Controlled Singular Volterra Integral Equations and Pontryagin Maximum Principle

TL;DR

This paper develops a Pontryagin maximum principle for controlled singular Volterra integral equations, including fractional differential equations, without requiring regularity, convexity, or additional conditions on controls or nonlinearities.

Contribution

It introduces a novel method to derive maximum principles for singular Volterra equations that relax common regularity and convexity assumptions present in prior work.

Findings

Established well-posedness and regularity results for singular Volterra equations.

Derived a Pontryagin maximum principle applicable without control regularity or convexity assumptions.

Extended the framework to include fractional order differential equations as special cases.

Abstract

This paper is concerned with a class of controlled singular Volterra integral equations, which could be used to describe problems involving memories. The well-known fractional order ordinary differential equations of the Riemann--Liouville or Caputo types are strictly special cases of the equations studied in this paper. Well-posedness and some regularity results in proper spaces are established for such kind of questions. For the associated optimal control problem, by using a Liapounoff's type theorem and the spike variation technique, we establish a Pontryagin's type maximum principle for optimal controls. Different from the existing literature, our method enables us to deal with the problem without assuming regularity conditions on the controls, the convexity condition on the control domain, and some additional unnecessary conditions on the nonlinear terms of the integral equation and the cost functional.