.34), that
\t-a)-*-i>\\z{a)\\*do.
C\ + / ! is similar to the constant in [13, equation (24)], K a bound for \\si/~pL(a)s/^i \\
in the interval x ^t^T.
Applying [13, Proposition 7] we obtain an a priori bound
of the form [13, (25)] in arbitrary intervals z^t^T',
T > x, thus the solution can
be extended by [13, Lemma 3].
Let us now consider case (II). We again use the integral equation (3.34). This
time, the space is C^Qt, T~\;H), consisting of all functions w(-) continuous in
223
Optimal controls in viscous flow problems
T < t S T' and such that
which is a Banach space under | • Wc^tXim- We consider (3.34) in the ball
Note that if w(-) e fi(l) then | w(r) ||H ^ C(r - T)~*, SO that
where we have made use of the beta integral formula
This estimate implies that, for T' — z sufficiently small, the operator M maps the ball
B{\) into itself. Moreover, if necessary reducing further T — x, M i s a contraction
map in 2?(1) so that existence of a solution w(-) follows from the contraction mapping
principle. The solution z(t) of (3.29) therefore belongs to H± for t>x, thus we can
use an a priori bound like [13, (25)] to extend the solution to T < t ^ T. The bound
follows from
|z(t)II* ^ C ( t -
(t-
da,
[13, Proposition 7] and the beta formula (3.35).
To prove (lib), we note first that using the beta integral (3.35) and the estimate
(3.31) we get
1
l l ^ ) l l ^ ^ C ( ^ ' l | l l
C
(
)
1
Using this result (for S = \ + ft) and Lemma 2.4, we estimate (3.29) as
II Z(') IILVT;*,) S "7= II Z0 \\H + C \\h \\L^,T,H)
v2
224
H. Fattorini and S. S. Sritharan
Similarly we estimate (3.29) to get
1
\\\\
f'
\\h\\
( t > t ; H ) }.
( t - a ) " ' | | Z(«T) 11**7
D
It is convenient to express solutions of (3.25)-(3.26) by means of the solution
operator S?(t, s), denned in 0 ^ s < t ^ T by
T,
(3.36)
z(t) the solution of the homogeneous equation (40) (h = 0) in s ^ t ^ T with initial
condition
z(s) = zoeH.
(3.37)
It follows from the considerations above that £f{t,s) is the only solution of the
integral equation
(3.38)
LEMMA 3.11. (a) The solution operator £f(t, s) belongs to ^{H; H) and to ££{H±, H±)
for each sf^t.
(b) The function (t, s)-*S?(t, s) is strongly continuous and bounded in T ^ s ^ t ^ T
as an ££(H; H)-valued function and also as anj£{flx.,tl^)-valued function.
(c) We have
srK
(d) If zoeH,S?(-,
0^s<t^T.
(3.39)
s)z0 e L2(s, T; H±) with
.
(3.40)
Proof, (a) follows from (3.30) and (3.32), (c) from (3.31) and (d) from (3.33), T
replaced by s in all three estimates. The continuity assertions in (b) are a rather
obvious consequence of the bounds on ^ ^ ( t , s), the integral equation (3.38) and
the equation obtained multiplying on the left by &/*. We omit the details.
•
The solution of the initial value problem (3.25)-(3.26) can be written in the form
)h{o) da.
(3.41)
This can be seen by combining the integral equation (3.29) (multiplied by si* on
the left) for the solution z(t) of the inhomogeneous problem with the integral equation
(3.38) for Sf{t,s).
Equation (3.41) can be used to derive another integral equation for £f(t,s). In
fact, noting that z(t) = S(t — s)z0, z0 e H± is a solution of the initial value problem
dtz(t) + (^ + Ut))z(t) = Ut)z{t),
z(s) = z0,
s^t^T,
Optimal controls in viscous flow problems
225
we deduce that
which implies the integral equation
(3.42)
The theory corresponding to the case a < i is simpler, since we can estimate using
Holder's inequality in the last integral. We omit the details.
As shown below, the solution operator £f{t, s) of the equation (3.25) depends
continuously on L{-).
LEMMA 3.12. Let £f(t, s) be the solution operator of the equation
t) = O,
(3.43)
where the operator function L\-) satisfies Hypothesis 3.9. Let M be a bound for
\ \ \ inx-^tST.
Then we have
\\s/iy(t,s)-s/i£f(t,s)\\^C'M(t-sy11,
Proof. Combining the integral equation (3.38) for j^i£^'(t,
equation for &(t, s), we obtain
x^s^t^T.
(3.44)
s) with the corresponding
Estimating,
^ t ,
s) -
c
x
(t - r ) - * - " || ^y{r, s) - st±&{r, s) || dr,
where C is the constant in (3.39) and K is a bound for || j ^ " " L ( 0 ^ ~ * I I in T ^ t ^ T.
Accordingly, (3.44) follows from [13, Proposition 7 ] .
The theory of the backwards adjoint initial value problem (3.27)-(3.28) runs along
similar lines. The associated integral equation is
+
^"S(r-s)h{r)dr.
(3.45)
Js
The solution operator Sf*(s, t), s <t which solves the homogeneous equation (3.27)
226
H. Fattorini and S. S. Sritharan
in T ^ s ^ t with the final condition Sf*(t, t% = £ is a solution of the integral equation
r,t)dr.
(3.46)
Taking adjoints in the integral equation (3.42), we deduce that £f(t, s)* and Sf*(s, t)
satisfy the same integral equation, thus
Sf{t, sf = 9"*{s, t),
0^s<t^T.
(3.47)
To deduce certain of the properties of £f*{s, t) we may use (3.45) and properties of
Hf(t, s), but it is more effective to operate directly with (3.44). We obtain in this way
the following result:
3.13. (a) £f*{s, t) belongs to £\H; H) and to ^(H^; Hp)for each s^t.
(b) The function (t, s)—*Sf*(s,t) is strongly continuous and bounded
inxf^s^tf^
as an JP(//; H)-valued function and also as an £C(Hp; Hp)-valued function.
(c) We have
LEMMA
C(t-s)-fl, T^s<t^T.
(3.48)
The results on the equations (3.25) and (3.26) will be applied to the case L(t) =
[DyjV]{t,y(t, t; £; U), that is, to the equation
dtz(t) + (^ + lDyJ>1(y(t, T; C U)))z(t) = £X{t),
(3.49)
z(x) = z0.
(3.50)
2
Let (C, U)eHix
L (T, T; H) be such that the trajectory y(-, T; f, U) (that is, the
solution of (2.1)-(2.2)) exists in i ^ t ^ T. We noted in [13, Lemma 4] that there
exists a ball B((£, U); p), p>0inHix
L 2 (T, T; H) where y(-, T; f, U) exists in x ^ t ^ T
2
as well, so that the map from H± x L (T, T; //) into C(T, T; H±) given by,
(3.51)
<D(C,(7)=J(-,T;C;1/)
is well defined and (again due to [13, Lemma 4]) Lipschitz continuous.
3.14. The map * : H± x L 2 (t, T; //) -* C([T, T]; J ^ ) is Frec/iet differentiable
in B((t U); p) and its Frechet derivative [D<D](£ U) is
THEOREM
[D<D(C, C/)](z0, X)(-) = z(-) e C([T, T]; 7/^),
w/iere z(-) is the solution o/(3.49)-(3.50), that is
z{t) = Sf{t, v, C U)z0 + I 9>{t, a; £ U)®X(o) da,
with <f{t, a;C; U) the solution operator o/(3.25)-(3.26).
Proof. Define
w{t) = s/*{y(t, T; f + z0; U + X)-y(t,
x; f; C7)-z(t)}.
This function satisfies the integral equation
w{t)={c, v, C; U)) - \PyJT\{y{a, i; C U))z(a)} da.
(3.52)
Optimal controls in viscous flow problems
227
Differentiability of JT(y) with respect to y implies that
p(y,y') = s/-i>{jr{y) _ jrfoy) - \_Dyjr\(y)(y> -y)}
satisfies
where S (y, y') -> 0 as y' ->• y. On the other hand, continuity of the differential [_DyjV"\ (y)
and the mean value theorem imply that d(y,y') is uniformly bounded if | | j ' — y\\±
sufficiently small. We rewrite the terms inside the curly bracket of the integrand of
(3.52) in the form
[Dyjr\{y{p, T; C; U))s/-*w(t) + piy(°, T; C, U),y((T, T; £ + z0; t/ + X)).
Using the above observations, [13, Lemma 4] and the dominated convergence
theorem, we obtain
|| w{t) \\H<o( || z 0 1| + + || X \\L^,T;F)) +
which via [13, Proposition 7] ends the proof.
(t - a)-*-"
|| w(a) \\H da
•
Computation of variations
We shall apply in this subsection the results of the first subsection, in particular
Theorem 3.6, to the functions
;CU),
= V(T,x;C;U),
(3.53)
(3.54)
where y(-, T;£; U) is the solution of (2.1)-(2.2). We assume that the conditions of
the Existence Theorem [13, Theorem 5] are satisfied, so that a solution U(-) of the
optimal control problem exists. In order to obtain a maximum principle, we shall
work not in the full space Jiad(x, T; U) of admissible controls but in a subspace Ji =
Jf(U,S, K) depending on the optimal control U and two parameters 5, K > 0 denned
below. This subspace is defined as follows. First, we introduce the Ekeland distance
dE in Jtai(x, T; U) by
dE(U('),V(-)) = mease(U,V),
(3.55)
where e(U, V)= {te [T, T]; U(t) # V(t)}. This distance can also be introduced in
Jl{x, T; U) (the space of strongly measurable fZ-valued functions), which becomes a
complete metric space [7] (no assumptions on U are needed). On the other hand,
•^ad(t, 1) U), although also a metric space under dE, may not be complete if U is
unbounded; in fact, the delimit of a sequence {[/"} <= Jiai(x, T; U) may fail to be in
L2(T,T;F).
(For instance, take U=R, Un(t) = 0 in x^t^r
+ 1/n and U"(t) =
l/(t - T) in T + 1/n S t ^ T.) The space M = Jf(t), d, k) is the subspace of Jtai{%, T; U)
consisting of all U(-) that satisfy
(a) dE{U,U)^5,
(b) \\U(t)\\F^K,
t a.e. in e(U, U).
228
H. Fattorini and S. S. Sritharan
Clearly, if U(-) e JC{U, 8, k), we have
II U{-) - U(-) \\L2(Z,T;F) ^ ^K.
(3.56)
By definition of optimal control, the trajectory y(t, x; £; U) exists in the whole interval
xi^tf^T. Accordingly, if we choose 8*K fLp, p the parameter in [13, Lemma 4], we
are guaranteed a global trajectory y(t, x; f; U) for every U e Jl{V, 8, k). We shall
assume that 8 is chosen in this way. On the other hand, no restrictions are placed
on K.
LEMMA 3.15. The space Jt{U, 8, k) is complete under the Ekeland distance (3.55).
Proof. Let {[/„(•)} be a Cauchy sequence in M{(j,8,k). Then {(/„(•)} is a Cauchy
sequence in Jt(x, T, U), thus by completeness there exists U(-) e Ji{x, T; U) with
dE(Un, [/)->0. We show below that U(-)e Jt$, 8, k). Selecting a subsequence, we
may assume that dE(Un,U)^2~".
Let
e n=
Then e1^e2^
\j'e(Uk,V).
• •• and meas (ek) ^ 2 1 "". Hence
OO
e = f~\ en
n=l
has measure zero. Now, if t $ e, then there exists n (depending on i) such that U(t) =
Uk(t) for k^n so that if U(t)¥=U(t) we have Uk(t)^U(t)
for k^n, hence
I Uk(t) \\F S K and a fortiori || U(t) \\F ^K.
•
Given the optimal control U(-), we define the set of variations (more precisely,
directional derivatives) S(T, x; £; s; U; V) e df(U), where the parameter s will belong
to a set e of full measure in [x, T ] to be determined later. These variations are
defined taking C/(A)(f) = C/s,i,K(t) in (3.5)—(3.6), where UStXtV{t) is the spike variation
of the control [/(•) denned by
V
if s - A < t S s,
U(t)
elsewhere,
where V is an element of the control set U. Obviously, dF(U, USyXtV) ^ A. If we set
U = U, A S 8 and take V as an element of the control set with || V j| ^ K, then we get
Let [/(•) e L J (t, T; F). A point s, T < s ^ T is a left Lebesgue point of [/(•) if and
only if
limA" 1
||l/(<x)-l7(s)|| F d<7-+0.
(3.57)
^-o J S _ A
The set e of the left Lebesgue points of an arbitrary function U(-)e Ll{x, T; F) has
full measure in T ^ t ^ T [17].
L E M M A 3.16. Let e be the set of Lebesgue
if see and t> s ,
Z(t,x;£s;U;V)=
points of the control [/(•) inx < s^T.
l i m A " 1 ^ , x;£ Us^v(t))-y(t,
A0
+
x;£ l/(t))]
Then,
(3.58)
Optimal controls in viscous flow problems
229
exists in the norm of H^. Convergence is uniform in the interval s + e g t ^ T for any
e > 0. The function S(t, z; £ s; U; V) is given by
E(t,z;£s;U;V)
.„ yt,s;U)@(V-U(s))
= \ v'
'
[0
ift>s,
~
for t < s,
where £f(t, s; U) is the solution operator of the linearised equation
8tz(t) + (^ + Dy^{y{t,
z; £ U))z(t) = 0.
(3.59)
Proof. The function S(-, z; £ s; U; V) satisfies (formally) the initial value problem
d,R(t, z; £ s; U; V)+{s/ + DyJ^(y(t, z; £ U))}E(t, z; £ s; U; V)
= 5D(t-s)@(V-U(s)),
z^t^T,
where SD(-) is the Dirac delta. Rigorously, this means that the function
*F(-, T; £ s; U; V) = s/*E(-, z; £ s; U; V) satisfies the integral equation
V(t,z;£s;U;V)
t, z; £ U))£/~ixP(r, z; £ s; U; V)} dr
+ H(t - s)j/*S(t - s)£(V- U(s)),
(3.60)
where H(t) is the Heaviside function. In the following lines we estimate
r
1
{y(t, v, £ Us,x,v)-y(t, v, C; U)} - S(t, v, (; s; U; V)
in the Hi-norm. The estimations will be*for
<D(t, T; £ s; X; U; V) = k^{t,{t, T; £ U,^,) - r,{t, v, £ U)} - V(t, T; £ s; U; V),
where tf(t, r, £ U) — s/*y(t, T; £ U). Since only t, s and X vary in this argument, we
write d>(t, T; £ s; X; U; V) — Q>(t, s; X) to lighten the notation. Using the integral equation (2.3) (or rather [13, equation (23)]) for y(t, T; £ Us^v) andy(t, z; £ U), we obtain
the following integral equation for O(t, s; X):
^r, z; £ Utti,v)) -s/-*Jf(j*-±tfc,
v, £ 17))]
r, z; £ C/)K"*T(r, z; £ s; U; V)} dr
(3.61)
Writing
X-1(^~"^(^~ii(r,r,£
U,,x,v))-s/->^-(st-*ti(r,r,C;
' * i i i r , z; £ U))^~^(r,
z; £ U))s/-*^(r,
U))}
z; £ s; U; V)
s; X) + X^p^r,
z; £ U), t,(r, z; £ [/,,,,„)),
230
H. Fattorini and S. S. Sritharan
where
p(ti, q) = st-vjfistf-^ij)
- stf-fJ^istf-^ii) - ^-^DyJT{^-^ii)sil-^{ti - tf),
we obtain
s/i+llS(t
<J>(t, s; A) = -
-
- r)DyJf{^--ti{r,
v, f; U))^-±<t>(r; s; X) dr
\ s/*+»S(t - r)X^p(t,(r, v, £ U), t,(r, t; f; t/ M ,J) dr
Taking norms,
(t-r)---ll\\a>(r,s;l)\\Hdr+\\S1(t,s,X)\\H+\\S2(t,s,X)\\H,
||*(t,s;A)||H^c|
(3.62)
where <51(t, s, A) (respectively 52(t, s, X)) is the second integral on the right-hand side
of (3.62) (respectively the combination of the third integral and the nonintegral term).
In the result below, s e [T, T] and {fi(t, s, X); 0 < X S <5} is a family of non-negative
functions in I}{z, T) with [i(t, s, X) = 0 for t^s- X, and {v(t, s, A); 0 < A § (5} is the
family whose elements are denned by
v(t,s,X)=
{t-r)~"fi{r,s,X)dr,
a < 1.
(3.63)
JT
Obviously, we also have v(t, s, X) = 0 for t ^ s — A. Consider the following possible
properties of {fx(t, s, A)}:
(a) {ju(-, s, A); 0 < A ^ <5} has equicontinuous integrals in [T, T ] .
(b) For any e > 0, {fi(- ,s,X);O<X^d}
is uniformly bounded in [s + e, T].
(c) /j(t, s, A) -»0 as A ->•(), a.e. in
s-^t^T.
(d) For any e > 0, v(t, s, A) -> 0 uniformly in s + e 5£ t ^ T.
3.17. (I) Assume that {ju(-, s, A)} satisfies (a). T/iew {v(-, s, A)} satisfies (a).
(II) Assume that {jx(-, s, A)} satisfies (b). 77ie« {v(-, s, A)} satisfies (b).
(III) Assume that {fi(-, s, A)} satisfies (a), (b) and (c). 77ien {v(-, s, A)} satisfies (a)
am/ (d).
LEMMA
Proo/. (I) Let p > 0. Pick any measurable set e ^ [T, T] with meas (e) ^ p. We have
{t-ryan{r,s,X)drdt=
v(t,s,X)dt=
Je Jt
fi{r,s,X)dr
JT
(t-r)-xdt.
Jen(r,7)
A very rough, but sufficient, estimate is obtained, assuming for each r the most
unfavourable case e = (r,r + p); we obtain
v(t,s, X)dt^Cpi-a
n(r,s,X)dr,
where the integral on the right-hand side remains bounded by equicontinuity of
Optimal controls in viscous flow problems
231
{ju(-, s, A)}. The proof of (II) is obvious. As for (III), assume that v(t, s, A) does not
tend to zero uniformly in the interval s + e^t^T.
Then there exists a sequence
{An}, An->0 and a sequence {tn} <= [s + e, T] (which we may assume convergent to
t e [s + e, T]) such that
(tH-r)-'n(r,s,kn)dr
(3.64)
does not tend to zero. Since the sequence {(tn — •) a; n = 1, 2 , . . . } has equicontinuous
integrals, (b) applied in the interval [s + e/2, T] guarantees that the sequence
{(fn — ')~"K', s, K)) has equicontinuous integrals in [s + e/2, T]. On the other hand,
since {(tn — •)""; n = 1, 2 , . . . } is bounded in [T, S + e/2], (a) implies that the sequence
{(fn — ')~"K' > s> ^«)} w m n a v e equicontinuous integrals in [T, S + e/2]. This and (c),
via the Vitali convergence theorem ([17, Chapter VI.3, Theorem 2]), allows us to
pass to the limit under the integral (3.63), which contradicts the fact that it does not
tend to zero. We note finally that (b) is a consequence of (c). •
3.17. (Ill) will be applied to (3.62) as follows. Applying [13, Proposition 7]
to (3.62) produces the estimate
LEMMA
|| <D(t,s, A) | | H g / i ( M , A)+ C I (t-ryi-i>Li(r,s,X)dr,
(3.65)
with n(t, s, A) = || d^t, s, A) ||H + || 52(t, s, A) || H , thus if we show that || S^t, s, A) || and
|| S2(t, s, A) || satisfy (a), (b) and (c) then <S>(t, s, A) will satisfy (d), which is in excess of
what we set out to prove. The other parts of Lemma 3.17 will be used in intermediate
results, and conclusion (a) on equicontinuity of the integrals will be used later.
We proceed to establish (a), (b), -(c) for the families {||51(-, s, A)||H} and
{||<52(-,s, A)|| H }. Write 52(t, s, A) = S21(t, s, X) + 522(t, s, A), where d21 is the integral
term. Condition (a) is obvious for b22 since
S22(t,s, A) = 0,
fort<s,
\\S22(t,s,l)\\HSC(t-TrK
for t>s.
To check (a) for S2l(-, s, A), let e be a measurable set in the interval s —
For any V(-)e L 2 (T, T; F), we have
k^t^T.
(t,s)
\\s/*S(t-r)aV{r)\\Bdrdt
e
Js-i.
I
\\s/*S(t-r)aV(r)\\dtdr
s~). Jer,(r,T)
(r
(meas(e))*^
s-X
Uen(i-.r)
u
\\stf*S(t - r)@V{r)\\2 dt\ dr,
J
where we have used in the last step the Cauchy-Schwarz inequality applied to the
functions 1, || st*S{- - r)®V{r) \\. Using Lemma 2.4 (I) and the fact that ® e £C(F; H),
|| S2l(t, s, A) \\dt ^ 4 = ( m e a s
V2
232
H. Fattorini and S. S. Sritharan
where the hypothesis that s is a left Lebesgue point of V(r) guarantees boundedness
of the last integral, which shows the equicontinuity condition (a). Finally, to show
(c) for 32(-, s, 2), we write
d2l(t,s,Z) = jf*S{t-s)A-1
fs S{s-r)<M{V-U(r))dr
(3.66)
Js-X
and use the fact that s is a left Lebesgue point of U(r) (thus of V(r) =V— U(r)).
We now turn to 3^-, s, X). We have
Wd.it, s,X)^C
P (t - r)-±-n~l
\\p(i,{r, T; £ U), t,{r, x; £ C/M,J)|| dr.
(3.67)
Since the family {t/(-, T;£; US^V)} is uniformly bounded, we can apply the local
Lipschitz condition and the local boundedness of the Frechet derivative postulated
in Hypothesis 2.2 to deduce that
x^t^T.
(3.68)
Note that the local Lipschitz continuity property shown in [13, Lemma 4] does not
help^ d i r e c t l y h e r e , s i n c e ||ti(t, v, C; U.iltV) -t,(t,
z; f; U)\\£C||
U.,liV - UWL^.T-.F)
=
O(A*); we have to produce an ad hoc estimate.
We obtain from the integral equation (2.3) (or rather [13, equation (23)]) that
=C
By [13, Proposition 7],
I {t-rr*-f\\52l(r,s,k)\\dr.
We have already show that {||<521(-, s, A)||} satisfies (a) and (b), thus an application of Lemma 3.17 (I) and (II) implies that the family
{2- 1 ||, / (-,T;6t/ s , /l ,J->7(-,T;C;t/)||} satisfies (a) and (b), thus by (3.68),
{\\p{tI-,x;C;U),ij(-,x;C;UsAyV))\\} satisfies (a) and (b) as well. The fact that
{/l~1|l7(-, T;£; USIX.V) — i(' ,*',& U)\\} satisfies (b) implies in particular that
|| 7(-, T; C; Us,x,v) —1(' > T; C', U) \\ = 0(1) as X -»0 for t ^ s + e, thus an application of
the definition of the Frechet derivative in the definition of p shows that
\\p(t,(t,x;C,U),fi(t,x;C;Us^v))\\^O
as A^O
in t ^ s + e for arbitrary e, thus a.e. in T ^ t ^ T. We then apply Lemma 3.7 (III) to
Optimal controls in viscous flow problems
233
(3.67) and deduce that {|| S^-, s, A) ||} satisfies (a) and (d). With this information on
hand and keeping in mind the corresponding properties of {||<52(-, s, X)\\} we go
back to (3.62), apply for the last time Lemma 3.17 (III) and thus end the proof of
Lemma 3.16.
•
We note that we have proved on the way the following result:
COROLLARY
3.18. The family
has equicontinuous integrals in [x, T].
This will be used below.
LEMMA 3.19. Let s be a left Lebesgue point of the control U(-) in x ^ s ^ T and also
a Lebesgue point ofY(-, £/(•))• Then
3 c (r, T; £ s; U, V) = ^lim+ A " \ ^ ( t , T, £ t/ M>r ) - <tf(r, T, £ I/))
(3.69)
exists for t> s. Convergence is uniform in the interval s + e ^ t ^ T/or any e > 0. The
function Ec(t, x; & s; (7, F) equals
Ec(t, T; £ s; U,V)=
<[D,«](ff,^(ff, T; £ I/)), H(<r, t; £ s; C7, F ) > H _ i X H i rfa
+ ([Ptpo](y(T, x; & U)), ST(T, s; f; U)@{V- U(s))H
+ Y(s,V)-Y(s,U(s)).
(3.70)
Proof. We have
', T; £ [/SiAiJ) - ^ ( ^ ( ^ ^; £ I/))}
j(ff, T; £ UStX,v)) - ®(ff, j(ff, T; £ I/))} dr
{Y(r,F)-Y(r,L/(r))}rfr.
(3.71)
Since s is a left Lebesgue point of Y(- ,[/(•)), the second integral has the limit
Y(s, V) — Y(s, [7(s)). The integrand of the first integral converges to
, ( f f , ^ ( f f , T; £ L/)), B(ff> t; £ t/, F ) > H _ i X ^ ,
as A —• 0 uniformly for r^.s + s,e any positive number. On the other hand, using the
mean value theorem, the local boundedness of [ O y 0 ] and the fact that
{ti(a, x; C; USiXiV)} is uniformly bounded, we deduce that
A- 1 ||e(r,.F(r,T;£ Us^v))-0(r,y(r,x;C,
U))\\ ^ CX^\\t,{r, T ; £ U.ti,v)-tl(r, i ; £ C/)||.
1
We know from Corollary 3.18 that {A" 1| tj(r, x; £ t/SiAjl)) - »/(r, T; £ C7) ||} has equicontinuous integrals in x ^ r ^ T, thus we can proceed in the same way as in the proof
of Lemma 3.17 (III). If our claim on the convergence of (3.70) is not true, then we
234
H. Fattorini and S. S. Sritharan
can select a sequence {AB}, An->0 and a sequence {£„} <= [s, s + e] such that
T, f; I/..ini0) - ntn, T, C; t/)) - He(t, t; f; s; U, V)
does not converge to zero, and we obtain a contradiction on the basis of Vitali's
Theorem as we did for the integral (3.63). •
In order to interpret the maximum principle, we must identify elements of
lim supn_»0O d(fc,/)([/"), where {V} is a sequence of controls that converge to the
optimal control U in the metric of the space J((U, S, K). The result below does this
by proving a sort of continuity of the variations Sc(-), S(-) with respect to U, for
which Lemma 3.20 below is an auxiliary result.
3.20. Let U e L2(x,T,F) be such that the trajectory y(t,v,£U)
exists in
z^t^T,
and let {[/"(•)} be a sequence in L2{x,T;F) such that {U"}->U in
L 2 (T, T; F). Then:
(a) y(t, v,£ U") exists in xfLt^L T for sufficiently large n,
(b) (t — s)~l>\\sfi&'{t, s; U)-rf*Sf{t,s;
£/)||->0 in the uniform operator topology
of &{H; H), uniformly
inx^s^t^T.
LEMMA
Proof, (a) is a consequence of [13, Lemma 4 ] . It is also shown there that
Un)-+y{- ,x\£; U) in C([T, T\; H); taking advantage of the continuity of
DyjV(y) in the J5?(//i;//.^-topology, we deduce that
J(-,T;£
in the norm of JZ\H;H), uniformly in x^tf^T.
Lemma 3.12 for L(t) = DyJf(y{t, T; C, U)). D
The result then follows from
The result in Lemma 3.20 holds also in the space Ji(tJ, S, K). Firstly, (a) is insured
for all n (see (3.56) and following comments). Secondly, any U e Jf(U, S, K) satisfies
|| l/(t)|| ^ max (K, \\ U(t) ||),
{||^(-)-^(-)llL^,r;F,} 2 ^ [
{2max(K,\\U(t)\\)}2dt.
(3.72)
Je(U,U)
LEMMA
3.21. Let {[/"} be a sequence in Jl{U, 8,
K)
with
f dE(U", U)<oo.
(3.73)
n=l
Then there exists a set e — e({U"}) of full measure in T^S^T such that, for each
s e e, (3.58) and (3.69) hold for U=Un andU=U
and such that
Ec(t, T; C; S; U", V)^Sc(t,
r, £ s; U, V), s e e,
S(t,r,£s;U",V)-+S(t,T;C,s;tJ,V), see,
(3.74)
(3.75)
everywhere in t>s.
Proof. We begin by taking
00
QO
d= H U e(U",U).
m=ln=m
We have meas (d) = 0. If cn is the set of all left Legesgue points of both Un(-) and
Optimal controls in viscous flow problems
235
Y(-, [/"(•)), c the set of all Lebesgue points of £)(•) and Y(-, £/(•)) and we define
e = cn[ f) cn\\d,
\n=l
/
then e has full measure in x ^ t ^ T. If s 6 e, then s e c , s e ne n , so that (3.58) and
(3.69) hold for all U" and for U. On the other hand, s$d, hence s$e{U", U) for
n 2: n0 (n0 depending on t), so that l/"(s) = U(s) for n 2: n0. Using the definition (3.57)
of B(t, T; £ S; U, V) for [7 = U" and 1/ = t/ and Lemma 3.20, it is clear that (3.75)
holds. As for (3.74), it follows from the previous consideration, the bound (3.39) for
y(t,s;C, U") (which is easily seen to be uniform in n) and the dominated convergence
theorem. •
The maximum principle
We apply Theorem 3.6 to the problem:
minimise fc(U) = W{T, x; £ U),
(3.76)
(3.77)
subject to f(U) = y(T, x; £ U) e Y.
The basic spaces will be J( = Ji(U, 5, K) and H. By virtue of [13, Lemma 4], / i s
continuous in the norm of L2(x, T; F); it follows from (3.72) that it is also continuous
in the metric of Jf(U, 3, K), which is more than needed in assumption (b) f o r / i n
Theorem 3.6. We must check that J^(t7, 9) = max (0, fc(U) — 0) is lower semicontinuous for every 9 S: ^m — e, and for this it is enough to prove that/(C/) itself is lower
semicontinuous in J?(U, 8, K). TO do this, take a sequence {{/"} e Ji(U, d, K) with
V->& in J?(U,S,k). Then it follows from (3.72) that [7"-+f/ in L2(x, T;F). We
can then apply [13, Lemma 4] to ..deduce that j ( - , x, £;[/")-> j ( - , T, £ U) in
C([x, T]); ffj). This makes it possible to take limits in the first term and the first
integral in the definition of the cost functional # ( • , T, £ Un); in the second integral
we use the weak lower semicontinuity Hypothesis 2.8 (note that we are only using
here strong lower semicontinuity; the hypothesis is only used in full in the existence
theorem [13, Theorem 5]. Under the condition that the target set Y is closed,
Theorem 3.6 provides a sequence {(/"(•)} e Ji(V, 3, K) with U"->U and a multiplier
(H, z) e R x H such that the Kuhn-Tucker inequality
/iHc + ( z , 3 ) H ^ 0
(3.78)
holds for arbitrary elements (S c , S) e lin inf,,.^ conv do(fc, f)(U"). Leaving aside for
the moment the question of whether the multiplier is nontrivial, we apply (3.78) for
the elements (Ec{t,s; U, V),E(t,s; U, V)) of liminf^^ conv do(fc, f){Vn) computed
in the previous subsection. We obtain
tiU><pQ-\{y(T, x; t U)), ST{T, s; £ U)®(V- U(s)))H + ^{Y(s, V) - Y(s, t))}
+ n I dDy@-]{a,y((T, v, £ U)), ST{t, s; £ U)@{V- U(s))}H_, XHi da
+ (,y(T,s;C,U)@(VU(s)))H^0, see,
VeJi,
where 5^(t, s; £ U) is the solution operator of the linearised equation
dtw(t) + (^ + [Pyjn(y(t,
t; £ U)))w(t) = 0.
(3.79)
236
H. Fattorini and S. S. Sritharan
Define a function p(s) by
p(s) = ST(s, T; £ U)* {z + nlD<po](y(T, t; £ U))}
S?(s, a; £ t>)*[P,e](ff, J;(<T, i; £ U)) da.
(3.80)
Using the relation (3.47) among the operators of a liner equation and its adjoint,
we recognise p(s) as the solution of the final value problem
- dsp(s) + (&? + XPyJf~\(y(s, T; £ t))*)p(s) = ^[^ y ®](s, J>(s, T; £ [/)), T ^ s ^ T,
(3.81)
/»(T) = z + /i[D^ 0 ](ji(r, T; £ t/)).
(3.82)
We can write (3.79) in the form
nY(s, U) + (®*p(s), U(s)} = min {^Y(s, V) + (®*p(s), F>; Ve U; \\ V\\F S K},
(3.83)
for see, e the set in Lemma 3.20. When U is bounded, the restriction || V\\F^ K is
automatically satisfied for K large enough. We obtain
fiY(s, U) + (!M*p(s), U(s)y = min {nY(s, V) + (®*p(s), F>; VeU}.
(3.83a)
We examine the condition that guarantees that the vector (n, z) in the Kuhn-Tucker
inequality (3.11) is not zero, namely the fact that the intersection in (3.12) should
contain an interior point. If e is the set in Lemma 3.20 which corresponds to U",
then Il(conv 8(fc,f)(U")) contains all limits of convex combinations of elements of
the form
•,s;C;Un)@{V-Un(s)},
ift^s,
if t < s.
It is easy to see that these limits of convex combinations include all elements of the
form
n(t)=
y(t,<j;C;Un)<M{V((j)-U"(a)}d(T,
(3.84)
where V(-) is an arbitrary strongly measurable function such that V(a) e U. However,
the U of all elements of the form (3.24) will in general have empty interior; for
instance, in the linear case Jfiy) = 0 then £f(t, s; £; U) = £f(t — s) and it is easy to
see using Lemma 2.4 that U c D(^), the elements of H having //^-norrn bounded,
thus U cannot even be dense in a ball H.
Thus to cause (3.12) to have an interior point, we have to rely on the target set
Y, or, rather, on the contingent cone KY(y"). In this way we also get a characterisation
of the concept of controllability.
When 17 is unbounded, we take a sequence {«;„}, /cB-> oo and obtain (3.83a) for a
multiplier {fin,zn) corresponding to each subset (Ve U; \\ V\\FKn} of the control set
U. To construct a multiplier satisfying (3.83a) for the entire control set U, we take
the weak limit (n, z) of (fin, zn). To show that (yu, z) # 0, we apply the same considerations used above for bounded control sets.
Optimal controls in viscous flow problems
237
4. The feedback problem
In this section and in the two subsequent ones, we analyse the feedback problem for
the case where the target set Y = H. This means that the final state is free of
constraints. Moreover, the analysis in these three sections will use additional properties of the nonlinearity .vF(-) as denned in [13, Theorem 3]. This would correspond
to the Navier-Stokes equations in two dimensional bounded and unbounded
domains. The main result of the present section is that the value function
•f(-, -):(0, T] x //->/? is a viscosity solution in the sense of Crandall and Lions
[5,6] to the Hamilton-Jacobi-Bellman equation (2.12).
The value function and its properties
We begin with a series of lemmas concerning the continuity properties of the
trajectory y and the value function 'f.
LEMMA 4.1. Let Hypotheses 2.1-2.3 be satisfied and also suppose that the nonlinear
operator N(-) satisfies in addition the estimate (17) of [13] with the constant Ct =
C^lj, T). Then the solution y of (2.1) satisfies the a priori estimate,
\\y(t)\\H + I
II^2J(S)IIH^^C8(||CIIH,
Vfe/7
I^
IIL2(T,T;I/)),
V U e L 2 (T, T; U).
and
(4.1)
Proof. A priori estimates are obtained assuming smooth solutions. Hence, taking
duality pairing with y in equation (2.1) we get:
ld_
~2~dt
Now, using the estimate (17) of [ 1 3 ] and Young's inequality,
1 d
t)
j ,
US
Mtt
'
\i r^*"
J
Mil
"•^lll.rilH
=
ll"'''1-'II.HII.ril.H
=
Z l l
c
" ' ~ N H
f
2 . W JW t f
\^w^)
From this we get
This gives the estimate for \\y(t) \\H due to Gronwall's inequality [13, Proposition 7].
Wen then use this result in equation (4.2) and integrate with respect to t from T to
t to get the estimate for JJ || ^y(s) \\2H ds.
•
LEMMA 4.2. Let Hypotheses 2.1-2.3 be satisfied and also suppose that the nonlinear
operator N(-) satisfies in addition the estimates (17)—(19) o / [ 1 3 ] with the constants
C- = C,(/T, T), i = 1, 2, 3. Then we have:
\\y{t,r,C;U)-y{t,r,{;U)\\2H+
I
and
\\s/*(y{s,r,C;U)-y(s,z;(;U))\\Bds
V Ue L 2 (T, T; U).
(4.3)
238
H. Fattorini and S. S. Sritharan
The solution y is strongly continuous in time: V £ e H and V U e L 2 (T, T; U),
||j>(f,T;C;l/)-C||H-*O
ast-T.
(4.4)
Moreover, for some Uo e U,
||^(t, T; C; t/o) - Cliff ^ C10( || CIU,-^. U0)\t—c\,
V£e £(,«?*) anrf V t e [ i , T]
(4.5)
V £ £ D(J/*)
(4.6)
and
flwrf
V t e [T, T]
Proof. Let j and j be trajectories of (2.1) with initial datum £ and £, respectively.
Then z = y — y solves
^09 =0
(4.7)
and
z(T) = f - ^ .
Taking duality pairing with z in equation (4.7), we get
\\A\
using the estimate of (19) of [13]. Now, using Young's inequality, we get
Thus,
||z||+||^z||^1A(t)IMI2f,
(4.8)
where \j/(t) = C|{ || sd^y ||H + || J / * J | | H } 2 and belongs to L 1 ^, T) due to Lemma 4.1.
From equation (4.8) we can obtain the estimate (4.3) in a similar way as in Lemma 4.1.
To obtain (4.4), note that from Lemma 4.1, yeL2(x, T; £>(j/*)) and hence from
the properties of s/ and JV{-) we deduce that yteL2(z, T;D{s^^^)). Hence, by a
well-known embedding theorem [15], y e C([T, T]; H) and this gives (4.4).
To obtain (4.5), this time we define z(t, T; £; U) = y(t, x; C; U) — £. Then z solves
zt + ^z + ^(y) - JT(C) = ®U0 -j*l-
JV(0
(4.9)
and
Z(T) = 0.
Taking inner products in (4.9) by z and using the fact that £ e D{jtf*), we can obtain
(4.5) as before using the estimates (17) and (19) of [13].
Finally to obtain (4.6) we take inner product in (2.1) by s/y and use estimate (18)
of [13]. •
Optimal controls in viscous flow problems
239
4.3. Let Hypotheses 2.1-2.3 and 2.5-2.8 be satisfied and suppose that the
estimate (17) o / [ 1 3 ] holds. If the initial datum for the evolution system (2.1) is such
that \\C\\H = R for some R>0, then the corresponding optimal control U satisfies
LEMMA
Proof Let us take a particular control Uo e U. Then we have from Lemma 4.1
||y(t, T; C UO) III + [ II s/*y{s) \\2H ds ^ C8(U0, T, R).
(4.10)
Now, since Uo is nonoptimal,
<po(y{T, T; f; U)) + \
£
9o(y(T,
[&{t, y(t, T; f; U)) + Y(t, t/(t))] it
T; f; l/0)) + I [0(t, j(t, T; f; 17O)) + Y(t, l/ 0 )] A.
(4.11)
Here j(r, T; £; £/) is the optimal trajectory corresponding to the initial data £. Now,
using the properties of q>0, 0 and Y and Lemma 4.1, we deduce that
<Po(y(T,
T; C; ^o))^Ci4(
II J(T,
T;C;1/0)||H)^C15(J?,
t/ 0 , T).
(4.12)
Similarly,
0(t,j(t,T;C;C/o))^^C16
and
|| J / * J | | | dt^C 1 6 (i?,L/ 0 ,T)
(4.13)
rr
Y(t, l/ 0 ) it ^ C17(C7O, T).
(4.14)
We use estimates (4.12)-(4-14) in (4.11) to deduce that
Y(t,U(t))dtSC18(R,U0,T).
This and Hypothesis 2.8 provide us with
\\U\\L2(uT;V)^C(R,U0,
T). •
(4.15)
THEOREM 4.4. Let Hypotheses 2.1-2.3 and 2.5-2.8 be satisfied and let the estimates
(17)—(19) o / [ 1 3 ] hold. Then the value function satisfies
(4.16)
and
C21['
,
VUoeU
\\s/*y(s,r,C,U0)\\2Bis,
and Vte[T, T\.
(4.17)
240
H. Fattorini and S. S. Sritharan
Proof. Let us first prove the continuity of the map
~\:H^R, re[0, T\.
Let U(t) be the optimal control corresponding to the initial data (T, £). Then
II krllz.2(t,7W) S C(||^|| H , T) due to Lemma 4.3. Moreover, let U be in general a nonoptimal control for the initial data (T, £). Let the trajectory which corresponds to
(T, C,U)be y(t, x;C,U) and that which corresponds to (T, ft U) be y(t, T; £;£/).
r\y, c - r\x, <n s n(y(T, r, C; u)) - <po(y{T, v,ftu))
+ j
{G(t,y(t,t;£U))-&(t,y(t,r,$;U))}dt.
(4.18)
Since cp0 is Frechet differentiable, it is locally Lipschitz,
\<Po(y(T))-<Po(y(T))\^C(\\y\\H,\\y\\H)\\y-y\\H.
Hence, using Lemmas 4.1-4.3, we get
\9oWT))-<Po(y(T))\£C(\\C\\H,K\\n)K-e\\HNow consider the integral term in (4.18). Using Hypothesis 2.7, we get
(4.19)
{S(t,yXt))-S(t,y(t))}dt
[
^ c 18 [ W^y-^yWniWs**y
WH + ^HH} dt
^c18
by the Schwartz inequality. Hence by Lemmas 4.1-4.3 we get
{@(t,y(t))-®(t,y(t))}dt^C(U\\H,
KWHUCSWH-
(4-20)
We thus combine (4.18)-(4.20) to get
\r[T,C]-*r[r,€]\£C(M\\H,K\\H)K-e\\H, VCZeH, Vte[0,T].
Let us now establish the continuity with respect to time. For this purpose, we
construct a special control [/(•) in the following way:
(Uo
if s e (T, t),
U(s)={ „
\u(s) Xse(t,T),
with ii^t^T.
Here U is the optimal control corresponding to the initial data (t, Q.
Let y and j be, respectively, the trajectories corresponding to
(4.21)
se(t,T),
y(t) = C
and
),
se(T,T),
(4.22)
Optimal controls in viscous flow problems
241
y{*) = £
Let the initial datum £ e H for the moment. Then since y is optimal and y is
nonoptimal, we have
r\x, Q - r\t, C] S <Po(y(T, r, £ u)) - n(y(T, t, £ u))
{&(s,y(s,r,£U))-G(s,y{s,t;£U))}ds
{0(s, y{s, T; C; 17O)) + Y(s, £/0)} ds.
(4.23)
Note that by the semigroup property,
y(s, v,£U)= y(s, t; y(t, x; £ Uo); U),
x^t^s.
Now, using the continuity of <p0 and Lemmas 4.1-4.3, we get
\My(T,t;y(t,T;£U0);U))-(p0(y(T,t;£U))\^C(U\\H)\\y(t,v,£U0)-C\\H(4.24)
Now we note that, using Hypotheses 2.7-2.8, we can estimate
<
ft
s, y(s, T; £ Uo)) + Y(s, Uo)} ds ^ C t
|| s/*f{s, v, £ U2o) \\2 ds + C2 • (t - x).
(4.25)
Note now that, using Hypothesis 2.7 and Lemmas 4.1-4.3, we can estimate
{&(s,y(s, t;y(t, x; £ Uo); U)) - &(s,y(s, t; £ U))} ds
t
;C;i/o)-CllH.
(4 26)
-
Using (4.24)-(4.26) in (4.23) we get (4.17). Thus, V £ e H, V [•, Q is continuous.
Now note that if f e /)(,«/*), then, using Lemma 4.2, we can deduce further that
\r[t,C]-rix,C]\£C(\\j**c\\H)-\t-i\. •
Remark 4.5. From this result and the estimates for y, we can conclude that the value
function f e C([0, T] x H). For each t e [x, T], V(t, •) is Lipschitz in C, e H and for
each C e D(J^*), f(; £) is Lipschitz in t e [T, T\.
We can also prove the Bellman principle of optimality which holds under rather
general conditions:
THEOREM
holds.
4.6. Let us assume that the existence Theorem 1 (or Theorem 5) of [13]
Then,forO^x^t^T,
r\%, Q = inf | I [0(s, y(s, T; f; I/)) + Y(s, C/(s))] is
f,y(t, T; C; C/)]; t/ e ^ a(i (T, t; t/)l
(4.27)
242
H. Fattorini and S. S. Sritharan
Proof. We can choose the control to be optimal in x ^ t ^ T, so that
;y(s, T; £ U)) + Y(s, (7(s))] ds
:, j(s, r, y(t, x; £ tf); t/(s)) + Y(s, t/(s))]
^
[ 0 ( S , J ( S , T ; C ; C / ) ) + Y ( S , U(s))-]ds
(r
+ inf <
[0(s, z(s, t; j ( t , T; C; C/)); C/(S)) + Y(s,
+ %(z(r, t;y(t, T; f; (7); 17)); U e ^ ( t , T; C/)|,
where z solves (2.1) with initial data y(t,x;£; U) at time t with a control U. The
second term on the right is simply the value function. Hence,
^[f, C] ^
[0(s, y(s, T; C; I/)) + Y(s, C/(s))]rfs+ -T[t, y(t, x; £ C/)]. (4.28)
Now note that V U e J?ad(0, T; U),
TT[T,f]g j [®(s,z(s,T;f;I/))+Y(s,l/(s))]ds
r
[0(s, z(s, t; z(t, x; £ U)); U(s)) + Y(s,
Now, we choose the part of U(-) in [£, T] to be the optimal control, so that the sum
of the second integral and the third term above will become equal to the value
function
r\x, Q S I [®(s, z(s, T; C, U)) + Y(s, Uis))] ds + r\t, z{t, v, £ I/)].
(4.29)
Note that the control U(-) in [T, t] is still arbitrary. Thus taking infimum over all
admissible controls,
r\x, Q ^ inf |
[0(s, z(s, x; £U))+ Y(s, C7(s))] ds
+ r\t, z(t, x; £ I/)]; t/ e ^ ad (T, t; i
Comparing (4.28) and (4.30) we deduce (4.27).
•
Optimal controls in viscous flow problems
243
Hamilton-Jacobi-Bellman equation and viscosity solutions
The main theorem established below is proved using the results of the previous
subsection. A similar theorem has been proved in [2, 3] for a semilinear evolution
problem where the nonlinearity Jf and the functional 0 are bounded (that is,
JV{-):H^>H
and ®():H—>/?). Proof in our case (where these operators are
unbounded) is similar and we only sketch the ideas.
4.7. The value function "V" :[t, T] x H'-* R is a viscosity solution to the
Hamilton-Jacobi-Bellman equation (2.12). That is, V f e Cl(\r, T] x H),
iff-cp
attains a local maximum at (t 0 , Co) e [T> T] x D(stf), then
THEOREM
Co,D^)^0
(4.31)
and if' "V' — p attains a local minimum at (t0, Co) e [T, T] X D(stf), then
-dttp{t0, Co) + Jtf(t0, Co, D&) ^ 0.
(4.32)
Proof. Let q> e C X ([T, T\; X H) and suppose that V — q> attains a minimum at (t0, Co)Then, for any e > 0 and V U e L2(t0, t0 + e; U),
' t0; C; U)] - T^[t0. Co] ^ ^(to + e5j(fo + e, t0; C; C7)) - ^(t 0 , Co)
— <p(r,y(r,to;£U))dr
Hence, substitution from equation (2.1), we get
r[t0
+ e, j ( t 0 + e, t0; C; t/)] - *Tto» Co]
[ °
<0
E
{ 3 ^ + <a*[D,<p], [/> - ([D,?.], stfy + ^{y(r)))H}
dr. (4.33)
Let us now suppose that
-dt<p(to, Co) + ^ ( t o , Co, DiV) g A < 0.
(4.34)
We now show that this leads to a contradiction. Note that (4.34) implies that
VUeU,
-dt<p{to, Co) + < - * * [ ^ ] , t/> - Y(t 0 , I/) + ([D c rt, ^Co +
^A<0.
^(CO))H
- ©(to, Co)
(4.35)
Let us now consider t such that to^t^to
+ E. Then for U(-)e C([f0, t 0 + e]; U) and
for the corresponding trajectory y(t) =y(t, t0; Co; U), the continuity of various terms
in the inequality (4.35) implies that
], U(t)y~Y(t,
U X/2<0,
(4.36)
244
H. Fattorini and S. S. Sritharan
for sufficiently small E > 0. Substituting (4.36) in (4.33), we get
*Tto, Co] ^ ^ / 2 + I °
{Y(t, U(t)) + 0(t,y(t))} dt + V\t0 + e, j>(*o + C t0; C, V)],
Jt0
(4.37)
with X < 0. However, since C([t 0 , to + «]; ^0 is dense in L 2 (t 0 , f0 + e; U), we deduce
using the continuity of 0, Y and "^[t, •] and the continuous dependence of the
trajectory on the control that (4.37) holds V U e L2(t0, t0 + e; U). Hence
TT[t0> Co] ^ AE/2 + inf { I °
{Y(t, U(t)) + &(t, y(t))} dt
+ -T[t0 + e,y(t0 + e, t0; f; I/)]; (7 6 L2(£0, t0 + e; f/)l.
(4.38)
This contradicts the Bellman principle of optimality (Theorem 4.6) and thus (4.34)
is false and (4.32) is proved.
It remains to prove (4.31). Let us again take <p e ( ^ ( [ T , T] X H) and suppose this
time that 'f — q> attains a maximum at (t0, Co)- Then, for any e > 0 and
V U e L2(t0, t0 + e; U),
t0 + e,y(t0 + s, t0; C; t / ) ] - y[t0,
Co] ^ <P(t0 + e,.H'o + e» t 0 ; C; U)) - <p(t0, Co)-
From this we get as before,
t0 + E, y(t0 + e,t0; f;
^.
(4.39)
Let us now suppose that
-dMto, Co) + Jfih, Co, D#) ^n>0.
(4.40)
Now, from the definition of the conjugate function, we have for each \i > 0, 3 Vo e U
such that
Y*(t0, -**[/)jrt)-Ai/2^<-«*[D^](t o ,f o ), V0}-Y(t0, Vo).
(4.41)
Substituting (4.41) in (4.40), we get
-8t<p(t0, Co) + <~^*ID^I
Vo} - Y(t 0 , V0)+([D<<p], ^Co + ^ ( C O ) ) H - ©(to, Co)
^ n/2 > 0.
(4.42)
Let us now choose a smooth control function U(-)e C([t0, to + e]; U) such that
U(to) = Vo. Then we can deduce using the continuity of various terms in the inequality
(4.42) that, for t in t0 ^ t ^ t 0 + e with e > 0 sufficiently small,
], u(t))-Y(t,
u(t))
], s/y(t) + ^{y(t)))H - @(t, y(t)) ^ n/4 > 0.
(4.43)
Optimal controls in viscous flow problems
245
Substituting (4.43) in (4.39) we get
r[t0,
Co] ^ W 4 + I °
{Y(t, 1/(0) + ®(t,y(t))} dt + f [t 0 + e,^(to + e, t0; £ I/)].
Since /*, e > 0, we get
°
o,Co]>
Jt0
(4.44)
This contradicts the Bellman principle of optimality, therefore (4.40) is false and this
proves (4.31). •
Remark 4.8. Note that if t~ — <p attains a maximum at
-V e C([T, T] x H) and q> e ( ^ ( [ T , T] x H), then we should have
(t0, Co), with
\_D(p](t0, Co) 6 S+ 'V[to,Co]-
(4.45)
Similarly, if (t0, Co) is a point of minimum for V — <p, then
[I>^](fo>Co)e d~^[to>Co].
(4.46)
To see (4.45), we set x--=(t, C) and consider the inequality
y\x]
— ^ [ j c o ] ^ P(*) — <p(*o) = ([^x^J^oX * ~ ^o)^ + &(?,x — xo)>
where
|^(y,x-Xo)L ^ Q
Now,
f ^ M - -T[J: 0 ] - ([J),y](x0), J: - x o ) g |
<
|^(y, •-*o)l
and hence from the definition of the super differential (2.15) we get (4.45). The result
(4.46) can be similarly verified.
Remark 4.9. Although we have proved that the value function is a viscosity solution
of the Hamilton-Jacobi-Bellman equation, the uniqueness question of such a solution
remains open. The primary difficulty here is due to the unboundedness of the
nonlinearity J/~.
5. Pontryagin maximum principle via Hamilton-Jacobi-BeJIman equations
We derive the maximum principle using a method introduced by Barron and Jensen
[4] (see also [3]). The main advantage of this method is that it closely follows the
formal derivation of the maximum principle from the Hamilton-Jacobi-Bellman
equation. However, unlike the classical procedure which requires C2 regularity on
the value function, this method only requires the value function to be continuous.
As remarked in Section 4, we only consider the case of constraint-free final state (i.e.
246
Y=H).
H. Fattorini and S. S. Sritharan
Let us begin with the evolution system
d,z + s/z + Jf(z) = <%U,
te[s, T],
(5.1)
(5.2)
where s e [x, T] and U is the optimal control in [T, T] for the initial data £ at T.
Note that in general z(t, s; £; U) would be an optimal trajectory only if s = x.
Let us define the function iV{-, •): [x, T] X H-+R as
fr
iT(s,C):=vo(z(T,s;C;O))+
{@(r,z(r,s;C;U)) + r[r,tJ)}dr,
(5.3)
Js
with
W(T,C) = <Po(O, V f e D ( j / ) .
(5.4)
Then we have the following theorem:
5.1. Let U e L2(x, T; U) be an optimal control and y(t, T; f; U) be the corresponding optimal trajectory. Then, for t almost everywhere in [r, T],
THEOREM
Y*(t, -3S*lDyir^t,y(t,v,C;
0)))=-<a*[Py1T\(t,$(t,r,C,
U)), U)-Y(t,
U).
(5.5)
Moreover, ifp(-) e C([x, T]; H) solves the adjoint system (2.13), then
p(t) = [,Dy^(t,y(t,
T; C; U)l
V t e [x, T\-
Proof. The proof of this theorem uses the fact that the value function is a viscosity
subsolution. We first show that the function if{-, •) has enough regularity properties
to serve as the test function for the viscosity solution technique.
•
LEMMA 5.2. V £ e D(srf), the function iV {•, f) is absolutely continuous in s e [T, T] and
V s e [T, T], iV'(s, •) is continuously Frechet differentiable for £ e
Proof. Let us define, for t e [s, T ] ,
y
¥(t) = W^(t,s;C,
U)y, for some yeH
and
$(t) = d,z(t, s; £ [/).
Then *F(-) solves
dt¥ + sd^ + [D^izflW
= 0,
t e [ s , T],
v
P(s) = y e ^ .
(5.6)
Similarly, <!>(•) solves
[Dz^"(z)]<D = 0,
t e [ s , T],
(I)(S) = J / C + ^ ( £ ) - ^ C / ( 5 ) .
(5.7)
The initial condition for (5.7) was derived in the following way. Let S(t) be the
holomorphic semigroup generated by — srf. Then, as observed in [13], the solution
of (5.1)—(5.2) is given by
z(t, s; t;U) = S(t-s)£+
P
S(t- r ) ( - JT{z(r, s; £ U)) + <MU(r)) dr.
Optimal controls in viscous flow problems
247
Then, for s almost everywhere in [T, t ] ,
Using Lemma 3.10, it is possible to conclude that if £ e D ( j / ) , then (5.6) and (5.7)
have, respectively, unique solutions *F, <J>e C([s, T\; H)nL2(s, T; £>(.*/*)) with
Now, for y e H,
<X_D^{s,Q,yy
= {lD(Po]{z(T)),V{T))H+
\ <[J> z 0](r, z(r)), ¥(r)> dr. (5.8)
Js
Hence, using the regularity of *F and z,
Thus [D;TT ](s, 0 e H, V £ e D ( ^ ) and V s e [T, T ] .
Now, if C e D(s/), then for s almost everywhere in [T, 7^],
(5.9)
/
x
Now using the properties of <1>, we conclude that 3 s '^ "(-, £) e L (0, T).
Let us now proceed with the proof of the theorem. Consider
1T(t, z(t, s; C, U)) = <PoMT, s; C, U)) + \
{0(r, z(r)) + Y(r, U(r))} dr
Jt + St
t + st
(®(r,z(r))+r(r,U(r)))dr
rt + st
= HT{t + 8t, z(t + St, s; C, U)) +
(0(r, z(r)) + Y(r, U(r))) dr.
Hence, for t almost everywhere in [s, T ] ,
- TT(t, z(t, s; t, U)) = - 0 ( t , s; C, U))-Y(t, U(t)).
That is, for T ^ s ^ f ^ T, and for t almost everywhere in [s, T ] ,
))H
t / > - Y ( t , f/)}=0,
(5.10)
with
TT(T, Z(T, 5; C, f/)) = <Po(z(T, s; C, f>))Thus
-dtnr{t, z{t)) + MT(t, z(t), [p,ir\(t,
At))) Z 0.
(5.11)
Now, note that for s e [T, T] and £ e D ( ^ ) , we have TT(s, f) ^ ^ (s, £) and TT(T, C) =
248
H. Fattorini and S. S. Sritharan
"V (T, £). In fact, along the optimal trajectory,
HT{t,y{t, r; C, U)) = ntj(t,
T; {, U)).
Hence Y — iV attains a maximum of zero along each point (t,y(t, T;£ £/)). Now,
using the fact that Y is a viscosity subsolution of (2.12),
0.
(5.12)
O.
(5.13)
Comparing (5.11) and (5.12) for s = T, we deduce that
Again comparing (5.13) with (5.10) for s = T, we deduce (5.5).
Let us now consider the solution *P e C([T, T~\; D(S/)) of the linear problem (5.6)
with y e D(s/). Taking duality pairing with the adjoint problem (2.13),
i-dtP, T(f)> + <j*p, *(*)> + <XP,JrY{f)p, T(t)> = <[Py&l(t,y), *>.
Now note that since "V,pe C([s, T\;H)nL2(s,
D(s/~i)), we have
T; D{sf±)) and dt*¥,dtpeL2(s,T;
for almost all t e [s, T]. We thus get
-dt(p, T) H + (5,T + ^»P + [
Noting that the second term on the left is zero, we integrate from t to T,
dr.
Here we used the fact that p{T) = [Dy<p0^\(y(T)). By the continuity of p and \(i, the
above result holds V t e [s, T].
We now compare this result with (5.8):
)), 1)H
([^olW)),
())H
|
< [ , ] ( , jKr)), «P(r)> dr
and conclude that
, Vte[s, T].
Since D(«s/) c: H is dense, we deduce that
/>(•) = [D,1T](-,K-, T, C, f/)) 6 C([0, T]; //).
Note that since ~f — if attains a maximum of zero at (t,y(t, x, C, U)) we also have,
Optimal controls in viscous flow problems
249
due to Remark 4.8,
[pyin(- J(-, *, c, u)) e a; n-j(-, T, C, U)).
This proves the theorem. Note also that, from (5.5), we get
(/(•) = VY*(- , -a*[pyir\(-
J(-> *, C, #)))=VY*(- , -«*/»(•))•
(5.14)
6. Verification theorem
In this section we complete the verification theorem. Recall from Lemma 4.3 that
"T(-, C) for £ e D(s/) is Lipschitz in time. Hence, by the Rademacher Theorem [14],
•f{-, Q is differentiate almost everywhere in [T, T]. Let t e [t, T] be such a point.
Let U(-) be the optimal control in the interval [t, T] corresponding to the initial
data C e D(>s/) at time t. Then, from the definition of the value function,
T
{0(r, y(r, t + e;C, U)) + Y(r, U(r))} dr + (po(y(T, t + e; f; U))
{0(r,y(r, t; £, U))+Y(r, U(r))} dr - <po(y(T, t; f, U))
T
{0(r, y(r, t + E;£ £/)) - &(r, y(r, t; £
+ {^0(j(T, t + e; C; I/)) - n(y(T, t; f, t/))}.
Hence,
, j(r, t; f, f/)),
Now substituting from the adjoint equation (2.13),
dtr(t,o^ -&(t,o-r(t, t)(t)) +
Substituting for $(t), and using the maximum principle (Theorem 2.11), we get
dtr-(t,Q£jr(t,C,p(t)).
(6.1)
Now, let U be the optimal control in [t + s, T~\ corresponding to initial data £ e
at t + e. Then, setting
C/(r) =
(U(r)
if r e ( t + e,T],
l{7(t + ) if
re(t,t
250
H. Fattorini and S. S. Sritharan
we get
{©(r, y(r, t + e;£U)) + Y(r, U(r))} dr + cpo{y(T, t + e;£ U))
{0(r, y(r, t; £ U)) + Y(r, U(r))} dr -
9o{y{T,
t; £ U))
{©(r, y(r, t + a;£ U)) - ©(r, y(r, t; £ I/))} dr
{&(r,y(r,t;C,U(t)))+Y(U(t))}dr
jt
+ {<po(y(T, t + e;C, U)) - <po(y(T, t; £ [/))}.
Thus, noting that
[Dcj](r, t; f; U)dty{t, t; f; U) = -9(r),
we get
dtr{t, O ^ - 0 ( t , 0 - Y(t, U{t)) + I <[Z),©](r, j(r, t; C,
This gives as before
(6.2)
Comparing (6.1) with (6.2) (and also noting Remark 4.8) we get (2.14) with
p(t)ed^'f(t,C)- Note also that the right-hand side of (6.2) is integrable in time
and hence
7. Concluding remarks
(1) The version of the maximum principle proven in Sections 4 and 5 is weaker
than that in Section 3 because it does not incorporate the target set.
(2) As remarked in [13], lack of lower semicontinuity of the cost functional (in
particular due to nonconvexity of Y(t, •)) will lead to Young measure-valued optimal
controls. Existence of such controls is studied in [12]. It is of interest then to develop
and extend the maximum principle and feedback analysis established in the present
paper to such controls.
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(Issued 29 April 1994)