### Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints

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This paper studies the attainable set at time for the control system $$\dot{y}\left(t\right)=f(y\left(t\right),u\left(t\right))\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{2.0em}{0ex}}u\left(t\right)\in U$$ 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.

We investigate the value function of the Bolza problem of the Calculus of Variations $$V(t,x)=inf\left\{{\int}_{0}^{t}L(y\left(s\right),{y}^{\text{'}}\left(s\right))ds+\varphi \left(y\left(t\right)\right):y\in {W}^{1,1}(0,t;{\mathbb{R}}^{n}),\phantom{\rule{0.277778em}{0ex}}y\left(0\right)=x\right\},$$ with a lower semicontinuous Lagrangian and a final cost $\varphi $, 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...

Abstract. The existence theorem of an invariant measure and Poincare's Recurrence Theorem are extended to set-valued dynamical systems with closed graph on a compact metric space.

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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