Abstract

This paper is devoted to regularity of minimizers and adjoint states for the Bolza optimal control problem under state constraints. It is well known that the adjoint state of the Pontryagin maximum principle may be discontinuous whenever the optimal trajectory lies partially on the boundary of constraints. Still we prove that if the associated Hamiltonian H(t,x,⋅)H(t,x,\cdot) is differentiable and the constraints are sleek, then every optimal trajectory is continuously differentiable. Moreover if for all xx on the boundary of constraints, ∂H∂p(t,x,⋅)\frac{\partial H}{\partial p}(t,x, \cdot) is strictly monotone in directions normal at xx to the set of constraints, then the adjoint state is also continuous on interior of its interval of definition. Finally, we identify a class of constraints for which the adjoint state is absolutely continuous or even Lipschitz on this open interval. This allows us to derive necessary conditions for optimality in the form of variational differential inequalities, maximum principle and modified transversality conditions. We also provide sufficient conditions for Lipschitz continuity of optimal controls and for normality of the maximum principle.

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Keywords

  • Optimal control
  • state constraints
  • constrained maximum principle
  • smooth minimizers
  • normal necessary conditions
  • absolutely continuous costate
  • Lipschitz optimal control

Mathematics Subject Classification

49J30, 49K30, 49N60

References

  1. A. V. Arutyanov, S. M. Aseev: Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952.https://doi.org/10.1137/S036301299426996XGoogle Scholar
  2. J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (1990).Google Scholar
  3. P. Cannarsa, H. Frankowska: Some characterizations of optimal trajectories in control theory, SIAM J. Control Optim. 29 (1991) 1322–1347.https://doi.org/10.1137/0329068Google Scholar
  4. F. H. Clarke: Optimization and Nonsmooth Analysis, Wiley-Interscience, New York (1983).Google Scholar
  5. A. Cernea, H. Frankowska: The connection between the maximum principle and the value function for optimal control problems under state constraints, Proceedings of CDC IEEE Conference, Bahamas, December 13-16 (2004) 893–989.Google Scholar
  6. A. Cernea, H. Frankowska: A connection between the maximum principle and dynamic programming for constrained control problems, SIAM J. Control Optim. 44 (2005) 673–703.https://doi.org/10.1137/S0363012903430585Google Scholar
  7. M. Delfour, J.-P. Zolesio: Shape analysis via oriented distance functions, J. Funct. Anal. 123 (1994) 129–201.https://doi.org/10.1006/jfan.1994.1086Google Scholar
  8. A. I. Dontchev, W. W. Hager: A new approach to Lipschitz continuity in state constrained optimal control, Syst. Control Lett. 35 (1998) 137–143.https://doi.org/10.1016/S0167-6911(98)00043-7Google Scholar
  9. A. Y. Dubovitskii, A. A. Milyutin: Extremal problems with constraints, USSR Comput. Math. and Math. Physiscs 5 (1965) 1–80.https://doi.org/10.1016/0041-5553(65)90148-5Google Scholar
  10. H. Frankowska, F. Rampazzo: Filippov’s and Filippov-Ważewski’s theorems on closed domains, J. Differ. Equations 161 (2000) 449–478.https://doi.org/10.1006/jdeq.2000.3711Google Scholar
  11. R. V. Gamkrelidze: Optimal processes with bounded phase coordinates, Izv. Akad. Nauk, USSR Sec. Mat. 24 (1960) 315–356.Google Scholar
  12. W. W. Hager: Lipschitz continuity for constrained processes, SIAM J. Control Optim. 17 (1979) 321–338.https://doi.org/10.1137/0317026Google Scholar
  13. A. D. Ioffe, V. M. Tichomirov: Theory of Extremal Problems, North-Holland, Amsterdam (1979).Google Scholar
  14. P. Loewen, R. T. Rockafellar: The adjoint arc in nonsmooth optimization, Trans. Amer. Math. Soc. 325 (1991) 39–72.https://doi.org/10.1090/S0002-9947-1991-1036004-7Google Scholar
  15. D. B. Luenberger: Optimization by Vector Space Methods, John Wiley & Sons, New York (1969).Google Scholar
  16. K. M. Malanowski: On the regularity of solutions to optimal control problems for systems linear with respect to control variable, Arch. Auto i Telemech. 23 (1978) 227–241.Google Scholar
  17. F. Rampazzo, R. B. Vinter: Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (2000) 989–1007.https://doi.org/10.1137/S0363012998340223Google Scholar
  18. I. A. Shvartsman, R. B. Vinter: Regularity properties of optimal controls for problems with time-varying state and control constraints, Nonlinear Anal., Theory Methods Appl., to appear.Google Scholar
  19. R. B. Vinter: Optimal Control, Birkhäuser, Boston (2000).Google Scholar

Suggested citation

H. Frankowska. “Regularity of Minimizers and of Adjoint States in Optimal Control under State Constraints.” Journal of Convex Analysis 13 (2006), No. 2, 299–328. https://doi.org/10.68381/jca13027

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press.