Abstract
We discuss control systems defined on an infinite horizon, where typically all the associated costs become unbounded as the time grows indefinitely. It is proved, under certain lower semicontinuity and controllability assumptions, that a linear time function can be subtracted from the cost, resulting in a modified cost, which is bounded on the infinite time interval. The cost evaluated over one sampling interval has a simple representation in terms of the initial and final states. Applying this representation we obtain an optimality result for control systems represented by ordinary differential equations whose cost integrand contains a discounting factor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bellman R, Bucy R (1964) Asymptotic control theory. SIAM Control 2:11–18
Brock WA, Haurie A (1976) On existence of overtaking optimal trajectories over an infinite time horizon. Math Op Res 1:337–346
Gale D (1967) On optimal development in a multi-sector economy. Rev Econ Studies 34:1–19
Rockefellar RT (1979) Convex processes and Hamiltonian dynamical systems. In: Krein J (ed) Convex Analysis and Mathematical Economics. Lecture Notes in Economics and Mathematical Systems 168:122–136
Rudin W (1974) Real and Complex Analysis. McGraw-Hill Series in Higher Mathematics. New York, McGraw-Hill
von Weizsacker CC (1965) Existence of optimal programs of accumulation for an infinite horizon. Rev Econ Studies 32:85–104
Author information
Authors and Affiliations
Additional information
Communicated by A. V. Balakrishnan
Rights and permissions
About this article
Cite this article
Leizarowitz, A. Infinite horizon autonomous systems with unbounded cost. Appl Math Optim 13, 19–43 (1985). https://doi.org/10.1007/BF01442197
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01442197