SENSITIVITY AND CONTROLLABILITY OF SYSTEMS GOVERNED BY INTEGRAL EQUATIONS VIA PROXIMAL ANALYSIS A. YEZZA

CANADIAN JOURNAL OF MATHEMATICS JOURNAL CANADIEN DE MATHEMATIQ UES

The Canadian Mathematical Society La SociCtt mathkmatique du Canada

Can. J. Math.Vol.45 (5). 1993 pp. 1104-1120

SENSITIVITY AND CONTROLLABILITY OF SYSTEMS GOVERNED BY INTEGRAL EQUATIONS VIA PROXIMAL ANALYSIS A. YEZZA

ABSTRACT In this paper, we are concerned with the basic problem defined in [9]. Formulas for aV(0) and P V ( 0 ) . respectively the generalized and asymptotic gradient of the value function at zero, correspondingto an L'-additive perturbation of dynamics are given. Under the normality condition, aV(0) NmS out to be a compact subset of L ~ , formedentirely of arcs, and V is locally finite and Lipschitz at 0. Moreover,estimations of the generalized directional derivative and Dini's derivative of V at 0 are derived. Supplementary conditions imply that Dini's derivative of Vat 0 exists, and V is actually saictly differentiableat this point.

1. Introduction. 1.1 Proximal Analysis. Let X be a real Banach space whose norm is denoted 11 . 11 and C a closed subset of X. The space X is denoted respectively H or Wn whenever it is a Hilbert space or finite-dimensional. When X = H or Wn, we denote its inner product by (., -), which gives the norm 11 . 11. We define the distance function fiom C, dc(.) by dc(x) := inf{llc - xll : c E C). Proximal analysis started with the notion of perpendiculars introduced by Clarke in the finite-dimensional case [4], which has given rise to a formula for the generalized gradient of the distance function from C, adc(.). The fact that the normal cone of C, Nc(.) is the w*-closed convex cone generated by adc(.), involves an important object which is called proximal normal vector. In fact, when X = H, a vector E X is said to be perpendicular to C at x E C, and we write ICat x, if = x' - x, where x is the unique nearest point from C to x'. The fact that ICat x or equivalently dc(x+ 0 such that for all [al,Y'] E epi V

1112

A. YEZZA

FIRST CASE: V ( a ) = Y(= J(x, v ) ). For an arbitrary y in Ac2(T,Rn) with y(0) = $(0), y(1) E C, and y remaining in G I ,and for any p E R;let a'(.)E L ~ ( TW) , be given by:

~'(= t ) j(tI - d ( t )- 4 ( t , t,~ ( 0PO))-,

4r(t,s,x(s),4 s ) ) ds.

Therefore Cy, PI E ~ d ( P ( c d )and ) , consequently [ a ' , ~ ' E ] epi V . Replacing [ a ' , ~ 'and ] [ a ,Y ]by their values in (5. l), we obtain (5.2)

X J ( Xv,) +

Il;' i((t)- ( - ~ ( t+) 4 ( t ,t. ~ ( t ~) ,( t )+)9br(t.s.x(s)v(s))ds} dt

Then Cy, p) = (x, v ) minimizes the RHS of this inequality over all possible Cy, p). Consider a second state m(.) = Ac2(T,R) solution of k ( t ) = ~ ( y(t), t , ~ ( t )(a.e.) ) and m(0) = 0, with y and p chosen as above. We can consider y as a second control v(.) in L2(T,Rn).Define the functional:

such that

and the functionf:Rn x W set for t in T:

--+

R, Cy, m )

~ ( := t )

-fi,m) := Xf Cy) + o[fCy)+ m - J(x,v)I2,and

9

F(s,x(s),v(s))ds.

From all this data, we deduce that the statelcontrol pair (y,m; p, v) = (x,rno; v,x) is a solution of the following problem:

(PI

jCy,m; p, v) : = f ( y ( l ) 4, 1 ) ) + Sd F(t,y(.),p(.>,v(.>) dt -* min y(t) = v(t)(a.e.), y(0) = +(O),y(1) E C and y remains in ol, m(t)= ~ ( y(t), t , ~ ( t )(a.e.), ) m(O)= 0 , ( P ,v) E 3-x L2(T,Rn).

PROXIMAL ANALYSIS

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This problem is very similar to the basic problem &0), and the technique of [9, Proof of Theorem 7.41 can be applied to yield a p E Ac2(T, Rn), X0 E (0, 1 }, and E Xaf (x(l)), with 1 = X0 + XOI 0 small enough, we can construct a sequence {[Pi,-Xi]) and an element [q,s] E L ~ ( TW) , x Rn such that


0, pi E -@(n f l dmV(0) if Xi = 0 (see last lemma). Define the index set A, := {i E I, : A; > 0). Then we can write

n

If A, = 0, then from the last line, the fact that p; E -@() ndmV(0)and the convexity of this last set, it follows immediately that E w { - @ ( ~ ~ ~ ~ V ( O since ) ) , E is chosen arbitrarily positive. Now if A, # 0,we can suppose without any loss of generality that the set of finite sequences {(P;/A;) : i E A,),>o C -M1(n is uniformly bounded in L2 relative to E. That is; there exists a constant M > 0 such that