bilinear optimal control of a reaction-diffusion equation: overcoming boundedness constraints on controls

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.