Approximate controllability of nonautonomous second-order nonlocal measure driven systems with state-dependent delay

Abstract

This article studies a new class of nonautonomous second-order nonlocal measure driven control systems involving state-dependent delay (SDD). The system presented in this paper can deal with a large class of hybrid systems with their Zeno behaviour having no restrictions. We use the theory of measure of noncompactness to develop our main results. The existence of a solution to the considered problem is obtained via the Mönch fixed point theorem. The approximate controllability of the aforementioned system is investigated under the assumption that the associated linear system is approximately controllable. To investigate our main results, the integral is taken in the Bochner–Stieltjes sense that makes the obtained results more general. In the end, an example is formulated for the practical usefulness of the established theory.

Keywords:

• Measure driven differential equations
• regulated functions
• nonautonomous second-order systems
• Bochner–Stieltjes integral
• approximate controllability

Acknowledgments

We are very grateful to anonymous referees and the editor for their constructive suggestions, which improve the quality of this manuscript.

Disclosure statement

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

Additional information

Funding

This work was supported by the Council of Scientific and Industrial Research [grant number 09/045(1551)/2017-EMR-I] and Institution of Eminence, University of Delhi [grant number IoE/FRP/PCMS/2020/27].