[Paper Review] Controllability of the Strongly Damped Impulsive Semilinear Wave Equation with Memory and Delay

Cristi Guevara, Hugo Leiva|arXiv (Cornell University)|Apr 9, 2017

Stability and Controllability of Differential Equations8 references3 citations

TL;DR

This paper establishes the approximate controllability of a strongly damped semilinear wave equation with memory, impulses, and delay using a novel approach that avoids fixed point theorems. By applying A.E. Bashirov et al.'s technique and leveraging delay to steer the control solution along a fixed curve in a short interval, the authors prove that the system can be steered arbitrarily close to any desired final state despite unbounded control sequences arising from memory and impulsive effects.

ABSTRACT

This article is devoted to study the interior approximated controllability of the strongly damped semilinear wave equation with memory, impulses and delay terms. The problem is challenging since the state equation contains memory and impulsive terms yielding to potential unbounded control sequences steering the system to a neighborhood of the final state, thus fixed point theorems cannot be used directly. As alternative, the A.E Bashirov and et al. techniques are applied and together with the delay allow the control solution to be directed to fixed curve in a short time interval and achieve our result.

Motivation & Objective

• To investigate the approximate controllability of a strongly damped semilinear wave equation with memory, impulses, and time delay in a bounded domain.
• To address the challenge that memory and impulsive terms may lead to unbounded control sequences, which prevent direct application of fixed point theorems.
• To develop a control strategy that ensures the system can be steered arbitrarily close to a desired final state using a bounded control input.
• To extend existing controllability results to systems with combined effects of memory, impulses, and delay.

Proposed method

• The authors use a fixed-point-free technique based on the work of A.E. Bashirov et al., which avoids reliance on fixed point theorems that fail under unbounded control sequences.
• They define a control solution via a time-delayed feedback mechanism that directs the system along a fixed curve in a short time interval near the final time.
• The system is reformulated in a Hilbert space setting using the state space $\mathcal{Z}^{1/2} = D((-Δ)^{1/2}) \times L^2(\Omega)$, with the evolution governed by a strongly continuous semigroup.
• A mild solution is constructed using the variation of constants formula, incorporating distributed control $u$, memory integral terms $\int_0^t M(t,s)g(w(s-r,x))ds$, and nonlinear terms $f$.
• The control input $u_\alpha^\delta$ is designed to steer the system in a final time interval $[\tau - \delta, \tau]$, ensuring convergence to the target state within $\epsilon$.
• Estimates on the operator norms and nonlinear terms are used to bound the distance between the controlled solution and the target state, proving approximate controllability.

Experimental results

Research questions

• RQ1 Can the strongly damped semilinear wave equation with memory, impulses, and delay be approximately controllable despite unbounded control sequences caused by memory and impulsive effects?
• RQ2 Does the A.E. Bashirov et al. fixed-point-free technique provide a viable alternative to standard fixed point theorems in this context?
• RQ3 Can the delay term be exploited to guide the control solution along a fixed curve in a short time interval to achieve approximate controllability?
• RQ4 Is approximate controllability preserved when the system includes both memory and impulsive perturbations?
• RQ5 Can the proposed method be generalized to other classes of semilinear evolution equations with similar features?

Key findings

• The system is approximately controllable in the space $\mathcal{Z}^{1/2}$ for any $\epsilon > 0$, meaning the solution can be steered within $\epsilon$-neighborhood of any desired final state.
• The control strategy avoids fixed point theorems by using a time-delayed feedback mechanism that ensures boundedness and convergence.
• The distance between the controlled solution $z^{\delta,\alpha}(\tau)$ and the target state $z_1$ is bounded by $\epsilon$, with $\|z^{\delta,\alpha}(\tau) - y^{\delta,\alpha}(\tau)\| < \epsilon/2$ and $\|y^{\delta,\alpha}(\tau) - z_1\| < \epsilon/2$.
• The method is robust to the presence of memory, impulses, and delay, as demonstrated by the construction of a control sequence that remains bounded despite these effects.
• The approach is extendable to other systems, including impulsive semilinear beam equations with memory and delay, and more general semilinear evolution equations in Hilbert spaces.
• The result holds under the assumption that the nonlinear terms satisfy a Lipschitz-type bound: $|f(t,w,v,u)| \leq a_0\sqrt{|w|^2 + |v|^2} + b_0$, ensuring sufficient regularity for the solution's existence and uniqueness.