Optimal control on a brain tumor growth model with lactate metabolism, viscoelastic effects, and tissue damage

In this paper, we study an optimal control problem for a brain tumor growth model that incorporates lactate metabolism, viscoelastic effects, and tissue damage. The PDE system, introduced by Cavalleri, Colli, Miranville, and Rocca in [Nonlinear Anal.: Real World Appl. 87 (2026), 104419], couples a Fisher–Kolmogorov-type equation for tumor cell density with a reaction-diffusion equation for the lactate, a quasi-static force balance governing the displacement, and a nonlinear differential inclusion for tissue damage. The control variables, representing chemotherapy and a lactate-targeting drug, influence tumor progression and treatment response. Starting from well-posedness, regularity, and continuous dependence results proven in the aforementioned article, we define a suitable cost functional and prove the existence of optimal controls. Then, we analyze the differentiability of the control-to-state operator and establish a necessary first-order condition for treatment optimality.