Mathematical Sciences: On the Existence of Optimal Solutionsfor Infinite Horizon Hereditary Control Problems

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Infinite horizon optimal control problems arise naturally in the study of models of economic growth. These models consist of minimizing an integral functional with an infinite upper limit. The admissible pairs of control functions and state functions must satisfy a differential equation, state constraints and control constraints. Given the admissible pair, the functional need not be finite, and as a consequence, the traditional concept of minimum becomes meaningless. To circumvent this difficulty, a variety of weaker notions of optimality have been introduced. The investigator has established the existence of these weaker notions of optimality under minimal convexity hypotheses for ordinary differential equation constraints. He now proposes to generalize these results for the class of differential equations with delays called hereditary differential systems. The difficulties encountered there are greater than in the case of ordinary differential equations, but the models of this type may be more realistic for problems involving resource management and economics. Research in optimal control problems has been an essential part of control theory, optimization and calculus of variations. Results of such research are often interesting for people interested in trajectory optimization for space vehicles and for scientists interested in resource management.