Displaying similar documents to “Interior sphere property for level sets of the value function of an exit time problem”

Interior sphere property of attainable sets and time optimal control problems

Piermarco Cannarsa, Hélène Frankowska (2006)

ESAIM: Control, Optimisation and Calculus of Variations

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This paper studies the attainable set at time for the control system y ˙ ( t ) = f ( y ( t ) , u ( t ) ) u ( t ) U showing that, under suitable assumptions on , 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-regularity for boundaries of attainable sets.

Viscosity solutions of the Bellman equation for exit time optimal control problems with non-Lipschitz dynamics

Michael Malisoff (2001)

ESAIM: Control, Optimisation and Calculus of Variations

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We study the Bellman equation for undiscounted exit time optimal control problems with fully nonlinear lagrangians and fully nonlinear dynamics using the dynamic programming approach. We allow problems whose non-Lipschitz dynamics admit more than one solution trajectory for some choices of open loop controls and initial positions. We prove a uniqueness theorem which characterizes the value functions of these problems as the unique viscosity solutions of the corresponding Bellman equations...

Feedback in state constrained optimal control

Francis H. Clarke, Ludovic Rifford, R. J. Stern (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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An optimal control problem is studied, in which the state is required to remain in a compact set S . A control feedback law is constructed which, for given ε > 0 , produces ε -optimal trajectories that satisfy the state constraint universally with respect to all initial conditions in S . The construction relies upon a constraint removal technique which utilizes geometric properties of inner approximations of S and a related trajectory tracking result. The control feedback is shown to possess...

On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients

Ludovic Rifford (2001)

ESAIM: Control, Optimisation and Calculus of Variations

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Let x ˙ = f ( x , u ) be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke’s generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov...

Regularity along optimal trajectories of the value function of a Mayer problem

Carlo Sinestrari (2004)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system.