The maximum principle in optimal control problems with phase constraints. Nondegeneracy and stability

This note contains the statements of several results concerning necessary optimality conditions for the optimal control problem of minimizing J(t_1,t_2,x(t_1),x(t_2)) subject to x'(t)in F(t,x(t)) a.e., x(t)in Gsubset bold R^n for all tin[t_1,t_2], (t_1,t_2,x(t_1),x(t_2))in Psubset bold R^{2n+2}. par The first theorem states a Hamiltonian form of the maximum principle which, in addition to known results [F. H. Clarke, {it Optimization and nonsmooth analysis}, Wiley, New York, 1983; [msn] MR0709590 (85m:49002) [/msn]; P. D. Loewen and R. T. Rockafellar, SIAM J. Control Optim. {bf 32} (1994), no.~2, 442--470], contains continuity and jump properties of the Hamiltonian along extremals. The latter properties are shown to imply certain nondegeneracy properties of the multipliers under an additional end-point controllability hypothesis. The last theorem contains a stability property of the multipliers with respect to perturbations F_alpha,G_alpha,J_alpha of the initial problem data.