Optimal control of a parabolic equation with a nonlocal nonlinearity
Résumé
This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity.
The system is described by a parabolic equation involving a nonlinear term that depends on the solution
and its integral over the domain. We prove the existence and uniqueness of the solution to the system and
the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional
and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order
necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality
are established