Volterra and Functional Differential Equations

ASYMPTOTIC EXPANSIONS OF A PENALTY METHOD FOR COMPUTING A REGULATOR PROBLEM GOVERNED BY VOLTERRA EQUATIONS

Goong Chen Department of Mathematics Pennsylvania state University University Park. Pennsylvania Ronald Grimmer Department of Mathematics Southern Illinois University Carbondale, Illinois

DOI: 10.1201/9781003420026-16

1. INTROOUCTION

In this study we Give a formal justification of a special penalty approach to solving an optimal regulator problem governed by a Volterra integro-differential equation. This approach was motivated by some numerical considerations.

Consider the following quadratic cost optional control problem

$$\min \text{J(u)}\equiv \frac{1}{2}{\displaystyle {\int }_0^T\left[<\text{y(t),My(t)}{\rangle }_{{\text{H}}_{\text{1}}}+<\text{u(t}),\text{Nu}\mid {\text{(t) >}}_{{\text{H}}_{\text{2}}}\right]}\text{dt}$$ (1.1) ...