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Measurable viability theorems and the Hamilton-Jacobi-Bellman equation. (English) Zbl 0836.34016

Let tP(t):[0,T]R d be an absolutely continuous set-valued map and (t,x)F(t,x):[0,T]×R d R d a set-valued map with closed convex values, measurable in t, continuous in x, and satisfying sup{|y|;yF(t,x),xR d }μ(t), t[0,T] for some Lebesgue integrable function μ. Then for any t 0 [0,T] and any x 0 P(t 0 ), the dynamical system (DS) x(t)F(t,x(t)), t[t 0 ,T], x(t 0 )=x 0 , has a solution x such that x(t)P(t) for all t[t 0 ,T] iff for almost all t[0,T] and all xP(t), F(t,x)DP(t,x)(1), where DP(t,x) is the contingent derivative of P at (t,x).

If, moreover, F has compact values and is locally Lipschitz in x, then any solution x of (DS) satisfies x(t)P(t) for all t[t 0 ,T] iff F(t,x)DP(t,x)(1).

The above results are used to study semicontinuous solutions of the Hamilton-Jacobi-Bellman equation u t +H(t,x,u x )=0, where H is measurable in t, locally Lipschitz in x and convex in the third variable.


MSC:
34A60Differential inclusions
49L25Viscosity solutions (infinite-dimensional problems)
93B03Attainable sets