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Exterior sphere condition and time optimal control for differential inclusions

Abstract : The minimum time function $T(\cdot)$ of smooth control systems is known to be locally semiconcave provided Petrov's controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of $T(\cdot)$ near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened set-up, we prove that the hypograph of $T(\cdot)$ satisfies, locally, an exterior sphere condition. As is well-known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming $T(\cdot)$ to be Lipschitz.
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Contributor : Estelle Bouzat Connect in order to contact the contributor
Submitted on : Thursday, June 21, 2012 - 1:50:11 PM
Last modification on : Monday, October 5, 2020 - 11:22:01 AM

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Piermarco Cannarsa, Khai T. Nguyen. Exterior sphere condition and time optimal control for differential inclusions. SIAM Journal on Control and Optimization, 2011, 49 (6), pp.2558-2576. ⟨10.1137/110825078⟩. ⟨hal-00710651⟩



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