Measurable viability theorems and the Hamilton-Jacobi-Bellman equation
Let \(t \rightsquigarrow P(t) : [0,T] \rightsquigarrow R^d\) be an absolutely continuous set-valued map and \((t,x) \rightsquigarrow F(t,x) : [0,T] \times R^d \rightsquigarrow R^d\) a set-valued map with closed convex values, measurable in \(t\), continuous in \(x\), and satisfying \(\sup \{|y |; y \in F(t,x),\;x \in R^d\} \leq \mu(t)\), \(t \in [0,T]\) for some Lebesgue integrable function \(\mu\). Then for any \(t_0 \in [0,T]\) and any \(x_0 \in P(t_0)\), the dynamical system (DS) \(x(t) \in F(t,x(t))\), \(t \in [t_0,T]\), \(x(t_0) = x_0\), has a solution \(x\) such that \(x(t) \in P(t)\) for all \(t \in [t_0,T]\) iff for almost all \(t \in [0,T]\) and all \(x \in P(t)\), \(F(t,x) \cap DP (t,x) (1) \neq \emptyset\), 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 \(x(t) \in P(t)\) for all \(t \in [t_0,T]\) iff \(F(t,x) \subset DP (t,x) (1)\). 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.
scientific article; zbMATH DE number 736405