Control of semilinear differential equations with moving singularities

TL;DR

This paper investigates controlling semilinear differential equations with a moving singularity, establishing conditions for controllability, and demonstrating the approach with numerical simulations and extensions to fractional differential equations.

Contribution

It introduces a novel controllability framework for equations with parameter-dependent singularities, using lower and upper solutions and a bisection algorithm.

Findings

Controllability conditions depend on the moving singularity.

The control functional is continuous over the parameter domain.

Numerical simulations validate the theoretical results.

Abstract

In this paper, we present a control problem related to a semilinear differential equation with a moving singularity, i.e., the singular point depends on a parameter. The particularity of the controllability condition resides in the fact that it depends on the singular point which in turn depends on the control variable. We provide sufficient conditions to ensure that the functional determining the control is continuous over the entire domain of the parameter. Lower and upper solutions technique combined with a bisection algorithm is used to prove the controllability of the equation and to approximate the control. An example is given together with some numerical simulations. The results naturally extend to fractional differential equations.