### G-85-03

# Optimization with Unbounded Time Interval for a Class of Non Linear Systems

## Alain Haurie

The aim of these notes is to give a comprehensive presentation of the recently developed theory of optimal control of non linear dynamic systems defined on unbounded time intervals.

In the realm of mechanical engineering the linear-quadratic regulator problem was the source of the development of a theory of optimal control of linear dynamical systems on unbounded intervals. In economic theory, the problem of optimal accumulation of capital was posited by F. Ramsey in 1928 and generated during the last decade interesting new developments in the theory of the optimal control of non linear systems over an infinite time horizon.

In these notes the most important results obtained by Cass, Shell, Halkin, Brock, Sheinkman, Haurie and Magill are fully developed. The mathematics are kept at a moderate level and therefore we will not present the very general theory developed by Rockafellar, which is based on advanced convex analysis.

First the control problem will be defined and the various infinite-horizon optimality concepts will be presented and compared. Then the necessary conditions known as the maximum principle will be extended to this class of problem. These conditions do not contain a set of transversality conditions, hence the extremal trajectories are not fully characterized by these conditions.

Fortunately, under some conditions, like biconvexity of the extended velocity set or concavity-convexity of the Hamiltonian etc..., a global asymptotic stability property can be established for optimally controlled autonomous systems. This restores the completeness of the set of necessary conditions and permits the computation of extremals.

The asymptotic stability property is also used in the extension of Mangasarian's sufficient conditions for optimality and existence of optimal trajectories over an infinite time horizon can be asserted under similar convexity assumptions. All these results constitute a rather complete theory.

An important extension of the stability results concerns the case where the criterion is discounted with a positive discount rate. Discounting is common practice in economics and operations research. Since a high discount rate makes the distant future unimportant, one can imagine that positive discounting may induce instability. Under a "steepness" or curvature assumption for the Hamiltonian function, the global asymptotic stability of optimal motion will still be guaranteed.

Several applications of this theory will be considered, most of them in the field of economics and ecology, and the possibility to extend these results to differential games and stochastic systems will be briefly discussed.

Published **February 1985**
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138 pages