Journal of Applied Mathematics & Bioinformatics, vol.1, no.1, 2011, 1-20
ISSN: 1792-6602 (print), 1792-6939 (online)
c International Scientific Press, 2011
°
Sufficiency in optimal control without
the strengthened condition of Legendre
Gerardo Sánchez Licea1
Abstract
In this paper we derive a sufficiency theorem of an unconstrained
fixed-endpoint problem of Lagrange which provides sufficient conditions
for processes which do not satisfy the standard assumption of nonsingularity, that is, the new sufficiency theorem does not impose the strengthened condition of Legendre. The proof of the sufficiency result is direct
in nature since the former uses explicitly the positivity of the second
variation, in contrast with possible generalizations of conjugate points,
solutions of certain matrix Riccati equations, invariant integrals, or the
Hamiltonian-Jacobi theory.
Mathematics Subject Classification : 49K15
Keywords: Optimal control, sufficient conditions for optimality, strong minima, singular extremals, strengthened Legendre-Clebsch condition.
1
Introduction
In the classical calculus of variations it is well-known that the nonnegativity of
the second variation along an arc x0 over the set of admissible variations becomes a second order necessary condition for optimality. The theory of Jacobi
1
Facultad de Ciencias UNAM, Departamento de Matemáticas, México D.F. 04510,
México, e-mail: gesl@ciencias.unam.mx
Article Info: Revised : March 25, 2011. Published online : May 31, 2011
2
Sufficiency in optimal control ...
concerns with a characterization of the nonnegativity of the second variation
over the set of admissible variations with the nonexistence of conjugate points
on the underlying time interval under consideration. In concrete, a smooth
nonsingular trajectory x0 satisfies the fact that its second variation is nonnegative over the set of admissible variations if and only if x0 satisfies the
condition of Legendre and there are no conjugate points on x0 in the underlying open interval. Moreover, the second variation is positive over the set of
nonnull admissible variations if and only if x0 satisfies the strengthened condition of Legendre and there are no conjugate points on x0 in the half-open
interval. One of the unfortunate features of Jacobi’s theory arises when the
arc under consideration is not nonsingular, that is, when it is singular, in this
case, the theory of Jacobi is not applicable. On the other hand, all the classical
sufficiency theorems in the theory of calculus of variations assume the strengthened condition of Legendre, and therefore the extremals under consideration
satisfying the classical sufficiency conditions must be nonsingular.
Because of this fact, Ewing mentions in [7], that in the theory of calculus
of variations, there is a gap between the necessary and sufficient conditions for
optimality. In fact, in [7], Ewing devotes an entire section to problems for which
Legendre strengthened condition fails. There he shows that we can partially
close this gap by an elementary device discussed in [6] of adding a penalty term.
This procedure is illustrated by means of three examples where the trajectory
being examined is singular, but one can obtain a solution directly from the
properties of the particular examples. However, this technique may not hold
in general. Ewing states that ‘although the use of the penalty term sheds
light on the theory, it provides no panacea for attacking particular examples.
Indeed there are no panaceas!’
In more recent years, the study of second order conditions for optimality
in the theory of calculus of variations and optimal control has provided an
extensive literature (see [1-5, 17-22, 24, 26-35] and references therein). In
[17] sufficient conditions in the calculus of variations are obtained by using an
appropriate form of local convexity. For optimal control problems, sufficiency
is derived in [25, 26] by applying the positivity of the second variation as well
as a generalized theory of Jacobi in terms of conjugate points, the insertion
of the original optimal control problem in a Banach space exhibits in [35]
Gerardo Sánchez Licea
3
an alternate sufficiency method, the construction of a bounded solution to a
matrix-valued Riccati equation, a verification function satisfying the HamiltonJacobi equation, and a quadratic function that satisfies a Hamilton-Jacobi
inequality become fundamental devices in sufficiency results given in [2, 5, 16,
18-21, 24].
A different approach which provides sufficiency in the classical isoperimetric calculus of variations fixed-endpoint problem is given by Hestenes in [15].
This method treats explicitly with the positivity of the second variation on the
set of nonnull admissible variations and it is implicitly based on the concept
of a directionally convergent sequence of trajectories which is in turn a generalization of the concept of directional convergence for vectors in the finite
dimension case. The development of this technique as it appears in [15], as well
as its application to more general problems, can be traced back to different
papers of the author and McShane (see [8-14, 23]). A generalization of this
method which covers optimal control problems can be found in [26, 31, 32].
In this paper we study an unconstrained fixed-endpoint optimal control
problem of Lagrange. The main contribution is based on two fundamental
facts. First, we show how by applying a similar technique of that given in [26,
31, 32], one can be able to obtain an itself-contained proof to a sufficiency result
which is in contrast with possible generalizations of conjugate points, solutions
of certain matrix Riccati equations, invariant integrals, theory of extremals, or
the Hamiltonian-Jacobi theory. Second, the new sufficiency theorem does not
include the standard assumption of nonsingularity, that is, the strengthened
Legendre-Clebsch condition is not imposed. In particular, we refer the reader
to [24] where the importance of this condition is fully explained.
The paper is organized as follows. In Section 2 we pose the problem we
shall deal with, introduce some notation and basic definitions, and state the
main result. In Section 3 we illustrate the usefulness of the new sufficiency
theorem by means of a simple example of a singular process which is a proper
strong minimum of the problem in hand. Section 4 is devoted to the proof of
the main sufficiency theorem together with the statement of an auxiliary result
on which the proof is strongly based. The auxiliary result, which implicitly
includes a possible generalization of the notion of a directionally convergent
sequence of trajectories, is established in Section 5.
4
2
Sufficiency in optimal control ...
The problem and the main result
The fixed-endpoint optimal control problem we shall study in this paper can
be stated as follows. Suppose we are given an interval T := [t0 , t1 ] in R, two
points ξ0 and ξ1 in Rn , and functions L and f mapping T × Rn × Rm to R
and Rn respectively.
Let X := AC(T ; Rn ) denote the space of absolutely continuous functions
mapping T to Rn , let U := L1 (T ; Rm ), set Z := X × U , and denote by Ze the
set of all (x, u) ∈ Z satisfying
a. L(t, x(t), u(t)) is integrable on T .
b. ẋ(t) = f (t, x(t), u(t)) a.e. in T .
c. x(t0 ) = ξ0 , x(t1 ) = ξ1 .
The problem we shall deal with, which we label (P), is that of minimizing
I over Ze , where
Z t1
I(x, u) :=
L(t, x(t), u(t))dt.
t0
For this problem, an admissible process is an element of Ze , that is, a couple
(x, u) comprising functions x ∈ X and u ∈ U which satisfy the constraints
of problem (P). An admissible process (x, u) is called a strong minimum of
problem (P) if there exists ǫ > 0 such that I(x, u) ≤ I(y, v) for all (y, v) ∈ Ze ,
(y, v) 6= (x, u), with ky − xk∞ < ǫ. If the inequality can be replaced by a strict
inequality then (x, u) is said to be a proper strong minimum of (P).
We shall assume throughout the paper that the functions L and f are
continuous and of class C 2 with respect to x and u on T × Rn × Rm .
For the theory to follow we shall find convenient to introduce the following
definitions.
• For all (t, x, u, p) ∈ T × Rn × Rm × Rn let
H(t, x, u, p) := hp, f (t, x, u)i − L(t, x, u).
• A triple (x, u, p) will be called an extremal if (x, u) is a process, p ∈ X,
ṗ(t) = −Hx∗ (t, x(t), u(t), p(t)) (a.e. in T ) and Hu (t, x(t), u(t), p(t)) = 0 (t ∈ T )
where ‘∗ ’ denotes transpose. Throughout the paper all derivatives such as Hx
5
Gerardo Sánchez Licea
and Hu will be gradient row vectors and all vector-valued functions such as
p ∈ X will be considered as column vectors.
• For a given p ∈ X define, for all (t, x, u) ∈ T × Rn × Rm ,
F (t, x, u) := L(t, x, u)−hp(t), f (t, x, u)i−hṗ(t), xi [= −H(t, x, u, p(t))−hṗ(t), xi].
With respect to F , define the functional J : Ze → R as
Z t1
J(x, u) := hp(t1 ), ξ1 i − hp(t0 ), ξ0 i +
F (t, x(t), u(t))dt.
t0
Consider the first variation of J along (x, u) ∈ X ×L∞ (T ; Rm ) over (y, v) ∈
Z given by
Z t1
′
J ((x, u); (y, v)) :=
{Fx (t, x(t), u(t))y(t) + Fu (t, x(t), u(t))v(t)}dt,
t0
and the second variation of J along (x, u) ∈ X × L∞ (T ; Rm ) over (y, v) ∈
X × L2 (T ; Rm ) given by
Z t1
′′
2Ω(t, y(t), v(t))dt
J ((x, u); (y, v)) :=
t0
where, for all (t, y, v) ∈ T × Rn × Rm ,
2Ω(t, y, v) := hy, Fxx (t, x(t), u(t))yi+2hy, Fxu (t, x(t), u(t))vi+hv, Fuu (t, x(t), u(t))vi.
Also, with respect to F , denote by E the Weierstrass excess function which
corresponds to
E(t, x, u, v) := F (t, x, v) − F (t, x, u) − Fu (t, x, u)(v − u)
for all (t, x, u, v) ∈ T × Rn × Rm × Rm .
• A process (x, u) is nonsingular if the determinant |−Huu (t, x(t), u(t), p(t))| =
|Fuu (t, x(t), u(t))| is different from zero for all t ∈ T . A process (x, u) satisfies
the strengthened condition of Legendre if the matrix −Huu (t, x(t), u(t), p(t)) =
Fuu (t, x(t), u(t)) > 0 for all t ∈ T .
• For all (x, u) ∈ X × L∞ (T ; Rm ), denote by Y (x, u) the class of all (y, v) ∈
X × L2 (T ; Rm ) satisfying
ẏ(t) = A(t)y(t) + B(t)v(t) a.e. in T, y(t0 ) = y(t1 ) = 0
6
Sufficiency in optimal control ...
where A(t) := fx (t, x(t), u(t)), B(t) := fu (t, x(t), u(t)) (t ∈ T ). Elements of
Y (x, u) will be called admissible variations along (x, u).
• For all x ∈ X and all u ∈ U let
Z t1
D1 (x) :=
ϕ(ẋ(t))dt
and
D2 (u) :=
t0
Z
t1
ϕ(u(t))dt
t0
where
ϕ(c) := (1 + |c|2 )1/2 − 1.
• Define D : Z → R by
D(x, u) := max{D1 (x), D2 (u)},
and denote by k · k = k · k∞ the supremum norm in X.
Let us now state the main theorem of the paper. It consists of a sufficiency
result for a proper strong minimum of problem (P) assuming, with respect to
a given extremal, the Legendre-Clebsch condition, the positivity of the second
variation along nonnull admissible variations, and two conditions related to
the Weierstrass excess function.
2.1 Theorem: Let (x0 , u0 , p) be an extremal with u0 ∈ L∞ (T ; Rm ) and
suppose that the there exist h, ǫ > 0 such that
i. Fuu (t, x0 (t), u0 (t)) ≥ 0 (a.e. in T ).
ii. J ′′ ((x0 , u0 ); (y, v)) > 0 for all nonnull admissible variations (y, v) along
(x, u).
iii. For all (x, u) ∈ Ze satisfying kx − x0 k < ǫ, E(t, x(t), u0 (t), u(t)) ≥ 0
(a.e. in T ) and
Z
t1
E(t, x(t), u0 (t), u(t))dt ≥ hD(x − x0 , u − u0 ).
t0
Then there exist ρ, δ > 0 such that, for all admissible processes (x, u) satisfying
kx − x0 k < ρ,
I(x, u) ≥ I(x0 , u0 ) + δD(x − x0 , u − u0 ).
In particular, (x0 , u0 ) is a proper strong minimum of (P).
7
Gerardo Sánchez Licea
3
Example
In this section we provide an example of a fixed-endpoint problem for which
an application of Theorem 2.1 shows that the singular extremal under consideration is in fact a proper strong minimum.
3.1 Example: Consider the problem of minimizing
Z 4
{2u21 (t) + |u2 (t)|3 − x1 (t) − x2 (t)}dt
I(x, u) =
0
subject to
a. 2u21 (t) + |u2 (t)|3 − x1 (t) − x2 (t) is integrable on [0, 4].
b. (ẋ1 (t), ẋ2 (t)) = (sin2 u2 (t), u1 (t)) a.e. in [0, 4].
c. (x1 (0), x2 (0)) = (0, 0) and (x1 (4), x2 (4)) = (0, −2).
For this case n = m = 2, T = [0, 4], ξ0 = (0, 0)∗ , ξ1 = (0, −2)∗ ,
L(t, x, u) = 2u21 + |u2 |3 − x1 − x2
and f (t, x, u) = (sin2 u2 , u1 )∗ .
Let x0 (t) = (0, −t2 /8)∗ , u0 (t) = (−t/4, 0)∗ (t ∈ T ). Clearly, (x0 , u0 ) ∈ Ze . We
have
H(t, x, u, p) = p1 sin2 u2 + p2 u1 − 2u21 − |u2 |3 + x1 + x2 ,
Hx (t, x, u, p) = (1, 1),
Hu (t, x, u, p) = (p2 − 4u1 , 2p1 sin u2 cos u2 − 3|u2 |u2 ).
Therefore, (x0 , u0 , p) with p(t) = (−t, −t)∗ (t ∈ T ) is an extremal. In addition,
F (t, x, u) = 2u21 + |u2 |3 + t sin2 u2 + tu1 , and so
Ã
!
4
0
Fuu (t, x, u) =
.
0 6|u2 | + 2t cos2 u2 − 2t sin2 u2
Observe that
hFuu (t, x0 (t), u0 (t))h, hi = 4h21 + 2th22
(t ∈ T )
so that (x0 , u0 ) does not satisfy the strengthened condition of Legendre but
2.1(i) holds. Now, observe that
!
!
Ã
Ã
0 0
0 0
A(t) =
, B(t) =
(t ∈ T )
0 0
1 0
8
Sufficiency in optimal control ...
and hence (y, v) ∈ Y (x0 , u0 ) implies that (ẏ1 (t), ẏ2 (t)) = (0, v1 (t)) a.e. in T . It
follows that
Z 4
′′
J ((x0 , u0 ); (y, v)) =
{4v12 (t) + 2tv22 (t)}dt > 0
0
for all (y, v) ∈ Y (x0 , u0 ), (y, v) 6= (0, 0). Hence 2.1(ii) holds.
Clearly, (x, u) ∈ Ze implies that x1 ≡ 0 and u2 (t) = k(t)π (t ∈ T ) for some
integrable function k(·) which maps T to the set of integers. Observing that
ϕ(c) ≤ |c|2 /2 for all c ∈ R2 , (x, u) ∈ Ze implies that
E(t, x(t), u0 (t), u(t)) = E(t, 0, x2 (t), −t/4, 0, u1 (t), k(t)π)
= 2u21 (t) + |k(t)|3 π 3 + tu1 (t) + t2 /8
= 2(u1 (t) + t/4)2 + |k(t)|3 π 3
≥ 2−1 [(u1 (t) + t/4)2 + k 2 (t)π 2 ]
≥ max{ϕ((0, u1 (t) + t/4)∗ ), ϕ((u1 (t) + t/4, k(t)π)∗ )}
= max{ϕ(ẋ(t) − ẋ0 (t)), ϕ(u(t) − u0 (t))}
(a.e. in T ).
Thus with any ǫ > 0 and h = 1, 2.1(iii) holds. By Theorem 2.1, (x0 , u0 ) is a
proper strong minimum of problem (P).
4
Proof of Theorem 2.1
In this section we shall prove Theorem 2.1. We first state an auxiliary result
(proved in Section 5) on which the proof of Theorem 2.1 is strongly based.
Implicit on the statement of the result it is inserted a possible generalization of the notion of a directionally convergent sequence of trajectories, firstly
introduced in a calculus of variations context by Hestenes in [15], page 155.
4.1 Lemma: Let {zq := (xq , uq )} be a sequence in Z, z0 := (x0 , u0 ) ∈ Z,
and suppose that
lim D(zq − z0 ) = 0 and dq := [2D(zq − z0 )]1/2 > 0 (q ∈ N).
q→∞
For all q ∈ N and almost all t ∈ T define
¸1/2 ·
¸1/2 ¾
½·
1
1
.
, 1 + ϕ(uq (t) − u0 (t))
wq (t) := max 1 + ϕ(ẋq (t) − ẋ0 (t))
2
2
9
Gerardo Sánchez Licea
For all q ∈ N and t ∈ T define
yq (t) :=
xq (t) − x0 (t)
,
dq
vq (t) :=
uq (t) − u0 (t)
.
dq
Then the following hold:
a. For some v0 ∈ L2 (T ; Rm ) and some subsequence of {zq }, again denoted
by {zq }, {vq } converges weakly to v0 in L1 (T ; Rm ). Moreover, {(ẋq , uq )} converges almost uniformly to (ẋ0 , u0 ) on T and hence wq (t) → 1 almost uniformly
on T .
b. There exist a function σ0 ∈ L2 (T ; Rn ) and some subsequence of {zq },
again denoted by {zq }, such that {ẏq } converges weakly in L1 (T ; Rn ) to σ0 .
Moreover, if we define
Z t
y0 (t) :=
σ0 (s)ds (t ∈ T ),
t0
then yq (t) → y0 (t) uniformly on T .
c. Suppose S ⊂ T is measurable and wq (t) → 1 uniformly on S. Let
Rq (·), R0 (·) be m × m real matrix-valued functions with Rq (·) measurable on
S, R0 (·) ∈ L∞ (S; Rm×m ), Rq (t) → R0 (t) uniformly on S, and R0 (t) ≥ 0
(t ∈ S). Then for some subsequence of {zq }, again denoted by {zq },
Z
Z
lim inf hRq (t)vq (t), vq (t)idt ≥ hR0 (t)v0 (t), v0 (t)idt.
q→∞
S
S
Proof of Theorem 2.1:
Let z0 := (x0 , u0 ). Assume that, for all ρ, δ > 0, there exists z = (x, u) ∈ Ze
with kx − x0 k < ρ such that
J(x, u) < J(x0 , u0 ) + δD(z − z0 ).
(1)
We are going to show that this contradicts (ii) of Theorem 2.1 and the statement will follow, since I(x, u) = J(x, u) on Ze .
Note that, for all z = (x, u) ∈ Ze ,
J(z) = J(z0 ) + J ′ (z0 ; z − z0 ) + K(z) + Ẽ(z)
where
Ẽ(x, u) :=
Z
t1
t0
E(t, x(t), u0 (t), u(t))dt,
(2)
10
Sufficiency in optimal control ...
K(x, u) :=
t1
Z
{M (t, x(t)) + hu(t) − u0 (t), N (t, x(t))i}dt,
t0
M (t, y) := F (t, y, u0 (t)) − F (t, x0 (t), u0 (t)) − Fx (t, x0 (t), u0 (t))(y − x0 (t)),
N (t, y) := Fu∗ (t, y, u0 (t)) − Fu∗ (t, x0 (t), u0 (t)).
By Taylor’s theorem,
1
M (t, y) = hy − x0 (t), P (t, y)(y − x0 (t))i,
2
N (t, y) = Q(t, y)(y − x0 (t)),
where
P (t, y) := 2
Z
1
(1 − λ)Fxx (t, x0 (t) + λ(y − x0 (t)), u0 (t))dλ,
0
Q(t, y) :=
Z
1
Fux (t, x0 (t) + λ(y − x0 (t)), u0 (t))dλ.
0
Let us begin by proving the existence of α0 , δ0 > 0 such that, for all z =
(x, u) ∈ Ze with kx − x0 k < δ0 ,
|K(x, u)| ≤ α0 kx − x0 k[1 + D(z − z0 )].
(3)
By using the inequality of Schwarz and the continuity of the functions P and
Q we may choose α, δ0 > 0 such that for all z ∈ Ze with kx − x0 k < δ0 ,
|M (t, x(t))+hu(t)−u0 (t), N (t, x(t))i| ≤ α|x(t)−x0 (t)|[1+|u(t)−u0 (t)|2 ]1/2
(t ∈ T ).
Set α0 := max{α, α(t1 − t0 )}. Then, for all z ∈ Ze with kx − x0 k < δ0 ,
Z t1
[1 + ϕ(u(t) − u0 (t))]dt ≤ α0 kx − x0 k[1 + D(z − z0 )]
|K(z)| ≤ αkx − x0 k
t0
and hence (3) holds with α0 and δ0 given above.
Now, by (1), for all q ∈ N there exists zq := (xq , uq ) ∈ Ze such that
kxq − x0 k < δ0 ,
1
kxq − x0 k < ,
q
1
J(zq ) − J(z0 ) < D(zq − z0 ).
q
(4)
Since zq ∈ Ze , observe that the last inequality implies that uq (t) 6= u0 (t) on a
set of positive measure and so D(zq − z0 ) > 0 (q ∈ N). Since (x0 , u0 , p) is an
extremal, it follows that J ′ (z0 ; w) = 0 for all w ∈ Z. With this in mind, by
(2), the second condition of 2.1(iii), and (3),
J(zq ) − J(z0 ) = K(zq ) + Ẽ(zq ) ≥ −α0 kxq − x0 k + D(zq − z0 )(h − α0 kxq − x0 k).
11
Gerardo Sánchez Licea
By (4) we obtain
¶
µ
α0
1 α0
<
D(zq − z0 ) h − −
q
q
q
and consequently D(zq − z0 ) → 0, q → ∞. Define dq , wq , yq and vq , as in
Lemma 4.1, that is, dq := [2D(zq − z0 )]1/2 ,
¸1/2 ·
¸1/2 ¾
½·
1
1
,
, 1 + ϕ(uq (t) − u0 (t))
wq (t) := max 1 + ϕ(ẋq (t) − ẋ0 (t))
2
2
yq (t) :=
xq (t) − x0 (t)
dq
and vq (t) :=
uq (t) − u0 (t)
.
dq
By Lemma 4.1a there exist v0 ∈ L2 (T ; Rm ) and some subsequence of {zq },
again denoted by {zq }, such that {vq } converges weakly in L1 (T ; Rm ) to v0 .
By Lemma 4.1b for some σ0 ∈ L2 (T ; Rn ) and some subsequence of {zq }, again
Rt
denoted by {zq }, if y0 (t) := t0 σ0 (s)ds (t ∈ T ), then
lim yq (t) = y0 (t) uniformly on T .
q→∞
The theorem will be proved if we show that J ′′ (z0 ; (y0 , v0 )) ≤ 0, (y0 , v0 ) ∈ Y (z0 )
and (y0 , v0 ) 6= (0, 0).
The fact that y0 (t0 ) = y0 (t1 ) = 0 follows by Lemma 4.1b. By definition of
the functional K, for all q ∈ N,
À¾
¿
Z t1 ½
K(zq )
M (t, xq (t))
N (t, xq (t))
dt.
=
+ vq (t),
d2q
d2q
dq
t0
In view of Lemma 4.1b,
1
M (t, xq (t))
= hy0 (t), Fxx (t, x0 (t), u0 (t))y0 (t)i,
2
q→∞
dq
2
lim
N (t, xq (t))
= Fux (t, x0 (t), u0 (t))y0 (t)
q→∞
dq
lim
both uniformly on T and, since {vq } converges weakly to v0 in L1 (T ; Rm ),
Z
K(zq ) 1 t1
1 ′′
+
J (z0 ; (y0 , v0 )) = lim
hv0 (t), Fuu (t, x0 (t), u0 (t))v0 (t)idt. (5)
q→∞
2
d2q
2 t0
Let us now show that for some subsequence of {zq }, again denoted by {zq },
Z
1 t1
Ẽ(zq )
≥
lim inf
hv0 (t), Fuu (t, x0 (t), u0 (t))v0 (t)idt.
(6)
q→∞
d2q
2 t0
12
Sufficiency in optimal control ...
By Lemma 4.1a, we may choose S ⊂ T measurable such that (ẋq (t), uq (t)) →
(ẋ0 (t), u0 (t)) uniformly on S. By Taylor’s theorem, for all t ∈ S and q ∈ N,
we have
1
1
E(t, xq (t), u0 (t), uq (t)) = hvq (t), Rq (t)vq (t)i
2
dq
2
where
Rq (t) := 2
Z
1
(1 − λ)Fuu (t, xq (t), u0 (t) + λ[uq (t) − u0 (t)])dλ.
0
Clearly,
lim Rq (t) = R0 (t) := Fuu (t, x0 (t), u0 (t))
q→∞
uniformly on S.
By 2.1(i), R0 (t) ≥ 0 (t ∈ S). Moreover, by the first condition of 2.1(iii) and
Lemma 4.1c, for some subsequence of {zq }, again denoted by {zq },
Z
Ẽ(zq )
1
lim inf
≥
hv0 (t), Fuu (t, x0 (t), u0 (t))v0 (t)idt.
q→∞
d2q
2 S
Since S can be chosen to differ from T by a set of an arbitrary small measure,
and the function
t 7→ hv0 (t), Fuu (t, x0 (t), u0 (t))v0 (t)i
belongs to L1 (T ; R), this inequality holds when S = T , and this establishes
(6). Thus, by (4), (5) and (6),
J(zq ) − J(z0 )
1 ′′
Ẽ(zq )
K(zq )
+ lim inf
= lim inf
≤ 0.
J (z0 ; (y0 , v0 )) ≤ lim
2
2
q→∞
q→∞
q→∞
2
dq
dq
d2q
In addition, if (y0 , v0 ) = (0, 0), then
lim
q→∞
K(zq )
=0
d2q
and so, by the second condition of 2.1(iii),
1
Ẽ(zq )
≤0
h ≤ lim inf
q→∞
2
d2q
contradicting the positivity of h.
Finally, to show that (y0 , v0 ) ∈ Y (z0 ), observe that, by Taylor’s theorem,
for all q ∈ N,
ẏq (t) = Aq (t)yq (t) + Bq (t)vq (t) (a.e. in T )
13
Gerardo Sánchez Licea
where
Aq (t) =
Z
1
Z
1
fx (t, x0 (t) + λ[xq (t) − x0 (t)], u0 (t) + λ[uq (t) − u0 (t)])dλ,
0
Bq (t) =
fu (t, x0 (t) + λ[xq (t) − x0 (t)], u0 (t) + λ[uq (t) − u0 (t)])dλ.
0
We know by Lemma 4.1a that there exists S ⊂ T measurable such that
Aq (t) → A0 (t) := fx (t, x0 (t), u0 (t)),
Bq (t) → B0 (t) := fu (t, x0 (t), u0 (t))
both uniformly on S. Since yq (t) → y0 (t) uniformly on S and {vq } converges
weakly to v0 in L1 (S; Rm ), it follows that {ẏq } converges weakly in L1 (S; Rn ) to
A0 y0 + B0 v0 . By Lemma 4.1b, {ẏq } converges weakly in L1 (S; Rn ) to σ0 = ẏ0 .
Hence,
ẏ0 (t) = A0 (t)y0 (t) + B0 (t)v0 (t) (t ∈ S).
Since S can be chosen to differ from T by a set of an arbitrary small measure,
there cannot exist a subset of T of positive measure on which the functions y0
and v0 do not satisfy the differential equation ẏ0 (t) = A0 (t)y0 (t) + B0 (t)v0 (t).
Consequently,
ẏ0 (t) = A0 (t)y0 (t) + B0 (t)v0 (t) (a.e. in T )
and this completes the proof.
5
Proof of Lemma 4.1
(a): Observe that ϕ(c)(2 + ϕ(c)) = |c|2 (c ∈ Rm ). Then for all q ∈ N,
Z
t1
t0
|vq (t)|2
dt ≤ 1.
wq2 (t)
(7)
Thus there exist v0 ∈ L2 (T ; Rm ) and some subsequence of {zq } (we do not
relabel) such that {vq /wq } converges weakly to v0 in L2 (T ; Rm ). Let h ∈
L∞ (T ; Rm ). Note that, for all q ∈ N,
Z
t1
t0
hh(t), vq (t)idt =
Z
t1 ¿
t0
À
À
Z t1 ¿
vq (t)
vq (t)
h(t),
h(t)[wq (t) − 1],
dt +
dt.
wq (t)
wq (t)
t0
14
Sufficiency in optimal control ...
By the inequality of Schwarz and (7),
¯Z t1 ¿
À ¯2 Z t1
¯
¯
v
(t)
q
¯ ≤
¯
dt
|h(t)|2 [wq (t) − 1]2 dt.
h(t)[w
(t)
−
1],
q
¯
¯
wq (t)
t0
t0
For all q ∈ N, set
¸1/2
·
1
(a.e. in T ),
w1q (t) := 1 + ϕ(ẋq (t) − ẋ0 (t))
2
¸1/2
·
1
(t ∈ T ).
w2q (t) := 1 + ϕ(uq (t) − u0 (t))
2
2
Since for i = 1, 2, wiq
(t) ≥ wiq (t) ≥ 1 for almost all t ∈ T , we have
Z
t1
Z
t1
2
[wiq (t) − 1]dt ≤
[wiq
(t) − 1]dt
t0
t0
¾
½Z t1
Z t1
ϕ(uq (t) − u0 (t))dt = D(zq − z0 ).
≤ max
ϕ(ẋq (t) − ẋ0 (t))dt,
0 ≤
t0
t0
Thus it is readily seen that
Z t1
Z
lim
[wq (t) − 1]dt = lim
q→∞
q→∞
t0
Observe also that
Z t1
Z
2
[wq (t) − 1] dt =
t0
lim
q→∞
m
Z
[wq2 (t)
q→∞
2
t1
[wq (t) − 1]dt.
t0
|h(t)|2 [wq (t) − 1]2 dt = 0.
t0
q→∞
t0
− 1]dt − 2
Z
t1
Since L (T ; R ) ⊂ L (T ; Rm ),
Z t1
Z
lim
hh(t), vq (t)idt = lim
∞
t0
[wq2 (t) − 1]dt = 0.
t1
t0
Consequently,
t1
À
Z t1
vq (t)
dt =
hh(t), v0 (t)idt,
h(t),
wq (t)
t0
t1 ¿
t0
that is, {vq } converges weakly in L1 (T ; Rm ) to v0 .
In order to prove that {(ẋq , uq )} converges almost uniformly to (ẋ0 , u0 ) on
T , for all x ∈ X, set
kxk1 :=
Z
t1
t0
|ẋ(t)|dt,
·
¸1/2
1
w(t) := 1 + ϕ(ẋ(t))
2
(a.e in T ).
15
Gerardo Sánchez Licea
Observe first that
Z t1
Z
2
2w (t)dt = 2(t1 − t0 ) +
t0
and
Z
t1
ϕ(ẋ(t))dt = 2(t1 − t0 ) + D1 (x)
t0
t1
t0
|ẋ(t)|2
dt =
2w2 (t)
Z
t1
t0
|ẋ(t)|2
dt =
2 + ϕ(ẋ(t))
By the inequality of Schwarz,
Z
2
kxk1 ≤
t1
t0
|ẋ(t)|2
dt
2w2 (t)
Z
Z
t1
ϕ(ẋ(t))dt = D1 (x).
t0
t1
2w2 (t)dt.
t0
From these relations we have
kxk21 ≤ D1 (x)[2t1 − 2t0 + D1 (x)] ≤ D(z)[2t1 − 2t0 + D(z)] (z = (x, u)).
Consequently, kxq − x0 k1 → 0, q → ∞, and so some subsequence of {ẋq }
converges pointwisely a.e. to ẋ0 . By Egoroff’s theorem, it converges to ẋ0
almost uniformly on T .
Similarly it is readily seen that
kuk21 ≤ D2 (u)[2t1 − 2t0 + D2 (u)] ≤ D(z)[2t1 − 2t0 + D(z)] (z = (x, u))
implying that some subsequence of {uq } converges almost uniformly to u0 on
T . Thus there is some subsequence of {zq } (we do not relabel) such that
lim (ẋq (t), uq (t)) = (ẋ0 (t), u0 (t)) almost uniformly on T .
q→∞
(b): As in (a), for all q ∈ N,
Z t1
t0
|ẏq (t)|2
dt ≤ 1.
wq2 (t)
(8)
Hence there exists a function σ0 ∈ L2 (T ; Rn ) such that some subsequence of
{ẏq /wq } converges weakly in L2 (T ; Rn ) to σ0 . We conclude, by an argument
similar to that used in the proof of (a), that there exists some subsequence of
{zq } (we do not relabel) such that {ẏq } converges weakly in L1 (T ; Rn ) to σ0 .
It remains to show that yq (t) → y0 (t) uniformly on T . We have
Z t
yq (t) =
ẏq (s)ds (t ∈ T, q ∈ N),
t0
16
Sufficiency in optimal control ...
and hence
lim yq (t) = y0 (t) :=
q→∞
Z
t
σ0 (s)ds pointwisely on T .
t0
In order to prove that this convergence is uniform observe that, by (8), given
a measurable set S ⊂ T ,
¯Z
¯2 Z
Z
Z
¯
¯
|ẏq (t)|2
2
¯ ẏq (t)dt¯ ≤
dt wq (t)dt ≤
wq2 (t)dt (q ∈ N).
¯
¯
2
S
S
S
S wq (t)
Moreover
Z
S
wq2 (t)dt
= m(S) +
Z
S
[wq2 (t) − 1]dt (q ∈ N).
Given a constant ǫ > 0, choose qǫ ∈ N such that
Z t1
ǫ2
(q ≥ qǫ ).
[wq2 (t) − 1]dt <
2
t0
Choose 0 < δ < ǫ2 /2 such that
¯Z
¯
¯
¯
m(S) < δ ⇒ ¯¯ ẏq (t)dt¯¯< ǫ (q < qǫ ).
S
Note that if q ≥ qǫ , then
¯Z
¯2
Z
¯
¯
m(S) < δ ⇒ ¯¯ ẏq (t)dt¯¯ ≤ m(S) +
t0
S
and so
t1
[wq2 (t) − 1]dt <
ǫ2 ǫ2
+ = ǫ2 ,
2
2
¯Z
¯
¯
¯
m(S) < δ ⇒ ¯¯ ẏq (t)dt¯¯ < ǫ (q ∈ N).
S
Thus the sequence of functions {yq } is equicontinuous on T . Consequently,
yq (t) → y0 (t) uniformly on T .
(c): By hypothesis we may assume that, for all t ∈ S and q ∈ N,
|Rq (t) − R0 (t)|wq2 (t) ≤ 1.
Hence
Mq := sup |Rq (t) − R0 (t)|wq2 (t) < ∞
(q ∈ N).
t∈S
Using the inequality of Schwarz it is easily seen that, for all t ∈ S and q ∈ N,
|hRq (t)vq (t), vq (t)i − hR0 (t)vq (t), vq (t)i| ≤ Mq
|vq (t)|2
.
wq2 (t)
17
Gerardo Sánchez Licea
Since Rq (t) → R0 (t), and wq (t) → 1, both uniformly on S, we have Mq → 0.
Therefore, by (7),
Z
Z
lim inf hRq (t)vq (t), vq (t)idt = lim inf hR0 (t)vq (t), vq (t)idt.
q→∞
q→∞
S
S
But for all t ∈ S,
hR0 (t)vq (t), vq (t)i = hR0 (t)v0 (t), v0 (t)i + 2hvq (t) − v0 (t), R0 (t)v0 (t)i
+ hR0 (t)(vq (t) − v0 (t)), vq (t) − v0 (t)i.
Since wq (t) → 1 uniformly on S, it is readily seen (see the proof of (a)) that
there is some subsequence of {zq } (again denoted by {zq }) such that {vq }
converges weakly to v0 in L2 (S; Rm ). Since R0 v0 ∈ L2 (S; Rm ), we have
Z
lim hR0 (t)v0 (t), vq (t) − v0 (t)idt = 0.
q→∞
S
Hence
lim inf
q→∞
Z
hRq (t)vq (t), vq (t)idt =
S
Z
hR0 (t)v0 (t), v0 (t)idt
Z
+ lim inf hR0 (t)(vq (t) − v0 (t)), vq (t) − v0 (t)idt.
S
q→∞
S
Since the last term is nonnegative the result follows.
References
[1] A. A. Agrachev, G. Stefani, and P. L. Zezza, Strong optimality for a
bang-bang trajectory, SIAM J. Control Optim., 41, (2002), 991-1014.
[2] V. M. Alekseev, V.M. Tikhomirov, and S. V. Fomin, Optimal Control,
Translated from the Russian by V. M. Volosov, Contemporary Soviet
Mathematics, Consultants Bureau, New York, 1987.
[3] R. Berlanga, J. F. Rosenblueth, Jacobi’s Condition for Singular Extremals: An Extended Notion of Conjugate Points, Appl. Math. Letters,
15, (2002), 453-458.
[4] R. Berlanga, J. F. Rosenblueth, Extended conjugate points in the calculus
of variations, IMA J. Math. Control Inform., 21, (2004), 159-173.
18
Sufficiency in optimal control ...
[5] F. H. Clarke, Methods of Dynamic and Nonsmooth Optimization, CBMSNSF Regional Conference Series in Applied Mathematics 57, Society for
Industrial and Applied Mathematics, Philadelphia, 1989.
[6] G. M. Ewing, Sufficient conditions for a non-regular problem in the calculus of variations,Bull. Am. Math. Soc., 43, (1937), 371-376.
[7] G. M. Ewing, Calculus of Variations with Applications, Dover, New York,
1985.
[8] M. R. Hestenes, Sufficient conditions for the problem of Bolza in the
calculus of variations, Trans. Amer. Math. Soc., 36, (1934), 793-818.
[9] M. R. Hestenes, On sufficient conditions in the problems of Lagrange and
Bolza, Ann. of Math., 37, (1936), 543-551.
[10] M. R. Hestenes, A direct sufficiency proof for the problem of Bolza in the
calculus of variations, Trans. Amer. Math. Soc., 42, (1937), 141-154.
[11] M. R. Hestenes, Sufficient conditions for the isoperimetric problem of
Bolza in the calculus of variations, Trans. Amer. Math. Soc., 60, (1946),
93-118.
[12] M. R. Hestenes, The Weierstrass E-function in the calculus of variations,
Trans. Amer. Math. Soc., 60, (1947), 51-71.
[13] M. R. Hestenes, An indirect sufficiency proof for the problem of Bolza in
nonparametric form, Trans. Amer. Math. Soc., 62, (1947), 509-535.
[14] M. R. Hestenes, Sufficient conditions for multiple integral problems in the
calculus of variations, Amer. J. Math., 70, (1948), 239-276.
[15] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John
Wiley, New York, 1966.
[16] A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, Translated from the Russian by K. Makowski, Studies in Mathematics and its
Applications, 6, North-Holland Publishing, Amsterdam, 1979.
Gerardo Sánchez Licea
19
[17] P. D. Loewen, Second-order sufficiency criteria and local convexity for
equivalent problems in the calculus of variations, J. Math. Anal. Appl.,
146, (1990), 512-522.
[18] K. Malanowski, Sufficient optimality conditions for optimal control subject to state constraints, SIAM J. Control Optim., 35, (1997), 205-227.
[19] K. Malanowski, H. Maurer and S. Pickenhain, Second order sufficient conditions for state- constrained optimal control problems, J. Optim. Theory
Appl., 123, (2004), 595-617.
[20] H. Maurer and S. Pickenhain, Second order sufficient conditions for control
problems with mixed control-state constraints, J. Optimal Theory Appl.,
86, (1995), 649-667.
[21] H. Maurer and H. J. Oberle, Second order sufficient conditions for optimal
control problems with free final time: the Riccati approach, SIAM J.
Control Optim., 41, (2002), 380–403.
[22] D. Q. Mayne, Sufficient conditions for a control to be a strong minimum,
J. Optimal Theory Appl., 21, (1977), 339-351.
[23] E. J. McShane, Sufficient conditions for a weak relative minimum in the
problem of Bolza, Trans. Amer. Math. Soc., 52, (1942), 344-379.
[24] A.A. Milyutin and N. P. Osmolovskiı̌, Calculus of Variations and Optimal Control, Translations of Mathematical Monographs, 180, American
Mathematical Society, Providence, Rhode Island, 1998.
[25] M. Popescu, Singular normal extremals and conjugate points for Bolza
functionals, J. Optimal Theory and Appl., 23, (2007), 243-256.
[26] J. F. Rosenblueth, Variational conditions and conjugate points for the
fixed-endpoint control problem, IMA J. Math. Control Inform., 16,
(1999), 147-163.
[27] J. F. Rosenblueth, A New Notion of Conjugacy for Isoperimetric Problems, Appl. Math. Optim., 50, (2004), 209-228.
[28] J. F. Rosenblueth, Conjugacy for normal or abnormal linear Bolza problems, J. Math. Anal. Appl., 318, (2006), 444-458.
20
Sufficiency in optimal control ...
[29] J. F. Rosenblueth, A direct approach to second order conditions for mixed
equality constraints, J. Math. Anal. Appl., 333, (2007), 770-779.
[30] Maria Do Rosário de Pinho, J. F. Rosenblueth, Mixed constraints in optimal control: an implicit function theorem approach, IMA J. Math. Control Inform., 24, (2007), 197-218.
[31] J. F. Rosenblueth, G. Sánchez Licea, A direct sufficiency proof for a weak
minimum in optimal control, Appl. Math. Sci., 4(6), (2010), 253-269.
[32] J. F. Rosenblueth, G. Sánchez Licea, Sufficient variational conditions for
isoperimetric control problems, Int. Math. Forum., 6, (2011), 303-324.
[33] G. Stefani, On sufficient optimality conditions for a singular arc, Proceedings of the 42nd IEEE Conference on Decision and Control, (2003),
2746-2749
[34] G. Stefani, Minimum-time optimality of a singular arc: second order sufficient conditions, Proceedings of the 43rd IEEE Conference on Decision
and Control, (2004), 450-454.
[35] G. Stefani and P. L. Zezza, Optimality conditions for a constrained optimal control problem, SIAM J. Control Optim., 34, (1996), 635-659.