On Optimality Conditions in Control Theory

TL;DR

This paper develops and extends optimality conditions for various control problems, including infinite horizon and Volterra integral equations, using needle variation methods and measure theory, with illustrative examples.

Contribution

It introduces a unified framework for optimality conditions across multiple control problem types, extending needle variation techniques to complex cases.

Findings

Derived necessary optimality conditions for infinite horizon control problems.

Extended needle variation method to Volterra integral equations with two-dimensional time.

Demonstrated conditions through illustrative examples.

Abstract

We study optimality conditions for various types of control problems like the standard optimal control problem, optimal multiprocesses, problems with infinite horizon or the control of Volterra integral equations. To derive necessary conditions the needle variation method of Ioffe & Tichomirov is the central tool. In the particular control problem with infinite horizon the question of a suitable setting arises. We propose the framework of continuous state trajectories converging at infinity. This requires a version of Riesz' representation theorem and the introduction of regular Borel measures on the extended real number line. The control of Volterra integral equations including an inner and outer time variable. Consequently, we deal with a two-dimensional time set. We extend the needle variation method of Ioffe & Tichomirov to this case. The obtained optimality conditions are demonstrated in illustrative examples.