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Zbl 0836.34016
Frankowska, H.; Plaskacz, S.; Rze\D{z}uchowski, T.
Measurable viability theorems and the Hamilton-Jacobi-Bellman equation.
(English)
[J] J. Differ. Equations 116, No.2, 265-305 (1995). ISSN 0022-0396

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 $\sup \{|y |; y \in F(t,x),\ x \in R^d\} \le \mu(t)$, $t \in [0,T]$ 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 $x \in P(t)$, $F(t,x) \cap DP (t,x) (1) \ne \emptyset$, where $DP(t,x)$ is the contingent derivative of $P$ at $(t,x)$.\par 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 $F(t,x) \subset DP (t,x) (1)$. \par 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.
[J.Kucera (Pullman)]
MSC 2000:
*34A60 ODE with multivalued right-hand sides
49L25 Viscosity solutions
93B03 Attainable sets

Keywords: viability; set-valued map; Hamilton-Jacobi-Bellman equation

Cited in: Zbl 1095.34002 Zbl 1090.34026

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