JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 104, No. 1, pp. 21–40, JANUARY 2000 Existence of Neighboring Feasible Trajectories : Applications to Dynamic Programming for State-Constrained Optimal Control Problems 1 H. FRANKOWSKA 2 AND R. B. VINTER 3 Communicated by D. Q. Mayne Abstract. In this paper, the value function for an optimal control problem with endpoint and state constraints is characterized as the unique lower semicontinuous generalized solution of the Hamilton– Jacobi equation. This is achieved under a constraint qualification (CQ) concerning the interaction of the state and dynamic constraints. The novelty of the results reported here is partly the nature of (CQ) and partly the proof techniques employed, which are based on new estimates of the distance of the set of state trajectories satisfying a state constraint from a given trajectory which violates the constraint. Key Words: Optimal control, state constraints, dynamic program- ming, Hamilton–Jacobi equation. 1. Introduction Consider the optimal control problem (P) minimize g(x(1)), over arcs x ∈W 1,1 ([0, 1] ; R n ) satisfying x ˙ (t) ∈F (t, x(t)), a.e. t ∈[0, 1], x(t) ∈A, 2200t ∈[0, 1], x(0)Gx 0 , 1 This research was supported by the Human Capital and Mobility Program of the European Union, Contract CHRX-CT94-0431. 2 Professor, Centre de Recherche Viabilite ´, Jeux, Contro ˆ le, CNRS and Universite ´ de Paris- Dauphine, Paris, France. 3 Professor, Centre for Process Systems Engineering and Department of Electrical and Elec- tronic Engineering, Imperial College, London, UK. 21 0022-3239000100-0021$18.000  2000 Plenum Publishing Corporation ps893$p221 08-02-:0 12:20:16