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Optimal potentials for problems with changing sign data. (English) Zbl 1409.49007

Summary: In this paper, we consider variational optimal control problems. The state equation is an elliptic partial differential equation of a Schrödinger type, governed by the Laplace operator with a potential, with a right-hand side that may change sign. The control variable is the potential itself that may vary in a suitable admissible class of nonnegative potentials. The cost is an integral functional, linear (but non-monotone) with respect to the state function. We prove the existence of optimal potentials, and we provide some necessary conditions for optimality. Several numerical simulations are shown.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
35J10 Schrödinger operator, Schrödinger equation
35J25 Boundary value problems for second-order elliptic equations

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References:

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