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Journal of Applied Nonlinear Dynamics 12(2) (2023) 339--351 | DOI:10.5890/JAND.2023.06.011

M. Navaneetha Krishnan$^1$, N. Barani Balan$^1$, L. Shangerganesh$^2$, J. Manimaran$^3$

$^1$ Department of Mathematics, Central University of Tamil Nadu, Thiruvarur - 610 005, India

$^2$ Department of Applied Sciences, National Institute of Technology Goa, Goa - 403 401, India

$^3 $ Department of Mathematics, Vellore Institute of Technology, Chennai - 600127, India

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Abstract

In this article, we investigate an optimal control problem for acid-mediated cancer invasion model which describes the normal cell density, the tumor cell density, the excess $H^+$ ion concentration, the extracellular matrix, and active metalloproteinases. The main objective of this paper is to minimize the growth of tumor cells by controlling the excess production of $H^+$ ions. First, we establish the existence of weak solutions by using the Faedo-Galerkin approximation method, then we prove the existence of optimal control. Further, we derive the necessary optimality condition for acid-mediated cancer invasion model. Finally, we illustrate the importance of the control term using some numerical simulations.

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