×

Existence of finitely optimal solutions for infinite-horizon optimal control problems. (English) Zbl 0579.49004

We investigate the existence of finitely optimal solutions for the Lagrange problem of optimal control defined on \([0,+\infty)\) under weaker convexity and seminormality hypotheses than those of previous authors. The notion of finite optimality has been introduced into the literature as the weakest of a hierarchy of types of optimality that have been defined to permit the study of Lagrange problems, arising in mathematical economics, whose cost functionals either diverge or are not bounded below.
Our method of proof requires us to analyze the continuous dependence of finite-interval Lagrange problems with respect to a prescribed terminal condition. Once this is done, we show that a finitely optimal solution can be obtained as the limit of a sequence of solutions to a sequence of corresponding finite-horizon optimal control problems. Our results utilize the convexity and seminormality hypotheses which are now classical in the existence theory of optimal control.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
49J45 Methods involving semicontinuity and convergence; relaxation
93C15 Control/observation systems governed by ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arrow, K. J., andKurz, M.,Public Investment, the Rate of Return, and Optimal Fiscal Policy, Johns Hopkins Press, Baltimore, Maryland, 1970.
[2] Intriligator, M. D.,Mathematical Optimization and Economic Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1971. · Zbl 1140.90302
[3] Chakravarty, S.,The Existence of an Optimum Savings Program, Econometrica, Vol. 30, pp. 178-187, 1962. · Zbl 0285.90038
[4] Carlson, D. A.,On the Existence of Optimal Solutions for Infinite-Horizon Optimal Control Problems, University of Delaware, Ph.D Thesis, 1983.
[5] Stern, L. E.,The Infinite-Horizon Optimal Control Problem, University of Rhode Island, PhD Thesis, 1980.
[6] Halkin, H.,Necessary Conditions for Optimal Control Problems with Infinite Horizons, Econometrica, Vol. 42, pp. 267-272, 1974. · Zbl 0301.90009
[7] Haurie, A.,Optical Control on an Infinite-Time Horizon: The Turnpike Approach, Journal of Mathematical Economics, Vol. 3, pp. 81-102, 1976. · Zbl 0329.90021
[8] Haurie, A.,Existence and Global Asymptotic Stability of Optimal Trajectories for a Class of Infinite-Horizon Nonconvex Systems, Journal of Optimization Theory and Applications, Vol. 31, pp. 515-533, 1980. · Zbl 0417.49023
[9] Brock, W. A., andHaurie, A.,On the Existence of Overtaking Optimal Trajectories over an Infinite-Time Horizon, Mathematics of Operations Research, Vol. 1, pp. 337-346, 1976. · Zbl 0367.49003
[10] Balder, E. J.,An Existence Result for Optimal Economic Growth, Journal of Mathematical Analysis and Applications, Vol. 95, pp. 195-213, 1983. · Zbl 0517.49002
[11] Cesari, L., LaPalm, J. R., andNishiura, R.,Remarks on Some Existence Theorems for Optimal Control, Journal of Optimization Theory and Applications, Vol. 3, pp. 296-305, 1969. · Zbl 0172.13002
[12] Cesari, L.,Optimization Theory and Applications: Problems with Ordinary Differential Equations, Springer-Verlag, New York, New York, 1983. · Zbl 0506.49001
[13] Cesari, L., andSuryanarayana, M. B.,On Recent Existence Theorems in the Theory of Optimization, Journal of Optimization Theory and Applications, Vol. 31, pp. 397-416, 1980. · Zbl 0417.49015
[14] Carlson, D. A.,The Controllability of Infinite-Horizon Optimal Control Problems, Nonlinear Analysis, Theory, Methods, and Applications, Vol. 8, 1986. · Zbl 0579.49004
[15] Kuratowski, K., andRyll-Nardzewski, C.,A General Theorem on Selectors, Bulletin de l’Academie Polonaise des Sciences, Series des Sciences Mathematique, Astronomique, et Physiques, Vol. 13, pp. 397-403, 1965. · Zbl 0152.21403
[16] Lee, E. B., andMarkus, L.,Foundations of Optimal Control Theory, John Wiley and Sons, New York, New York, 1967. · Zbl 0159.13201
[17] Yano, M.,A Note on the Existence of an Optimal Capital Accumulation in the Continuous-Time Horizon, Journal of Economic Theory, Vol. 22, pp. 421-429, 1981. · Zbl 0511.90038
[18] Feinstein, C. D., andLuenberger, D. G.,Analysis of the Asymptotic Behavior of Optimal Control Trajectories: The Implicit Programming Problem, SIAM Journal on Control and Optimization, Vol. 19, pp. 561-585, 1981. · Zbl 0472.49008
[19] Berkovitz, L. D.,Optimal Control Theory, Springer-Verlag, New York, New York, 1974. · Zbl 0295.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.