bilinear optimal control, reaction-diffusion equations, essen-tially bounded controls, optimality conditions, Applied mathematics. Quantitative methods, T57-57.97

We study an optimal control problem for bilinear reaction-diffusion equa-tions. The novelty of our approach lies in considering controls from the space of essentially bounded functions without imposing a priori bounds on the admissible set. We introduce an auxiliary problem with controls in Lp spaces $(p > 1 + N /2, where N $ is the spatial dimension) and demon-strate that both formulations are equivalent in terms of optimal solutions. Under suitable assumptions on the system parameters, we establish the ex-istence of optimal controls and derive first-order optimality conditions. Our theoretical findings are supported by numerical simulations that validate the practical effectiveness of the proposed approach. This work provides a new framework for handling bilinear control problems without artificial constraints, offering potential applications in population dynamics and eco-logical management.