Abstract
In this paper, we consider a bilinear optimal control with final observation for the heat equation. We first give the existence and uniqueness results of weak solutions under an integral constraints on the control and the state. We prove the existence and uniqueness of the solution to a quadratic boundary optimal control problem by means of compactness Sobolev space embedding. Then we provide a characterization of the optimal control via the Euler-Lagrange first-order optimality conditions. The unique optimal control is characterized in terms of the solution of the optimality system, which consists of the state system coupled with an adjoint system.
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