Control of semilinear differential equations with moving singularities

TL;DR

This paper addresses the controllability of semilinear differential equations with moving singularities, where the singular point $\theta(\lambda)$ depends on the control variable $\lambda$. It studies the functional $\varphi(\lambda)=\psi_\lambda(u_\lambda)$ with $\psi_\lambda(u)=\int_{0}^{\theta(\lambda)} u(s)\,ds$ and proves that it is well-defined and continuous under natural regularity and growth assumptions. Using lower and upper solutions together with a bisection algorithm, the authors establish the existence of a $\lambda^*$ satisfying $\psi_{\lambda^*}(u_{\lambda^*})=p$ and provide a practical method to approximate it, complemented by a numerical example. The results extend to fractional differential equations via Caputo derivatives, highlighting potential applications to singular models and PDEs.

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.

Figures (2)

• Figure 1: Error decay in the bisection method.
• Figure 2: Approximate solution $u_\lambda$ vs exact solution $u_{\lambda_{\text{exact}}}$

Theorems & Definitions (21)

• Lemma 1.1
• proof
• Remark 1.1
• Theorem 1.2: Arzelà-Ascoli theorem
• Theorem 1.3
• Theorem 1.4: Grönwall inequality
• Lemma 2.1
• proof
• Lemma 2.2
• proof