Fundamental solutions for semi-linear neutral retarded integro-differential systems and applications to control problems

Abstract

In this work, we establish the theory of fundamental solution, including the existence and some essential properties, for neutral linear integro-differential equations with finite delay. In this way, we are able to represent the mild solutions of semi-linear neutral retarded integro-differential equations by fundamental solutions through the Laplace transformation method. Then, as applications, we investigate the problems of approximate controllability, optimal control and time optimal control for this kind of integro-differential evolution systems based on the theory of fundamental solution, resolvent operators and fractional powers, and sufficient conditions for these control problems are obtained, respectively. Finally, an example is presented to illustrate the achieved results.

Keywords:

• Neutral integro-differential system
• resolvent operator
• fundamental solution
• approximate controllability
• optimal control

2010 AMS Subject Classifications:

• 34K40
• 44A10
• 45K05
• 49J20
• 93B05

Acknowledgments

We would like to thank the referees greatly for the careful review and the valued comments on this article.

Disclosure statement

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

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work is supported by NSF of China, Science and Technology Commission of Shanghai Municipality (STCSM) [grant number 18dz2271000].