Multivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints

This paper deals with the Cauchy initial value problem for a (finite-dimensional) differential inclusion with a Carathèodory and Lipschitz continuous (with respect to the state variable) right- hand side whose values are compact but not necessarily convex. Moreover, the state variable is constrained to range over the closure K of an open set. Under an extension of the condition of inner pointing vector field condition -an extension which allows for zones of the boundary of K where the whole dynamics points outwards- the authors prove the lower semicontinuity of the solution map which assigns to the initial data the set of all solutions restricted to an arbitrary time interval. This is used to show the continuity of the value function of an infinite horizon optimal control problem with integral and state constraints. In the appendix it is also shown that the value function is the unique viscosity solution of a suitable boundary value problem including the usual Bellman equation.