Abstract
The dynamic programming approach to optimal control theory attempts to characterize the value functionV as a solution to the Hamilton-Jacobian-Bellman equation. Heuristic arguments have long been advanced relating the Pontryagin maximum principle and dynamic programming according to the equation (H(t, x * (t), u * (t), p(t)),−p(t))=√V(t,x * (t)), where (x*, u*) is the optimal control process under consideration,p(t), is the coextremal, andH is the Hamiltonian. The relationship has previously been verified under only very restrictive hypotheses. We prove new results, establishing the relationship, now expressed in terms of the generalized gradient ofV, for a large class of nonsmooth problems.
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References
[BC] A. E. Bryson and Y.-C. Ho,Applied Optimal Control, Blaisdell, Waltham, MA, 1969.
[C] F. H. Clarke,Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
F. H. Clarke, P. D. Loewen, and R. B. Vinter, The Value Function in Dynamic Optimization, Monograph (in preparation).
F. H. Clarke and R. B. Vinter, The relationship between the maximum principle and the dynamic programming technique,SIAM J. Control Optim. (to appear).
[CV2] F. H. Clarke and R. B. Vinter, On connections between the maximum principle and the dynamic programming technique, inFermat Days: Mathematics for Optimization (J.-B. Hiriart-Urruty, ed), Elsevier (North Holland), Amsterdam, 1986.
[FR] W. H. Fleming and R. W. Rishel,Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975.
[PBGM] L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze, and E. F. Mischenko,The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, 1962.
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Vinter, R.B. New results on the relationship between dynamic programming and the maximum principle. Math. Control Signal Systems 1, 97–105 (1988). https://doi.org/10.1007/BF02551239
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DOI: https://doi.org/10.1007/BF02551239