Finite element approximation of bilinear optimal control for degenerate Keller-Segel model

In this paper, we propose a new numerical scheme for a nonlinear degenerate chemotaxis model subject to bilinear optimal control. This model introduces major analytical and numerical challenges due to the lack of regularity in a complex framework. For that, we develop a finite element scheme for the direct and adjoint systems. First, many energy estimates are established to prove the stability of discrete solutions for both problems. Next, we solve the associated optimal control problem using a gradient optimization algorithm to ensure accurate reconstruction of the control. Finally, several numerical experiments are performed to demonstrate the stability and effectiveness of the proposed approach in regulating chemotactic aggregation. Consequently, the reliability of the algorithm confirms the influence of the bilinear control on the chemical dosing strategies.