An optimal control problem with final observation for systems governed by nonlinear Schrödinger equation

Abstract

In this paper, an optimal control problem with final observation for systems governed by nonlinear time-dependent Schr?dinger equation is studied. The existence and uniqueness of the solution of considered optimal control problem are proved. The first variation of objective functional is obtained and a necessary optimality condition in the variational form is given.

A Necessary Optimality Condition on the Control of a Charged Particle

“…Then we deduce from above that Φ(x, t) satisfies integral identity (11), that is, Φ is a unique solution of problems ( 8)- (10) in C 0 (Λ T , L 2 (X)) in the meaning of Definition 2. Also, since w = Φ − z, it is written that f or any t ∈ Λ T by (25). Also, since g(s, t) = 2 χ (u(s, t) − κ(s, t)), we can easily say that function Φ provides estimate (72), which completes the proof.…”

“…Proof. We have proven above that problems ( 15)-( 17) have a unique solution w in C 0 (Λ T , L 2 (X)) in the meaning of Definition 3 satisfying estimate (71), and also, problems ( 18)-( 20) have a unique solution z in W 2,1 2 (Ω) satisfying estimate (25). Hence, since w = Φ − z, we come to the conclusion that problems ( 8)-( 10) have a unique solution…”

“…Under the assumed conditions, from the definition of F in problems ( 15)-( 17) and estimate (25), it seems that F ∈ L 2 (Ω). Definition 3.…”

“…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. Salmanov [27] gives the existence and uniqueness theorems for a solution of an OCP with a Lions-type functional for the NLSEwSGT.…”

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

“…The objective functionals can be diversely chosen with regard to our purpose such as final, boundary or Lions-type functional [12]. 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.…”