Viscosity solutions and optimal control problems with integral constraints

We discuss optimal control problems with integral state-control constraints. We rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system, and prove that the value function, although possibly discontinuous, is the unique viscosity solution of the constrained boundary value problem for the corresponding Hamilton-Jacobi equation. The state constraint is the epigraph of the minimal solution of a second Hamilton-Jacobi equation. Our framework applies, for instance, to systems with design uncertainties. © 2000 Elsevier Science B.V. All rights reserved.