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State constraints in optimal control. The degeneracy phenomenon. (English) Zbl 0873.49015
Summary: We study the degeneracy phenomenon arising in optimal control problems with state constraints. It is shown that this phenomenon occurs because of the incompleteness of the standard variants of Pontryagin’s maximum principle for problems with state constraints. The new maximum principle containing some additional information about the behavior of the Hamiltonian at the endtimes is developed. As application we obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle. The results obtained include the optimal control problems of systems described by differential inclusions and ordinary differential equations.

MSC:
49K15 Optimality conditions for problems involving ordinary differential equations
49J24 Optimal control problems with differential inclusions (existence) (MSC2000)
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[1] Aphanasyev, A.P.; Dikusar, V.V.; Milutin, A.A.; Chukanov, C.A., Necessary conditions in optimal control, (1990), Nauka Moscow, in Russian
[2] Arutyunov, A.V., First-order necessary conditions in a problem of optimal control with phase constraints, (), 46-58, (in Russian) · Zbl 0733.49023
[3] Arutyunov, A.V.; Arutyunov, A.V., On the theory of the maximum principle in optimal control problems with phase constraints, Dokl. akad. nauk SSSR, Sov. math. dokl., 39, 11-14, (1989), English transl. · Zbl 0684.34073
[4] Arutyunov, A.V., Perturbations of extremal problems with constraints and necessary optimality condition, J. sov. math., 54, 1342-1400, (1991) · Zbl 0726.49015
[5] Arutyunov, A.V.; Aseev, S.M.; Blagodatskikh, V.I., First order necessary conditions for optimal control problem for differential inclusion with state constraints, Mat. sbornik, 184, 3-32, (1993), (in Russian) · Zbl 0834.49013
[6] Arutyunov, A.V.; Aseev, S.M., Maximum principle for optimal control problems with state constraints. nondegeneracy and stability, Dokl. Russian akad. nauk, 334, 134-137, (1994), (in Russian) · Zbl 0839.49014
[7] Arutyunov, A.V.; Blagodatskikh, V.I.; Arutyunov, A.V.; Blagodatskikh, V.I., The maximum principle for differential inclusions with phase constraints, (), 200, 3-25, (1991), English transl. · Zbl 0823.49015
[8] Arutyunov, A.V.; Tynyanskii, N.T., On the maximum principle in a problem with phase constraints, Izv. akad. nauk SSSR, tekh. kibern., 4, 60-68, (1984), (in Russian)
[9] Aseev, S.M., Smooth approximation of differential inclusions and time-optimal problem, Crm-1610, (1989), Montreal · Zbl 0801.49020
[10] Clarke, F.H., Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0727.90045
[11] Dubovitskii, A.Ya.; Dubovitskii, V.A.; Dubovitskii, A.Ya.; Dubovitskii, V.A., Necessary conditions for a strong minimum in optimal control problems with degenerate endpoint constraints and phase constraints, Uspekhi mat. nauk, Math. surveys, 40, 242, 175-176, (1985), English transl. · Zbl 0186.48905
[12] Dubovitskii, A.Ya.; Dubovitskii, V.A., The principle of the maximum in regular optimal control problems in which the endpoints of the phase trajectory Lie on the boundary of the phase constraint, Avtom. telemekh., 12, 25-33, (1987), (in Russian) · Zbl 0164.06402
[13] Dubovitskii, A.Ya.; Dubovitskii, V.A., Conditions for pointwise nontriviality of the maximum principle in a regular optimal control problem, (), 60-88, (in Russian) · Zbl 0186.48905
[14] Dubovitskii, A.Ya.; Dubovitskii, V.A., The maximum principle for trajectories whose endpoints Lie on the phase boundary, (1988), (in Russian)
[15] Dubovitskii, A.Ya.; Milutin, A.A.; Dubovitskii, A.Ya.; Milutin, A.A., Extremal problems with constraints, Zh. vychisl. mat. mat. fiz., Comput. math. phys., 5, 1-80, (1965), English transl.
[16] Ioffe, A.D.; Tikhomirov, V.M., Theory of extremal problems, (1979), North-Holland Amsterdam · Zbl 0194.13902
[17] Loewen, P.D.; Rockafellar, R.T., The adjoint arc in nonsmooth optimization, Trans. amer. math. soc., 325, 39-72, (1991) · Zbl 0734.49009
[18] Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mischenko, E.F., The mathematical theory of optimal processes, (1962), Wiley New York · Zbl 0102.32001
[19] Warga, J., Optimal control of differential and functional equations, (1972), Academic Press New York · Zbl 0253.49001
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