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Zbl 1082.93003
Krastanov, Mikhail; Quincampoix, Marc
Local small time controllability and attainability of a set for nonlinear control system. (English)
[J] ESAIM, Control Optim. Calc. Var. 6, 499-516 (2001). ISSN 1292-8119; ISSN 1262-3377/e

A closed set $S \subseteq \Bbb R^n$ is small time locally attainable (STLA) by a dynamical system described by a set-valued mapping $R(x,t)$ if and only if for any time $T>0$, a neighbourhood $\Cal O$ of $S$ exists such that for all $x \in {\Cal O}$ there exists $\tau \in [0,T]$ such that $R(x,\tau)\cap S\ne \emptyset$. The intention of this paper is to provide a unified treatment of the STLA problem first for a general dynamical system governed by a set valued mapping $R:[0,+\infty)\times {\Bbb R}^n \mapsto {\Bbb R}^n$ with closed nonempty values, continuous in the variable $x$ and satisfies the semi-group property $$ R(R(x,t),s)\subseteq R(x,t+s) $$ for all positive $s$ and $t$. In this context the authors associate with the initial conditions $x_0$ an $R$-trajectory to be any continuous function $x(\cdot) : [0,+\infty) \mapsto \Bbb R^n$ such that $x(0)=x_0$ and $x(t)\in R(x,t)$ for all $t>0$. These results are then used to obtain specific conditions for the case when $R$ is related to the reachable set mapping associated with a differential inclusion $x'(t) \in F(x(t))$. Higher order sufficiency conditions for STLA are obtained under the assumption that $F(x)=f(x)+g(x)U$ which covers the case of a control system governed by $$ x'(t)=f(x(t))+g(x(t))\cdot u(t) \tag{*} $$ where $f:\Bbb R^n \mapsto \Bbb R^n$ and $g:=(g_1 , \dots, g_l):\Bbb R^n \mapsto (\Bbb R^n)^l$ and $u(t) \in U\subseteq \Bbb R^l$. Necessary and sufficient conditions for STLA are obtained for linear control systems when the set $S$ is a hyperplane. The minimal time function $\Phi (x_0)$ is minimum over all $\tau$ such that there exists a solution $y(\cdot)$ to (*) starting from $x_0$ and reaching $S$ in time $\tau$, namely $x(\tau) \in S$. Under the sufficiency conditions for STLA it is shown that the function $\Phi$ is Hölder continuous in a neighbourhood of a boundary point of $S$.
[Andrew C. Eberhard (Melbourne)]

MSC 2000:: *93B05 Controllability
93B03 Attainable sets
93C05 Linear control systems
93C10 Nonlinear control systems
49J53 Set-valued and variational analysis

Keywords: attainability; controllability; local variations; polynomial control; linear controls

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