Optimal control problem for cancer invasion parabolic system with nonlinear diffusion

ABSTRACT

In this paper, we study a distributed optimal control problem of a coupled nonlinear system of reaction–diffusion equations. The system consists of three partial differential equations to represent cancer cell density, matrix-degrading enzymes concentration and oxygen concentration, and an ordinary differential equation to describe the extracellular matrix concentration. Our aim is to minimize the growth of cancer cells by controlling the production of matrix-degrading enzymes. First, we prove the existence and uniqueness of solutions of the direct problem. Then, we prove the existence of an optimal control. Finally, we derive the first-order optimality conditions and prove the existence of weak solutions of the adjoint problem.

Keywords:

• Cancer invasion
• optimal control
• adjoint problem

Acknowledgments

The authors thank the anonymous referee for the comments and suggestions which improved the quality of the article. The first author is thankful to the Ministry of Human Resources Development (MHRD) and National Institute of Technology Goa, India for awarding Junior Research Fellowship.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the second author is supported by the DST-SERB Early Career Award File No. ECR/20l6/000624.