is subdifferentiable at x(t) and the condition -p (t)~ ~B (x (t)) is satisfied. 6. The concept of a generalized solution is not too convenient for work, in view of its.
CONDITIONS NECESSARY FOR OPTIMALITY WITH PHASE CONSTRAINTS
IN THE CONVEX BOLZA PROBLEM
OF A GENERAL FORM
S. Yu. Yakovenko
UDC 519.95
i. Let Y: [0, T] × R n × R n ~ R I U { + ~ } b e a normal convex integrand in the sense of Rockafellar [I]: the function F(t, x, v) be convex and semicontinuous in the collection of its vector arguments and, in a measurable fashion, depend on a scalar. By the convex Bolza problem with the left end fixed at the point a ~ R n and linear terminal term b ~ R n* is meant the extremal problem of the form T
I F (t, x (0, J (0) dt + ~ i~f,
x (0) = a,
(1)
0
where the extremum is found in the class A of vector-functions a b s o l u t e l y c o n t i n u o u s on [ 0 , { ~ p , v > - - F ( t , x , v ) : v ~ R ~*} T]. By t h e H e m i l t o n i a n o f p r o b l e m ( 1 ) i s m e a n t a f u n c t i o n H ( t , x , p ) = s u p convex in p and concave in x with values in RI[I{~ ~}. 2. Proposition [2]. Let the pair of vector-functions (x(.),p(.))~A × A everywhere on [0, T] the system of Hamiltonian subdifferential inclusions ~(t) ~ ~ H ( t , x ( t ) , p (t)),
and t h e b o u n d a r y c o n d i t i o n s
satisfy almost
(2)
~(t)~ ~ (--H)(t,x(t),p(t))
x(O) = a, p ( T ) = --b.
Then the trajectory x(.) achieves a minimum in problem (i). 3. In order to formulate conditions necessary for optimality, let us introduce some notation. By the symbol L> let us denote the Banach space of vector-functions on the interval [0, T], each coordinate of which belongs to LP = LP(0, T). By X~ let us denote the set of trajectories satisfying the implicitly given phase constraints: X; = {x(.)~L~: 3v(.) ~ L I, ~ ( . ) ~ L~: F ( t , x ( t ) , v ( t ) ) < ~(t)}. Further, let / E (L~)* be an arbitrary functional. By the H e w i t t - Y o s i d a theorem [3], there exists a decomposition f = r + s, where r is an integral functional with vector density q(.) ~ L ~ , and s is a purely singular functional on L ~ . With each such functional f, there is connected a vector function of the distribution p(.): by definition,Vy~Rn =/[y.~[t.T](.)], where ~ (-) ~ L ~ is the characteristic function of the interval I. Relying on an analogy with the regular (s = 0) case, we will write q(t) = regp(t). 4. Definition. A p a i r of measurable vector-functions (x(-), p(-)) is c a l l e d a g e n e r a l i z e d solution of the Hamiltonian system (2) on the interval [0, T] if: i) x(.) ~ A , p(') has bounded variation on [0, T]; 2) there exists a functional / ~ (L=)* for which the function p(') + p(T) is a distribution function; 3) the singular component functional f achieves its minimum on XF on the trajectory x(.); 4) almost everywhere on [0, T] the subdifferential inclusion ~(t) ~H (t,x(~,p(t)), regp(t)~ ~ (--~)(t,x(t),p(~) is satisfied. Remark. If the trajectory x(-) belongs to the interior of the set XF , then, as follows from condition 3), s = 0 and the generalized solution is typical. 5. THEOREM i. Let the trajectory x(.) ~ A achieve a minimum in assume that there exists some other trajectory x0(.) ~ A such that for all y ( . ) ~ L ~ such that ;IY(')I;~ 0 and C < ~ for x0(t) + y(t), x0(t)) condition). Then
i) there exists a vector-function p(.) such that the pair (x('), p(-)) is a generalized solutfon of system (2) with boundary condition x(0) = a, p(T) = -b.
Institute of Control Problems. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. I, pp. 92-93, January-March, 1990. Original article submitted December 9, 1988.
82
0016-2663/90/2401-0082512.50
© 1990 Plenum Publishing Corporation
2) for each
t ~ [0, T] the Bellman function T
Bt(X)=y(.)inf{! F(s,y(s),~(s))ds..~-: is subdifferentiable
g(.)~A,
g(t)=x}
at x(t) and the condition -p (t)~ ~B ~ (x (t)) is satisfied.
6. The concept of a generalized solution is not too convenient for work, in view of its complexity. Nevertheless, for such solutions, one succeeds in establishing an analogue of the Lyapunov property. Let us consider on the Cartesian square @ = (R~× Rn*)2 of the phase space of system (2), a smooth function S (xl, pi, x~. p 2 ) = . By its upper derivative due to system (2), we mean the function NS: [0, T ] × ~ - ~ R ~U{-oo};DS(t,x~,p~,x 2,p2) =sup{ ÷ } [the upper bound is taken over a l l ~ ~ # ; H (t, xl, p~), qi ~ ~x (--If) (t, xi, p2, t=l,2, considering that sup f;~ = - - ~ ]. It is easy to show that the upper derivative is e v e r y w h e r e nonpositive. Therefore, for any pair w~ (.) = (xi (-),p~ (.))~ A ~ of regular solutions of (2), the result of substituting S (t, ~(t), w~ (t)) is an (absolutely continuous) nonincreasing function of time t on the interval [0, T]. If one moves from the regular to the generalized solutions of (2), then the function S(t) computed on a pair of such solutions now ceases to be absolutely continuous. However, the assertion regarding monotonicity remains valid: we have THEOREM 2. Let wi(.) = (xi(.) , Pi(')), i = I, 2 be a pair of generalized system (2) on the interval [0, T]. Then for any t~< t2, t~~ [0, T],
solutions of
t2 Q
7. Theorem 2 describes, in an oblique fashion, the R a d o n - N i k o d y m derivative for a singular component functional f, that is, a finitely additive measure dp(t). In [4] the analog of Theorem 1 was proved for the Bolza problem with a Lagrangian F(t, x, v) + indx(t)(x) such that X F = L ~, and the second addend is the indicator function of the closed convex subset X (t)~ R ~. In this paper, it was shown that the R a d o n - N i k o d y m derivative of the measure dp(t) at the points of its singularity [for x(t)=~ 0~~ (t) ] is directed along the normal to X(t). Hence, one can also deduce inequality (3). 8. Theorem 2 allows one to apply the technique of Lyapunov functions (see [5, 6]) to studying generalized solutions of system (2), and to obtain in such a manner, results describing the asymptotics of the extremals in problem (i) without additional assumptions on the character of the phase constraints. 9. The generalization of Theorem 2 holds in the case of systems with a bihomogeneous Hamiltonian: the role of the Lyapunov function, in this case, is played by the expression
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