Interior sphere property of attainable sets and time optimal control problems

Cannarsa, Piermarco; Frankowska, Hélène

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 350-370

This paper studies the attainable set at time $T>0$ for the control system

$$\dot{(}t)=f(\left(t\right),u\left(t\right))u\left(t\right)\in U$$

showing that, under suitable assumptions on $f$, such a set satisfies a uniform interior sphere condition. The interior sphere property is then applied to recover a semiconcavity result for the value function of time optimal control problems with a general target, and to deduce $C^{1,1}$-regularity for boundaries of attainable sets.

Classification : 26B25, 49K15, 93B03
Keywords: control theory, attainable sets, minimum time function, semiconcave functions

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