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Zbl 1210.34022
Rios, V.R.; Wolenski, P.R.
Proximal characterization of the reachable set for a discontinuous differential inclusion. (English)
[A] Ancona, Fabio (ed.) et al., Geometric control and nonsmooth analysis. In honor of the 73rd birthday of H. Hermes and of the 71st birthday of R. T. Rockafellar. Proceedings of the conference, Rome, Italy, June 2006. Hackensack, NJ: World Scientific. Series on Advances in Mathematics for Applied Sciences 76, 270-279 (2008). ISBN 978-981-277-606-8/hbk

Consider the differential inclusion $$\dot{x}(t) \in F(x(t))\quad \text{for a.e. } t\in I=[0,\infty), \tag1$$ where $F$ is a multifunction mapping $\Bbb R\sp{n}$ into the subsets of $\Bbb R\sp{n}.$ The reachable set of $F$ at time $t\ge 0$ is defined as $$R\sb{F}(t)=\{x(t): x(\cdot)\,\, \text{solves (1) on } [0,t] \text{ with } x(0)\in M\}$$ and $G(R\sb{F})= \{(t,x): t\ge 0$, $x\in R\sb{F}(t)\}$, where $M\subset\Bbb R\sp{n}$ is a compact set. It is supposed that {\parindent6mm \item{(1)} $F(x)$ is a nonempty, compact and convex set for all $x\in\Bbb R\sp{n}$; \item{(2)} there is a constant $c >0 $ so that $\sup\{\|v\|:v\in F(x)\}\le c(1+\|x\|)$ for all $x\in\Bbb R\sp{n}$; \item{(3)} $F(\cdot)$ is upper semicontinuous; \item{(4)} there is a constant $c >0 $ such that $ H\sb{F}(y,y-x) - H(x,y-x)\le K\|x-y\|\sp{2} $ for all $x, y\in\Bbb R\sp{n}.$ \par} Here, $H\sb{F}(x,p)=\sup\{\langle v,p\rangle: v\in F(x)\}.$ The proximal normal cone $N\sb{S}\sp{P}$ of a closed set $S\subset\Bbb R\sp{n}$ at $x\in S$ is defined as the set of elements $z\in\Bbb R\sp{n}$ for which there exists $\sigma =\sigma(z,x)\ge$ such that $ \langle z, y-x\rangle \le\sigma \|y-x\|\sp{2}\,\, \text{for all}\,\, y\in S.$ Theorem. Assume $F$ satisfies (1)--(4). Then the graph of its reachable set $G(R\sb{F})$ is the unique closed subset $S$ of $I\times\Bbb R\sp{n}$ satisfying the following for all $(\theta,\zeta)\in N\sb{S}\sp{P}(t,x)$ and all $(t,x)\in S$: {\parindent6.5mm \item{(a)} $\theta + H \sb{F}(x,\zeta)\ge 0$; \item{(b)} $\theta +\limsup\sb{y\to x} H\sb{F}(y,\zeta)\le 0$; \item{(c)}\, $\lim\sb{T\to 0}S\sb{T}= M.$ \par} Here, $\limsup\sb{y\to x}$ is the limsup of $\delta \to 0$ with $y= x+\delta \zeta$, $S\sb{T}=\{x\in\Bbb R\sp{n}: (T,x)\in S\}.$
[Oleg Filatov (Samara)]

MSC 2000:: *34A60 ODE with multivalued right-hand sides
34A36 Discontinuous equations

Keywords: reachable set; dissipative Lipschitz maps; differential inclusion

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