Optimal control of second-order integral equations

TL;DR

This paper develops necessary and sufficient optimality conditions for control problems involving second-order Fredholm and Volterra integral equations, which do not follow Pontryagin's principle, with applications to bilinear control problems.

Contribution

It introduces first and second order optimality conditions for non-Pontryaginian integral control problems involving second-order Fredholm and Volterra equations.

Findings

Established first order necessary conditions for the control problems.

Derived second order necessary and sufficient conditions.

Applied the theoretical results to bilinear Volterra control problems.

Abstract

We analyze optimal control problems for multiple Fredholm and Volterra integral equations. These are non Pontryaginian optimal control problems, i.e. an extremum principle of Pontryagin type does not hold. We obtain first order necessary conditions for optimality, and second order necessary and sufficient conditions. We illustrate with applications to first order and second order Volterra bilinear control problems.