On an Optimal Control Problem with Discontinuous Integrand

We consider an optimal control problem for an autonomous differential inclusion with free terminal time and a mixed functional which contains the characteristic function of a given open set M ⊂ ℝ n in the integral term. The statement of the problem weakens the statement of the classical optimal control problem with state constraints to the case where the presence of admissible trajectories of the system in the set M is physically allowed but undesirable due to safety or instability reasons. Using an approximation approach, necessary conditions for the optimality of an admissible trajectory are obtained in the form of Clarke’s Hamiltonian inclusion. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of the problem with a state constraint, the obtained necessary optimality conditions may degenerate. Conditions guaranteeing their nondegeneracy and pointwise nontriviality are presented. The results obtained extend the author’s previous results to the case of a problem with free terminal time and more general functional.