Optimal bilinear control of nonlinear stochastic Schrödinger equations driven by linear multiplicative noise.(English)Zbl 1431.60053

Summary: We analyze the bilinear optimal control problem of quantum mechanical systems with final observation governed by a stochastic nonlinear Schrödinger equation perturbed by a linear multiplicative Wiener process. The existence of an open-loop optimal control and first-order Lagrange optimality conditions are derived, via Skorohod’s representation theorem, Ekeland’s variational principle and the existence for the linearized dual backward stochastic equation. Moreover, our approach in particular applies to the deterministic case.

MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35Q40 PDEs in connection with quantum mechanics 49K20 Optimality conditions for problems involving partial differential equations 35J10 Schrödinger operator, Schrödinger equation
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