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Constructive solution of a bilinear optimal control problem for a Schrödinger equation. (English) Zbl 1153.49023
Summary: Often considered in numerical simulations related to the control of quantum systems, the so-called monotonic schemes have not been so far much studied from the functional analysis point of view. Yet, these procedures provide an efficient constructive method for solving a certain class of optimal control problems. This paper aims both at extending the results already available about these algorithms in the finite-dimensional case (i.e., the time-discretized case) and at completing those of the continuous case. This paper starts with some results about the regularity of a functional related to a wide class of models in quantum chemistry. These enable us to extend an inequality due to Łojasiewicz to the infinite-dimensional case. Finally, some inequalities proving the Cauchy character of the monotonic sequence are obtained, followed by an estimation of the rate of convergence.

MSC:
49K40 Sensitivity, stability, well-posedness
49N90 Applications of optimal control and differential games
35J10 Schrödinger operator, Schrödinger equation
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