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Interior sphere property of attainable sets and time optimal control problems

Piermarco CannarsaHélène Frankowska — 2006

ESAIM: Control, Optimisation and Calculus of Variations

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.

Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

Gianni Dal MasoHélène Frankowska — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We investigate the value function of the Bolza problem of the Calculus of Variations
 V ( t , x ) = inf 0 t L ( y ( s ) , y ' ( s ) ) d s + ϕ ( y ( t ) ) : y W 1 , 1 ( 0 , t ; n ) , y ( 0 ) = x , with a lower semicontinuous Lagrangian and a final cost ϕ , and show that it is locally Lipschitz for whenever is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize the value...

A Second-Order Maximum Principle in Optimal Control Under State Constraints

Frankowska, HélèneHoehener, DanielTonon, Daniela — 2013

Serdica Mathematical Journal

A second-order variational inclusion for control systems under state constraints is derived and applied to investigate necessary optimality conditions for the Mayer optimal control problem. A new pointwise condition verified by the adjoint state of the maximum principle is obtained as well as a second-order necessary optimality condition in the integral form. Finally, a new sufficient condition for normality of the maximum principle is proposed. Some extensions to the Mayer optimization problem...

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