Abstract
This paper investigates infinite horizon optimal control problems with fixed left endpoints with nonconvex, nonsmooth data. We derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality that indicate the relationship between the maximum principle and dynamic programming. The necessary conditions under consideration are extensions of those of [8] to an infinite horizon setting. We then present new sufficiency conditions consistent with the necessary conditions, which are motivated by the useful result by [26] whose sufficiency theorem is valid for nonconvex, nondifferentiable Hamiltonians. The sufficiency theorem presented in this paper employs the strengthened adjoint inclusion of the value function as well as the strengthened maximum principle. If we restrict our result to convex models, it is possible to characterize minimizing processes and provide necessary and sufficient conditions for optimality. In particular, the role of the transversality conditions at infinity is clarified.
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References
Aseev, S.M., Kryaziimskiy, A.: The Pontryagin Maximum Principle and Transversality Conditions for a Class of Optimal Control Problems with Infinite Time Horizons. SIAM J. Control Optim. 43, 1094–1119 (2004)
Balder, E.J.: An Existence Result for Optimal Economic Growth Problems. J. Math. Anal. Appl. 95, 195–213 (1983)
Bates, G.R.: Lower Closure Existence Theorems for Optimal Control Problems with Infinite Horizon. J. Optim. Theory Appl. 24, 639–649 (1978)
Baum, R.F.: Existence Theorems for Lagrange Control Problems with Unbounded Time Domain. J. Optim. Theory Appl. 19, 89–116 (1976)
Bell, M.L., Sargent, R.W.H., Vinter, R.B.: Existence of Optimal Controls for Continuous Time Infinite Horizon Problems. Internat. J. Control 68, 887–896 (1997)
Benveniste, L.M., Scheinkman, J.A.: Duality Theory for Dynamic Optimization Models of Economics: The Continuous Time Case. J. Econ. Theory 27, 1–19 (1982)
Clarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley & Sons, New York (1983)
Clarke, F.H., Vinter, R.B.: The Relationship between the Maximum Principle and Dynamic Programming. SIAM J. Control Optim. 25, 1291–1311 (1987)
Feinstein, C.D., Luenberger, D.G.: Analysis of the Asymptotic Behavior of Optimal Control Trajectories: The Implicit Programming Problem. SIAM J. Control Optim. 19, 561–585 (1981)
Halkin, H.: Necessary Conditions for Optimal Control Problems with Infinite Horizon. Econometrica 42, 267–272 (1974)
Mangasarian, O.L.: Sufficient Conditions for the Optimal Control of Nonlinear Systems. SIAM J. Control Optim. 4, 139–151 (1966)
Michel, P.: On the Transversality Condition in Infinite Horizon Optimal Problems. Econometrica 50, 975–984 (1982)
Mlynarska, E.: Dual Sufficient Optimality Conditions for the Generalized Problem of Bolza. J. Optim. Theory Appl. 104, 427–442 (2000)
Nowakowski, A.: The Dual Dynamic Programming. Proc. Amer. Math. Soc. 116, 1089–1096 (1992)
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mischenko, E.F.: The Mathematical Theory of Optimal Processes. John Wiley & Sons, New York (1962)
Rockafellar, R.T.: Conjugate Convex Functions in Optimal Control and the Calculus of Variations. J. Math. Anal. Appl. 32, 174–222 (1970)
Rockafellar, R.T.: Optimal Arcs and the Minimum Value Function in Problems of Lagrange. Trans. Amer. Math. Soc. 180, 53–83 (1973)
Rockafellar, R.T.: Existence Theorems for Generalized Control Problems of Bolza and Lagrange. Adv. Math. 15, 312–333 (1975)
Sagara, N.: Nonconvex Variational Problem with Recursive Integral Functionals in Sobolev Spaces: Existence and Representation. J. Math. Anal. Appl. 327, 203–219 (2007)
Seierstadt, A., Sydsæter, K.: Sufficient Conditions in Optimal Control Theory. Internat. Econ. Rev. 18, 367–391 (1977)
Takekuma, S.-I.: Support Price Theorem for the Continuous Time Model of Capital Accumulation. Econometrica 50, 427–442 (1982)
Takekuma, S.-I.: On Duality Theory for the Continuous Time Model of Capital Accumulation. Hitotsubashi J. Econ. 25, 145–154 (1984)
Vinter, R.B.: New Results on the Relationship between Dynamic Programming and the Maximum Principle. Math. Control, Signals Systems 1, 97–105 (1988)
Ye, J.J.: Nonsmooth Maximum Principle for Infinite–Horizon Problems. J. Optim. Theory Appl. 76, 485–500 (1993)
Zeidan, V.: A modified Hamilton–Jacobi Approach in the Generalized Problem of Bolza. Appl. Math. Optim. 11, 97–109 (1984)
Zeidan, V.: First and Second Order Sufficient Conditions for Optimal Control and the Calculus of Variations. Appl. Math. Optim. 11, 209–226 (1984)
Zeidan, V.: New Second-Order Optimality Conditions for Variational Problems with C 2-Hamiltonians. SIAM J. Control Optim. 40, 577–609 (2000)
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Sagara, N. (2008). Value Functions and Transversality Conditions for Infinite Horizon Optimal Control Problems. In: Le Thi, H.A., Bouvry, P., Pham Dinh, T. (eds) Modelling, Computation and Optimization in Information Systems and Management Sciences. MCO 2008. Communications in Computer and Information Science, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87477-5_30
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DOI: https://doi.org/10.1007/978-3-540-87477-5_30
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