EXISTENCE AND OPTIMAL CONTROL ANALYSIS OF ACID-MEDIATED TUMOR INVASION MODEL
Abstract
The distributed optimal control problem of a highly nonlinear coupled system of reaction-diffusion equations is investigated in the study. Normal cell density, tumor cell density, excess H+ ion concentration, and chemotherapy drug concentration are all represented by partial differential equations (PDEs) in the coupled system of acid-mediated tumor invasion model. It is a usual factor to formulate an optimal control problem by introducing control interventions while considering the tumor invasion model with drug chemotherapy. However, in our model, we consider a constant drug injection rate as a control variable based on biological motivation. The major goal of our optimal control problem is to reduce the overall amount of medicine supplied while minimizing cancer cell proliferation. First, we prove the existence of solutions to the direct problem using the Faedo-Galerkin approximation method, deriving a priori estimates, and then passing to the limit in the approximate solutions using monotonicity and compactness arguments. We introduce a functional to minimize and to establish the existence of optimal control for the proposed optimal control problem. Using the Lagrangian framework, we derive the adjoint problem and necessary optimality condition associated with our problem. Finally, we prove the existence of weak solutions to the adjoint system.
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