Interior approximate controllability of second-order semilinear control systems

Abstract

In our manuscript, we investigate the interior approximate controllability for the subsequent semilinear second-order system in $L_2\left(\mathrm{\Omega }\right)$ $\begin{array}{rl}{s}^{″}\left(\sigma \right)+A_{0}s\left(\sigma \right)& =B_{0}u\left(\sigma \right)+\xi (\sigma ,s(\sigma \left)\right),0\le \sigma \le \varrho \\ s\left(0\right)& =s_0,s^{\prime }\left(0\right)=s_1,\end{array}$ where $A_0$ is the linear and unbounded operator, $B_0$ is a bounded linear operator and ξ is a nonlinear operator defined on appropriate spaces. The proposed problem can be converted into an equivalent first-order semilinear control system, then the approximate controllability results for the proposed system are obtained from the study of the approximate controllability of the reduced first-order system. The Leray–Schauder alternative theorem and principle of contraction are used in the proof of our main theorems.

Keywords:

• Interior approximate controllability
• semilinear systems
• Leray–Schauder alternative theorem
• contraction principle

Acknowledgments

The authors would like to extend their deepest sincerest gratitude to all the people who helped there in any manner. This article is dedicated in the memory of Lt. Dr. Simegne Tafesse, unfortunately he passed away in a road accident. Dr. Tafesse was very dedicated professor in Department of Mathematics, Haramaya University, Ethopia. As an ardent researcher, he will be remembered for his empirical contribution in the field of Mathematical control theory.

Disclosure statement

No potential conflict of interest was reported by the author(s).