Optimal control problem for higher-order non-instantaneous impulsive fractional system with non-local conditions

Abstract

The purpose of this article is to study the optimal control problem of higher-order non-instantaneous impulsive fractional system with non-local conditions by virtue of cosine operator theory in the Hilbert space. Initially, the sufficient conditions for the existence of a mild solution of the proposed system are verified by applying the fixed point theorem in infinite dimensional space. Further, we formulate and prove the existence results for optimal control of the proposed system with the corresponding cost functional by using Balder's theorem. Finally, an example is provided to illustrate the main results.

Keywords:

• Frational calculus
• fixed point theorem
• mild solution
• non-instantaneous impulsive equations
• optimal control problem

Acknowledgements

The authors would like to thank the helpful comments and suggestions of the Associate Editor and the anonymous reviewers, which have improved the presentation of this paper.

Disclosure statement

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

Additional information

Notes on contributors

Dhanalakshmi Kasinathan

Dhanalakshmi Kasinathan received M.Sc degree from Sacred Heart College of Arts & Science College, Tiruppattur. Currently pursuing PhD in Mathematics at The Gandhigram Rural Institute (Deemed to be University), Gandhigram. The research interests includes dynamical system & fractional differential equations, stochastic differential equations, stability & control theory.

Ravikumar Kasinathan

Ravikumar Kasinathan received M.Sc degree from Ramakrishna Mission, Vivekanandha College, Madras University and PhD from PSG College of Arts & Science, Coimbatore, Bharathiar University, India. Currently working as an Assistant Professor at PSG College of Arts & Science, Coimbatore. The research interest includes control theory, stochastic differential/integro-differential systems.

Ramkumar Kasinathan

Ramkumar Kasinathan received M.Sc degree from Ramakrishna Mission, Vivekanandha College, Madras University and PhD from PSG College of Arts & Science, Coimbatore, Bharathiar University, India. Currently working as an Assistant Professor at PSG College of Arts & Science, Coimbatore. The research interest includes control theory, stochastic differential systems.

Dimplekumar Chalishajar

COL. Dimple Chalishajar did his Ph.D. from the University of Baroda and the Indian Institute of Science, India. His fields of interest are control theory, dynamical systems/inclusions, fractional-order systems, time and state delay systems PDEs, mathematical biology, functional analysis, etc. He has published 108 research articles in several peer-reviewed international journals with one monogram. He is on the editorial board of more than 12 international journals and has been a reviewer in more than 25 reputed international journals. He has reviewed 225 research articles so far to help the mathematical community. He has been invited to deliver an expert lecture by several universities nationally and also recognised internationally. He has delivered 80 research talks so far. He has an overall teaching experience of 25 years and he has been involved in the research for the last 27 years. He is an author/co-author of five books with one monogram. He worked as a professor and head of the department for 12 years in India and since 2008, he has been serving at Virginia Military Institute (VMI), USA. He is also actively involved in several administrative committees at VMI like, the Chair of the VMIRL Liaison Committee, Undergraduate Research Committee, Faculty Development Committee, etc.