Solvability and optimal control for second-order stochastic differential systems under the influence of delay and impulses

Abstract

We explore a new kind of second-order impulsive stochastic differential system with delays. By using the Hausdorff measure of noncompactness and M$\ddot{o}$nch fixed point theorem, we investigated the existence of mild solution for the considered systems. Further, we proved the existence of optimal control of the proposed system by using minimizing sequence, Lagrange functional, and Marzur lemma. An example is provided to validate the obtained results.

Keywords:

• Second-order stochastic systems
• measure of noncompactness
• optimal controls
• infinite delay

AMS Subject Classifications:

• 34K30
• 65C30
• 93E20
• 34K45
• 47D09

Acknowledgments

We are very thankful to the anonymous reviewers and associate editor for their constructive comments and suggestions which help us to improve the manuscript.

Disclosure statement

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