Appl Math Optim 30:1-14 (1994)
_ ptlied .
lilaiktmad
Optimindim
9 1994Springer-VerlagNew York Inc.
Second-Order Necessary Conditions for Differential
Inclusion Problems
H. Zheng
Department of Mathematics, University of British Columbia,
Vancouver, British Columbia, Canada V6T 1Y4
Communicated by F. H. Clarke
Abstract. We study second-order necessary conditions for optimality in the
unbounded differential inclusion control problem and recover the accessory
problem in optimal control theory.
Key Words. Optimal control, Unbounded differential inclusion, Secondorder necessary condition, Second variation.
AMS Classification. 49K24.
1. Introduction
Consider the following differential inclusion control problem:
(Po) minimize 9(x(T))
subject to
x'(t) ~ F(t, x(t)),
x(O) ~ Co,
x(T) ~ C 1.
First-order necessary conditions for an arc ~ to be optimal in (Po) are well known
(see [4,1 and [7-1). Second-order necessary conditions may be developed in different
ways such as variations of extremals for the classical Lagrangian problem, the
implicit function theorem for the optimal control problem [11], etc. Since the
second-order theory in finite-dimensional optimization has been well developed
by Rockafellar and others (see I-2-1 and [10]), another possible way to get
second-order conditions in (Po) is to reduce it to a finite-dimensional minimization
problem and then apply the known optimality conditions, for example, those in
[2,1 and [10]. We follow this approach here.
2
H. Zheng
Problem (Po) is equivalent to minimizing the extended real-valued function
G(x) over all x in R", where G is defined by
G(x) :=
g(x) + %~(.(x) +
~c~(X).
Here R(T) is the reachable set at time T to the differential inclusion (see [1])
(x'(t) ~ F(t, x(t))
(D)
a.e.
t E [0, T],
<ix(O)e Co,
and ~Ps(X) denotes the indicator of a subset S in R", defined by
Us(X) = 0
when
xeS
and
~ds(X) = + oe
when
x r S.
If an arc s solves problem (PD), then its endpoint s
minimizes G. Hence
the optimality conditions in [2] tell us that some generalized directional derivatives
of G at ~(T) must be nonnegative. To express these derivatives in terms of those
of g, WR(T), and qJc,, we must check a constraint qualification, so we are naturally
led to study the Clarke normal cone to the reachable set R(T) at the optimal
endpoint. It seems there is no easy way to do this.
In this paper we first study the Clarke normal cone to the reachable set via
proximal analysis and get necessary conditions for optimality in (PD) with bounded
differential inclusions. Then we discuss unbounded differential inclusions with the
reduction method in [8]. Finally we apply the results to the optimal control
problem and recover the accessory problem in [12].
2. Background Material
Let S be a closed subset of R" and let F be a set-valued function from R" to R".
Given s s S and (x, u) E gph V, recall the following definitions:
9 The proximal normal cone to S at s given by
eNs(s):= {~ e R"[3M > O, s.t. (~, g - s) <_ Mlg - sl 2, W e S } .
9 The limiting proximal normal cone to S at s is given by
LNs(s) = {~ = k~lim ~kI~kePNs(Sk), Sk S--->S}.
9 The Clarke normal cone to S at s is given by [4, Theorem 2.5.6]
Ns(s ) = 6-6 LNs(s).
9 The derivative cone to S at s is given by
Ds(s)'.-- {~R"lVtkJ, O, 3r ~ ~, s.t. s + tk~k~S }.
Second-Order NecessaryConditions for DifferentialInclusion Problems
3
9 The second-order derivative set to S at s relative to ~ ~ Ds(s) is
D2(s, 0 . ' = {~o~ R"lVtk $ O, 30k ~ ~, s.t. S + tkr + t2COke S}.
9 The first-order derivative of F at (x, u) is a set-valued function dF(x, u),
defined by
gph dF(x, u) = Dgphr(X, u).
9 The second-order derivative of F at (x, u) relative to (y, v) ~ Dgphr(X, u) is a
set-valued function dE['(x, u; y, v), defined by
gph d2F(x, u; y, v) = Dgphr(X
2
, U; y, V).
The first three concepts are described in 1-4] and 1,5]; the last four are discussed
in I-2] under the name of adjacent cone and adjacent derivative. The term
"derivative cone" was first used by Rockafellar I-9]. The main results in this paper
are based on the following necessary conditions for general optimization problems
in finite-dimensional spaces. In the statement we use the notation
Dig(x, y) = lim inf
g(x + hy') - g(x)
h
h,~ O , y " ~ y
D~g(x, y, z) = lira inf
h ~o,z, ~z
g(x + hy + hZz ') - g(x) - hO t g(x, y)
h2
Notice that when g is of class C 2,
D~ g(x, y) = g'(x)~y,
O~g(x, y, z) = g'(x)Tz + 89
Theorem 1. Let g: R" ~ R be Lipsehitzian in some open set containing s and let
Sa, $2 be nonempty closed subsets of R" containing ~. I f ~c solves the following
minimization problem,
(P)
minimize g(x)
over all x ~ $1 c~ $2,
and also satisfies the constraint qualification
Ns,(~) n (-Us2(X)) = {0},
then we have the first-order necessary condition:
D t g(2~, y) > O,
Vy E Ds,(22) c~ Ds2(~).
Furthermore, if equality holds for some ~, then we have the second-order necessary
condition:
D~g(~, y,^ z) > O,
2
^
^
2
^
Vz ~ DsI(X,
y)
~ Vs,(X,
~).
4
H. Zheng
Proof.
We apply some results in [2]. Let
6(x) := o(x) + 'esl(X) + ~es2(X).
Then 92 is a minimum point of G, so by the Fermat rule [2, Theorem 6.1.9] we
have D, G(92,y) _> 0 for all y in R". Since 9 is Lipschitzian around 92 and 92 satisfies
the constraint qualification, by Theorem 6.3.1, Proposition 6.2.4, and Corollary
4.3.5 of [2], we have
D TG(2, y) = D~g(s y) + ~FD~r162) + ~Fos2(~)(y).
The first-order necessary condition follows.
The second-order condition is obtained by combining Proposition 6.6.3 and
Theorems 6.6.2 and 4.7.4 from [2].
[]
3. Hypotheses
A tube about an arc x is a relatively open subset fl of [0, T] x R" containing the
graph of x. An arc x is admissible for (PD) if it is a trajectory of F and satisfies
both end constraints. An admissible arc x is a local solution to (PD) if there is some
tube ~ about x such that any other admissible arc y lying in f~ satisfies
g(y(T)) > g(x(T)).
From now on let 2 be a given local solution to (Po) relative to a tube f~. The
following hypotheses about (PD) are assumed throughout the paper:
(H1) On the tube f~ the multifunction F has nonempty, compact, convex
images. A nonnegative integrable function k exists such that
F(t, x) c k(t) cl B,
F(t, y) c_ F(t, x) +
V(t, x)~ f/;
k(t)ly - xl
cl B,
v(t, x), (t, y)e f~,
where B is the open unit ball in R".
(H2) The multifunction F is measurable with respect to the a-field 5 ~ x
generated by products of Lebesgue subsets of [0, T] with Borel subsets
of R".
(H3) The integrable function k above can be chosen so that the following
additional condition is satisfied: for every arc y satisfying
(y(t), y'(t)) E DgphV(t .)(Yc(t), Yc'(t)),
a constant ~o > 0 exists such that
d(~'(t) + ~y'(t), V(t, 2(0 + o~y(t))) <_ ct2k(t)
for all 0 < ~ < "o.
(H4) The subsets C o, C 1 of R" are nonempty and closed.
(H5) The function g: R" --* R is Lipschitzian near ~(T).
Second-Order NecessaryConditionsfor DifferentialInclusion Problems
5
Here the assumption (H3) is used in deriving a second-order approximation
to the reachable set. In some sense it means "the derivative of F " is Lipschitz on
the tube f~, as we will see later.
Remark. Without loss of generality, we may suppose that the domain of F is ft.
Otherwise we could define a new multifunction ff by restricting F to f~. The arc
would remain a local solution to the associated problem (PD), for which we are
about to provide necessary conditions. Since these necessary conditions involve
only the local behavior of P along ~, their conclusions for/V are identical to the
desired results for F. Similarly, we may suppose that the sets Co and C 1 are
compact.
4.
The Clarke Normal Cone to the Reachable Set
Under the above assumptions, we know the reachable set R(T) is a closed set [5].
In order to apply Theorem 1 to the function G described in the introduction, we
must first verify the constraint qualification. This requires that we describe the
Clarke normal cone to the reachable set R(T) at if(T).
Definition. A trajectory x for (D) is said to be proper if, for any sequence Xk of
trajectories for (D) having the property that xk(T) ~ x(T), there is a subsequence
along which IIxk - xllo~ ~ 0.
Let us denote by H the Hamiltonian associated with the multifunction F,
that is, H(t, x, p ) = max{(v, p)lv e F(t, x)}. We know H(t,., .) is locally Lipschitz
[4, Proposition 3.2.4], and denote by 8H(t, x, p) the Clarke subgradient set of
H(t,., .) with respect to (x, p).
Theorem 2.
Let 2 be a local solution for (PD); suppose 2 is proper. Then
NRtT)(Yc(T)) ~_ 6-6 Q(Yc),
where
Q(x) = {p(T)](-p', x') e t~H(t, x, p), p(O) 9 LNco(x(O)) }.
(1)
Proof Let x be a trajectory of (D), and ~ 9 PNRtT)(X(T)). Then a constant M > 0
exists, such that x(T) minimizes the smooth function ( - ~, y(T)) + Mry(T) - x(T)12
over all y(T)~R(T). According to Theorem 1.2 of I-8] (see also Theorem 4.1,
footnote, of [5]), this implies that an arc p satisfying the Hamiltonian inclusion
and initial condition defining Q(x) and - p ( T ) = --4 exists, so ~ = p(T) 9 Q(x).
Since r is arbitrary, this shows that PNRtT)(X(T)) ~ Q(x).
It follows that, for any ~ in LNR(T)(X(T)) , there must be a sequence of
trajectories x k and a sequence of adjoint arcs Pk such that
Xk(T) ~ Yc(T),
pk( T) ~
6
H. Zheng
with
( - P'k, xl) e ~?H(t, x k, Pk),
Pk(O)~ LNco(Xk(O))"
Since 2 is proper, there is a subsequence of {Xk} along which x k uniformly converges
to 2. Now H is locally Lipschitzian, so IP~(t)l < kl(t), ]x~(t)] < k~(t) almost everywhere for some integrable function ka. By Theorem 3.1.7 of [4], applied to F = 8H,
a further subsequence Pk uniformly converging to an adjoint arc p exists. Since
the graph of LNco is closed, we also have p(0) ~ LNco(2(O)), that is, p(T) ~ Q(2). So
LNR(r)(2(T)) ~_ Q(2), as required.
[]
5.
Approximation to the Reachable Set
In order to apply Theorem 1 to problem (PD), we also need to know the derivative
cone and the second-order derivative set to the reachable set R(T) at 2(T). The
first-order approximation has been discussed thoroughly in [6], where the following result is given.
Theorem 3.
Let q(T) be the reachable set of the following system:
y'(t) e dF(t, 2(0, 2'(t))y(t)
a.e.
t e [0, T],
(2)
y(O) ~ Dco(2(O)).
Then we have
q(T) ~_ DR(T)(2(T)).
Now we turn to the second-order derivative set to R(T) at 2(T) relative to
y(T) e q(T).
Theorem 4. Let the arc ~ satisfy (2), and let r2(T ) denote the reachable set of the
following system:
z'(t) e dZF(t, 2(0, 2'(t); y(t), y(t))z(t)
a.e.
t6[O,T],
z(0) e D20(2(0); 9(0)).
Then we have
r2(T ) ~_ D2tT)(2(T), yI(T)).
Proof Take any z(T)e r2(T), and any sequence tk > 0 converging to 0. Since
z(0) E D~o(2(0), p(0)), by the definition of the second-order derivative set there must
be a sequence s k ---,z(O), such that 2(0)+ tkY(O) + t~Ske Co. Let zk(t):= 2(t)+
Second-Order Necessary Conditions for Differential Inclusion Problems
7
tkY(t) + t~z(t) + t~(s k --Z(0)). Then z k is an arc on [0, T] and Zk(O) 9 C o. F o r k
sufficiently large, we have the estimate
d(Z'k(t), F(t, zk(t)) )
t2
<-Iz'(01
d(Yc'(t) + tkf/(t), F(t, 2(0 + tkf~(t)))
+
q
+ k(t)(Iz(t)t + Isk -
z(0)l)
(3)
< ]z'(t)l + k(t) + k(t)(lz(t)[ + 1).
(The last inequality relies on assumption (H3).) Fix t 9 [0, T] outside the measure
zero set implicitly specified below. Since
z'(t) 9 d2F(t, 2(t), 2'(t); 37(0, y'(t))z(t)
for the same t, + 0, a sequence {(~k, ~k)} which converges to (z(t), z'(t)) exists such
that
Yc'(t) + tkf/(t ) + t2(k 9 F(t, Yc(t) + tkf~(t) + t2~,).
Therefore
d(Z'k(t), F(t, zk(t)) )
t~
-< I~k -- z'(t)[ + k(t)(I ~k -- z(t)] + ]Sk -- z(O)[)
~0
as
k -4 oo.
U p o n writing pk:= ~T d(Z'k(t), F(t, Zk(t)) ) dt, we have, by (3) and the dominated
convergence theorem, that
lira P~ = O.
k-~oo t~
By Filippov's theorem [4, T h e o r e m 3.1.6], arcs x k on [0, T] and a constant c _> 0
exist such that the Xk are trajectories o f f with Xk(O) = Zk(O) 9 Co and [Ixk - Zk][o~ <
Cpk. Each endpoint Xk(T) lies in R(T) and we have
Xk(T ) -- ~(T) -- tkY(T )
Xk(T) -- Zk(T )
q
q
-~ z(T)
as
+ z ( T ) + s, - z(O)
k ~ oo.
So we must have z(T) 9 D~(r)(,2(T), f~(T)), as required.
[]
8
H. Zheng
6. Second-Order Necessary Condition for (PD)
Now we are ready to state the necessary conditions for problem (PD)"
Theorem 5. Let 92be a local solution to (Po) and satisfy the constraint qualification
(CQ)
~-6 Q(2) c~ (-Nc,(92(T))) = {0}.
Then we have the first-order necessary condition
D,g(Yc(T), y(T)) > O,
Vy(r) e rx(T) c~ Dc,(92(T)).
Furthermore, if equality holds for some f~(T), then we have the second-order necessary
condition
D~g(92(T), ~(T), z(r)) > O,
Vz(T) 9 rz(r) c~ D~,(2(T), p(W)).
Proof
If 92 is proper, the result follows directly from Theorem 1 and the results
above. Theorem 4 assures that assumption (CQ) implies the constraint qualification of Theorem 1, while the estimates of certain derivative sets in terms of rl(T)
and r2(T) are described in Theorems 3 and 4.
If 2 is not proper, instead of problem (PD) we may study the auxiliary problem
(Po) with state variable ~ = (x o, x)e R "+ 1 and data O(Yc)= x'~ + g(x), Go = {0} x Co,
C1 = [ - 1 , 1] x C1, if(t, )2) = {Ix - x(t)l '~} x F(t, x).
It is easy to see that ~ = (0, 92) is a local solution for problem (PD). We
show it is also proper: Let 2 k be a sequence of trajectories for the differential
inclusion (2) having the property J2,(T) -* ~(T), that is, So
r Ix,(t) - 92(014 dt ~ O, and
x,(T) -~ 92(T). The sequence x k is uniformly bounded and equicontinuous, so it has
a subsequence converging uniformly to some limit ~, but the integral condition
above forces )2 = 92, so ~ is proper. Therefore N~(7~(Tc(T))~_ d-6 O(Tc). Here 0 is
expressed by (1) and the extended Hamiltonian H is given by
n(t, Yc, ~) -= PolX - 92(014 + H(t, x, p).
The function H does not depend on Xo, so Po is a constant, and we have (~(}) c_
R x Q(92). Our hypothesis ~ Q(2) c~ (-Nc,(2(T))) = {0} implies that g-d (~(~) c~
(--Nc,(Yc)) = {0}. By Theorem 1, we have
DTO(~(T), p(T)) > O,
Vy(T) e fl(T) c~ Del(Yc(T)).
Furthermore, if equality holds for some ~(T), then we have
D~O(~c(T), ~(T), ~(T)) >_ O,
V~(T) e ~2(T) c~ D~,(I(T), ~(T)).
It is easy to verify that if y(T) e rl(T) (or z(T) e r2(T)), then (0, y(T)) ~ ~I(T) (or
(0, z(T)) ~ ~2(T)). Substituting these into the above expressions yields the required
result.
[]
The first-order necessary conditions implicit in Theorem 5 are not directly
comparable with those in the literature [4], [7] because they are in primal rather
Second-Order NecessaryConditionsfor DifferentialInclusion Problems
9
than dual form. In other words, they assert the nonexistence of admissible descent
directions for the objective function directly, instead of passing to a dual description involving an adjoint arc p('). I n Section 8 below we make explicit the
relationship between the necessary conditions of Theorem 5 and Pontryagin's
maximum principle for a problem in which F(t, x) = f(t, x, U) arises from a model
in optimal control.
7.
Unbounded Differential Inclusions
To get optimality conditions for unbounded differential inclusions, we first quote
a definition in [8]:
Definition. Let F: ~) ~ R" be a multifunction with closed values, and suppose F
is s x ~ measurable on ~. Consider a point (~, ~) in ~.
(a) The multifunction F is called sub-Lipschitzian at (~, ~) if, for every constant
p ___0, constants e > 0 and a ___0 exist such that
F(t,y) ~ p cl B ~ F(t, x) + a l Y - x l el B
for all t ~ (~ - e, ~ + e) n [0, T] and all x, y in ~ + eB.
(b) The multifunction F is called integrably sub-Lipschitzian in the large at
(~', ~) if constants e > 0 and fl > 0, together with a nonnegative function
a(t) integrable on (? - e, ~ + e), exist such that
F(t, y) n p el B ~ F(t, x) + (~(t) + flP) IY - x l el B
for all t ~ (~ - e, ~ + e) n [0, T], all x, y in 2 + eB, and all p _> 0.
Now for the differential inclusion control problem (PD) we retain hypotheses
(H2)-(H5) and replace (H1) with the following hypothesis:
(H6) The multifunction F has nonempty, closed, convex images on ~2, and
one of the following two conditions holds:
(a) The arc ~ is Lipschitzian, and the multifunction F is sub-Lipschitz
at every point (t, 2(0) in gph 2.
(b) The multifunction F is integrably sub-Lipschitzian in the large at
every point (t, 2(0) in gph 2.
We have the following optimality conditions for problem (PD):
Theorem 6.
Under hypotheses (H2)-(H6), the conclusions of Theorem 5 still hold.
Proof By Proposition 2.4 of [8] integrable functions R and q~ on [0, T] and a
relatively open subset ~ of fl containing the graph of ~ on which the truncated
multifunction/~ defined by
if(t, x):= F(t, x) n (2'(t) + R(t) el B)
10
H. Zheng
satisfies the hypothesis (H1) on ~ exist. The arc 2 is also a local solution for the
new problem (~'D), SO the conclusions of Theorem 5 can be applied to (PD) with
the corresponding constraint qualification and ~I(T) and ~2(T).
From the proof of Proposition 2.4 of [8] we know a constant Ro > 0 exists
such that R(t) > Ro for all t 9 [0, T]. Fixing t 9 [0, T] outside of a measure zero
set and taking a small neighborhood of (2(0, 2'(t)), we see immediately that
t1 9 F(t, ~) r
tl 9 F(t, ~)
whenever 119 2'(t) + RoB. So the graph of F and that of P are the same near the
point (2(0, 2'(0). Since the derivative of F at (2(t), 2'(0) is only related to the local
behavior of the graph of F at that point, we have
dF(t, 2(0, 2'(t)) = dff(t, 2(t), 2!(0).
So fl(T) = rx(T). Similarly we have ~2(T) = r2(T).
Finally, note that any ( - p ' , 2') 9 O/~(t, 2, p) implies that (p, 2') > (p, v) for
all v 9 F(t, 2). Formula (3.4) in [8] tells us that
~?fl(t, 2, p) c_ OH(t, 2, p).
So we have ~6 Q(~)___ ~6 Q(2). We get the desired result by putting all these
together.
[]
Remark. If the multifunction F satisfies assumption (H1), it will satisfy assumption (H6) just by taking ~ = k and fl = 0. Thus Theorem 6 is a true extension of
Theorem 5.
8. Application to Optimal Control Problems
Theorem 6 is the second-order necessary condition for a general multifunction F.
It can be used to study many problems, such as closed-loop control systems,
implicit dynamical systems, and so on (see [6]). Here we just discuss its application
to the optimal control problem below:
(Po)
minimize 9(x(T))
subject to
x'(t) = f(t, x(t), u(t)),
x(0) 9 C o,
x(T) 9 C1,
where u(t) is a measurable selection of U. Denote the reachable set at time T of
the control system (Pc) by/~(T), that is,
/~(Z) .'= {x(T)lx'(t) = f(t, x(t), u(t)), x(O) 9 Co, u(t) 9 U}.
Let f~ be a tube about an arc 2. Assume that U is nonempty and closed, and the
function f : f~ x U ~ R" satisfies the following conditions:
Second-Order NecessaryConditionsfor DifferentialInclusionProblems
11
9 The function f ( ' , x, u) is measurable on [0, T].
9 The function f ( t , . , ") is twice continuously differentiable, and an integrable
function k exists such that, at any point (t, x, u)~ ~ • U, we have ]f~[,
ILl--- k(t).
Consider the multifunction F(t, x):= f(t, x, U). Clearly, F satisfies assumptions (H2) and (H6)(b), and F has nonempty images on f~. Here we explicitly
suppose that F(t, x) is convex for all (t, x)~ f~ (this is a natural requirement for
the existence of optimal solutions, see [5]), that F(t, x) is closed (if U is compact,
this automatically holds), and that F satisfies (1-I3) (see Theorem 1 for a condition
under which (H3) holds).
Let Pl(T) be the reachable set of the following system:
Theorem 7.
y'(t) = fx(t)y(t) + f,(t)v(t)
a.e. t,
y(0) s OCo(~(0)),
vEL~
and
v(t)eDv(f~(t)).
Then we have _
5(T) ~ D~(,~(~(T)).
Here f~(t) denotes f'~(t, 2(t), f~(t)), and fu(t) is defined similarly.
Proof. According to Theorem 3, it suffices to show that, for fixed t ~ [0, T] outside
some measure zero set, any ct ~ R", v e Dv(f~(t)), we have
f~(t)a + f,(t)v ~ dF(t, Y~(t), ~'(t))a.
This is equivalent to, Vt k ,~ 0, 3~k -~ ~, (k --~ fx(t) O~"~- fu(t) v, Uk ~ U, such that
f(t, ~(t), ft(t)) + tk(k = f(t, YC(t)+ tk~k, Uk).
NOW we prove the latter statement. By assumption v ~ Fv(~(t)), so a sequence Vk ~ V
exists such that UR:= fi(t) + tkV k E U. Let ~k "~- 0~and
(k = (f(t, YC(t) + tka, Uk) -- f(t, Yc(t), ft(t)))/tk.
Using a first-order Taylor's expansion, we see immediately that (k ~ f,(t)a + f,(t)v.
[]
Theorem 8.
Let ~( T) ~ Pl ( T) be as described in Theorem 7 with control ~, and define
H(t) = 89
fl(t)) + fxu(t)(f~(t), ~(t)) + 89
~(t)).
Let P2(T) be the reachable set of the following control system:
z'(t) = ff ,(t)z(t) + ff~(t)w(t) + H(t)
a.e.
z(O) ~ D2o(s ~(0)),
w ~ L ~176 T] and w(t) ~ D~(fi(t), ~(t)).
t ~ [0, T],
12
H. Zheng
Then we have
P2(T) Z D~m02(T), .~(T)).
Proof
[]
Similar to Theorem 7.
We can now state a second-order necessary condition for the optimal control
problem (Pc). This result follows from Theorems 1, 2, 7, and 8.
Theorem 9.
Let 2 be a local solution for (Pc) with the corresponding control f~. I f
,2 satisfies the constraint qualification
c--6 Q(~) n (-Ncl(Yc(T))) = {0},
then we have
D,g(2(T), y(T)) >_ O,
Vy(T) e P~(T) n Dc,(2(T)).
Furthermore, if equality holds for some ~(T), then we have
D~g(~(T), f;(T), z(T)) >_ O,
Vz(T) 6 Pz(T)
n
Dc2,(2(T), y(T)).
Recall that the set Q(2) is defined in (1). In particular, the constraint qualification in Theorem 9 is expressed in terms of a Hamiltonian inclusion.
Recall [12] that the admissible pair (2, ~i) is said to be normal if the only
solution of the following system (.) is p -- 0:
- i f ( t ) = fx(t)rp(t),
p(0) e Nco(2(0)),
- p ( T ) e Nc~(2(T)),
(,)
max (p(t), f (t, ~2(t),u) ) = (p(t), f (t, 2(0, ~i(t))).
The final result in this paper shows how our constraint qualification (CQ) can be
replaced by this normality condition. It shows that Theorem 9 accurately recovers
the admissible directions and the accessory problem in Theorem 2.2 of [12].
Theorem 10. Suppose the optimal pair (2, fi) is normal and the control set U is
convex. Then the necessary conditions of Theorem 9 hold regardless of (CQ). In
particular, if the function g is twice continuously differentiable near the point ~(T),
and the arc y is a trajectory of the following system for some function v E L ~ [0, T],
y'(t) = fx(t)y(t) + fu(t)v(t),
y(O) e Dco(2(0)),
y(T) ~ Dc~(2(T)),
g'(~(T))ry(T) = O,
v(t) ~ U -- ~(t),
(t)
Second-Order NecessaryConditionsfor DifferentialInclusion Problems
13
then we have
g'(Yc(T))rz(T) + 89
2 0
for any arc z satisfying the following equation:
z'(t) = fx(t)z(t) + H(t),
z(O) e O~0(~(0), y(O)),
(~)
z(T) e D2~(2(T), y(T)).
Proof
Since U is a convex set, for all 0 < ~ < 1, we have fi(t) + ~v(t)~ U and
d(2'(t) + ey'(t), F(t, 2(0 + ey(t)))
<- If(t, 2(0 + ey(t), ~(t) + cw(t)) - 2'(t) -- ey'(t)]
~2k(t).
So assumption (H3) holds in this case.
Following the trick in [3], we want to show that if (2, fi) is normal, then our
constraint qualification (CQ) holds for the new differential inclusion control
problem (Po) with state variable 2 = (Xo, x) in R" + ~ and data 0(~) = g(x), if(t, s =
~ { ( l u - f i ( t ) l 4, f ( t , x , u ) ) : u ~ U } ,
~ o = { 0 } x Co, and C~ = [ - 1 , 1 ] x C~. If
Yc = (Xo, x) is admissible for (Po), by the closedness and convexity of f(t, x, U) and
the measurable selection theorem, a control u exists such that (x, u) is admissible
for (Pc)- Since (2, fi) is a local solution for (Pc), we have 0(~(T))> #(2(T)). So
= (0, 2) is a local solution for (PD). The extended Hamiltonian is
/7(t, 2,/3) = max{polU - ~i(t)[* + (p, f(t, x, u))}.
uEU
By Theorem 2.8.6 of [4] its generalized gradient ~3/7(t,~, p) is the convex hull of
the vectors (0, fx,(t, 2, fi)Tp, [fi __ f~(t)[4, f(t, 2, fi)) over all fi such that
max{po[u -- fi(t)[4 + (p, f(t, 2, u))} = polfi - O(t)[4 + (P, f(t, 2, ~)).
u~U
Now consider a Hamiltonian adjoint arc (Po, P) associated with }. /7 does not
depend on Xo, so Po is a constant. Since (0, - p ' , 0, 2') is in ~3/7, the only fi would
be fi(t). So Q(}) is the set of all (po(T), p(T)) such that po(t) is a constant, p satisfies
-if(t) = fx(t)Tp(t), p(O) ~ Nc0(2(0)), and
max{polu -- ~(t)l 4 + (p(t), f(t, 2(t), u))} = (p(t), f(t, 2(0, ~(t))).
u~U
It is easy to show that 0(~) is a closed convex set.
Now any/3(T) ~ 0(~) n (-Nc~(~c(T)) must have Po = 0, so p satisfies the system
(,). By the normality of (2, fi) we have p = 0. So the constraint qualification for
(PD) holds. It is easy to show that if y(T) E Pl(T) (or z(T) e P2(T)), then we have
(0, y(T)) ~ rl(T) (or (0, z(T)) ~ f2(T)); also note that 0 e D2(ft(t), v(t)). By applying
Theorem 5 we get the required conclusion.
[]
14
H. Zheng
To complete the comparison between Theorem 10 and Theorem 2.2 of [12],
let C O = {Xo} and C 1 = {x e R"lqb(x)= 0}. Then by Propositions 4.3.7 and 4.7.5
of [2] the endpoint constraints become y ( 0 ) = 0, ~'(2(T))y(T)= 0 in (t) and
z(O) = O, ~'(2(T))z(T) + 89
y(T)) = 0 in (~). So (t) is equivalent to (2.3),
(2.4), and (2.5) in [12]. If p is an adjoint arc [12, Theorem 2.1] with 2 o = 1,
that is, - i f ( t ) = fx(t)rp(t), p(T) = 9'(2(T)) + qY(2(T))rV, then (p(t)rz(t)) ' = p(t)rH(t).
These observations allow us to rewrite the central inequality of Theorem 10 as
0 <_ 89
= 89
+ p(T)rz(T) - ~TO)'(2(T))z(T)
+ ~T~)"(2(T))y(T) + f o P(t)rH(t) dt.
Thus Theorem 10 agrees with Theorem 2.2 of [12].
Acknowledgment
I would like to thank Professor Locwen for his advice and encouragement during the preparation of
this paper.
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Accepted 15 February 1993