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Proximal characterization of the reachable set for a discontinuous differential inclusion. (English) Zbl 1210.34022
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 (ISBN 978-981-277-606-8/hbk). Series on Advances in Mathematics for Applied Sciences 76, 270-279 (2008).
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 $y= x+\delta \zeta$, $S\sb{T}=\{x\in\Bbb R\sp{n}: (T,x)\in S\}.$ For the entire collection see [Zbl 1154.49002].

34A60Differential inclusions
34A36Discontinuous equations