On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation

Abstract

In this paper, an optimal control problem for Schrödinger equation with complex coefficient which contains gradient is examined. A theorem is given that states the existence and uniqueness of the solution of the initialboundary value problem for Schrödinger equation. Then for the solution of the optimal control problem, two different cases are investigated. Firstly, it is shown that the optimal control problem has a unique solution for α > 0 on a dense subset G on the space H which contains the measurable square integrable functions on (0, l) and secondly the optimal control problem has at least one solution for any α ≥ 0 on the space H.

A Necessary Optimality Condition on the Control of a Charged Particle

“…As can be seen, all of the aforementioned works are concerned with OCPs for standard Schrödinger equations (linear or nonlinear), that is, the Schrödinger equation does not contain any specific gradient term. But in [13], the authors prove the existence of the optimal solution for an OCP with a Lions-type functional for the LSEwSGT. In [25,26], the existence of the optimal solution and necessary optimality conditions are given for OCPs with a final functional for the NLSEwSGT.…”

The solvability of the optimal control problem for a nonlinear Schrödinger equation

“…In the studies [13][14][15][16][17][18][19][20][21][22], the objective functional is considered as a final functional and the controlled system is generally stated by the Schrödinger equation. In [23][24][25][26][27], the OCPs with Lions functional has been studied and the controlled system is stated by linear or nonlinear Schrödinger equations. Also, in [28][29][30], the OCPs for systems whose state is expressed by the Schrödinger equation with the boundary functional has been studied.…”