Existence of the optimal controls for a controlled elliptic system with an \(L^0\) term in the cost functional (Q6556591)

The primary aim of this paper is to establish the existence, regularity, and necessary conditions for optimal control in Problem (P). Demonstrating the existence of optimal controls is fundamental for characterizing and deriving them. Typically, a Cesari-type condition, which naturally generalized optimal control problems involving linear state equations with convex cost functionals, is required to ensure the existence of optimal (classical) controls.\N\NSeveral difficulties arose when considering the existence of the optimal controls for Problem (P). First, the control \( u(\cdot) \) is allowed to take values in \( (-\infty, +\infty) \), an open and unbounded range. To address this, a series of approximation problems, where the control took values in a compact set, is studied first. Second, since the map \( u \mapsto \Vert u \Vert_{0} \) is not convex and therefore not weakly lower semicontinuous from \( L_{p}(\Omega) \) to \( \mathbb{R} \), proving the existence of optimal controls in the usual way is infeasible. Consequently, a relaxed version is studied initially, where the existence and necessary conditions for the optimal relaxed controls are derived. It is proven that under our assumptions, every optimal relaxed control is classical. Additionally, \( F \) in the cost functional is an abstract strictly convex function, necessitating classification and detailed discussion.\N\NTo characterize the optimal controls, a necessary condition of Pontryagin's type is given. For the approximation problems, each optimal control coincide with an optimal relaxed control, whose necessary conditions are obtained using convex variational methods. For Problem (P), due to the non-smooth nature of the functional \( u \mapsto \Vert u \Vert_{0} \), spike perturbations are used to deduce the necessary condition. It is then shown that the optimal control is bounded and achieve \( C^{1,\lambda} \)-regularity of the optimal state.