scholarly journals A Note on Optimal Control Problem Governed by Schrödinger Equation

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Yusuf Koçak ◽  
Ercan Çelik ◽  
Nigar Yıldırım Aksoy
Keyword(s):  
Optimal Control ◽  
Variational Inequality ◽  
Schrödinger Equation ◽  
Optimal Control Problem ◽  
Existence And Uniqueness ◽  
Schrodinger Equation ◽  
Optimal Solution ◽  
Necessary Condition ◽  
Existence And Uniqueness Theorem ◽  
The Cost

AbstractIn this work, we present some results showing the controllability of the linear Schrödinger equation with complex potentials. Firstly we investigate the existence and uniqueness theorem for solution of the considered problem. Then we find the gradient of the cost functional with the help of Hamilton-Pontryagin functions. Finally we state a necessary condition in the form of variational inequality for the optimal solution using this gradient.

scholarly journals THE OPTIMAL CONTROL PROBLEM FOR ONE-DIMENSIONAL NONLINEAR SHRODINGER EQUATIONS WITH A SPECIAL GRADIENT TERM

2019 ◽  
pp. 49-66
Author(s):  
G. Yagub ◽  
N. S. Ibrahimov ◽  
M. Zengin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Quality Criterion ◽  
Necessary Condition ◽  
Gradient Term ◽  
Complex Coefficient ◽  
Existence And Uniqueness Theorem ◽  
One Dimensional ◽  
Nonlinear Part

In this paper we consider the optimal control problem for a one-dimensional nonlinear Schrodinger equation with a special gradient term and with a complex coefficient in the nonlinear part, when the quality criterion is a final functional and the controls are quadratically summable functions. In this case, the questions of the correctness of the formulation and the necessary condition for solving the optimal control problem under consideration are investigated. The existence and uniqueness theorem for the solution is proved and a necessary condition is established in the form of a variational inequality. Along with these, a formula is found for the gradient of the considered quality criterion.

scholarly journals On a system of Hamilton-Jacobi-Bellman inequalities associated to a minimax problem with additive final cost

2003 ◽  
Vol 2003 (72) ◽  
pp. 4517-4538 ◽  
Author(s):  
Silvia C. Di Marco ◽  
Roberto L. V. González
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Existence And Uniqueness ◽  
Original Problem ◽  
Auxiliary Problem ◽  
Minimax Problem ◽  
Dynamical Programming ◽  
Minimax Optimal Control ◽  
Hamilton Jacobi Bellman ◽  
The Cost

We study a minimax optimal control problem with finite horizon and additive final cost. After introducing an auxiliary problem, we analyze the dynamical programming principle (DPP) and we present a Hamilton-Jacobi-Bellman (HJB) system. We prove the existence and uniqueness of a viscosity solution for this system. This solution is the cost function of the auxiliary problem and it is possible to get the solution of the original problem in terms of this solution.

scholarly journals The Problem of the Optimal Control with a Lower Coefficient for Weakly Nonlinear Wave Equation in the Mixed Problem

2020 ◽  
Vol 13 (2) ◽  
pp. 314-322
Author(s):  
Gunay Ismayilova
Keyword(s):  
Optimal Control ◽  
Wave Equation ◽  
Variational Inequality ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Necessary Condition ◽  
Weakly Nonlinear ◽  
Necessary Condition Of Optimality

In this paper, we consider the problem of determining the lowest coefficient of weakly nonlinear wave equation. The problem is reduced to the optimal control problem, in the new problem. In the this existence theorem of the optimal control and, the Fre ́echet differentiability of the functional is proved. Also the necessary condition of optimality is derived in view of variational inequality.

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