Abstract
We consider an optimal control problem in which the dynamic equation and cost function depend on the recent past of the trajectory. The regularity assumed in the basic data is Lipschitz continuity with respect to the sup norm. It is shown that, for a given optimal solution, an adjoint arc of bounded variation exists that satisfies an associated Hamiltonian inclusion. From this result, known smooth versions of the Pontryagin maximum principle for hereditary problems can be easily derived. Problems with Euclidean endpoint constraints are also considered.
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References
T. S. Angell and A. Kirsch, On the necessary conditions for optimal control of retarded systems, Appl. Math. Optim. 22 (1990), 117–145.
J. P. Aubin and F. H. Clarke, Shadow prices and duality for a class of optimal control problems, SIAM J. Control Optim. 17 (1979), 567–587.
H. T. Banks, Variational problems involving functional differential equations, SIAM J. Control 7 (1969) 1–17.
H. T. Banks and A. Manitius, Application of abstract variational theory to hereditary systems-a survey, IEEE Trans. Automat. Control 19 (1974) 524–533.
Z. Bien and D. H. Chyung, Optimal control of delay systems with a final functional condition, Internat. J. Control 32 (1980), 539–560.
P. Billingsley, Probability and Measure, 2nd ed. Wiley, New York, 1986.
J. B. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed sets in Banach spaces, part I: theory, Canad. J. Math. 38 (1986), 431–452.
F. H. Clarke, Optimal control and the true Hamiltonian, SIAM Rev. 21 (1979), 157–166.
F. H. Clarke, Optimization and Nonsmooth Analysis. Classics in Applied Mathematics, Vol. 5. Society for Industrial and Applied Mathematics, Philadelphia, PA (original edition by Wiley, 1983), 1990.
F. H. Clarke, Methods of Dynamic and Nonsmooth Optimization. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 57. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1989.
F. H. Clarke and G. G. Watkins, Necessary conditions, controllability and the value function for differential-difference inclusions, Nonlinear Anal. TMA 10 (1986), 1155–1179.
F. Colonius, The maximum principle for relaxed hereditary differential systems with function space end condition, SIAM J. Control Optim. 20 (1982), 695–712.
F. Colonius, Optimal Periodic Control. Lecture Notes in Mathematics, Vol. 1313. Springer-Verlag, Berlin, 1988.
J. Hale, Theory of Functional Differential Equations. Springer-Verlag, New York, 1977.
B. K. Kim and Z. Bien, On function target control of dynamical systems with delays in state and control under bounded state constraints, Internat. J. Control 33 (1981), 891–902.
P. D. Loewen, The proximal normal formula in Hilbert space, Nonlinear Anal. TMA 11 (1987), 979–995.
L. W. Neustadt, Optimization. Princeton University Press, Princeton, NJ, 1976.
L. S. Pontryagin, V. Boltianski, R. Gamkrelidze, and E. Mischenko, The Mathematical Theory of Optimal Processes. Wiley, New York, 1962.
J. Warga, Controllability of nondifferentiable hereditary processes, SIAM J. Control Optim. 16 (1978), 813–831.
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Clarke, F.H., Wolenski, P.R. Necessary conditions for functional differential inclusions. Appl Math Optim 34, 51–78 (1996). https://doi.org/10.1007/BF01182473
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DOI: https://doi.org/10.1007/BF01182473