Abstract. Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, weshow that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka
In the classical time optimal control problem, it is well known that the so-called Petrov condition is necessary and sufficient for the minimum time function to be locally Lipschitz continuous. In this paper, the same regularity result is obtained in the presence of nonsmooth state constraints. Moreover, for a special class of control systems we obtain a local semiconcavity result for the constrained minimum time function.
The Maximum principle in control theory provides necessary optimality conditions for a given trajectory in terms of the co-state, which is the solution of a suitable adjoint system. For constrained problems the adjoint system contains a measure supported at the boundary of the constraint set. In this paper we give a representation formula for such a measure for smooth constraint sets and nice Hamiltonians. As an application, we obtain a perimeter estimate for constrained attainable sets.
Abstract.We consider an optimal control problem for a system of the formẋ = f (x, u), with a running cost L. We prove an interior sphere property for the level sets of the corresponding value function V . From such a property we obtain a semiconcavity result for V , as well as perimeter estimates for the attainable sets of a symmetric control system.
Mathematics Subject Classification. 93B03, 49L20, 49L25
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