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<p>In this paper, we attempted to synthesize a nonlinear optimal control procedure for the quantum equation. To this end, to analyze the quantum optimal control system derived from the time-dependent Schrödinger equation, we developed a linearization method. Using a new approach based on the embedding process, we first formulated the system in variational form. Then, by defining a positive Radon measure, we represented the problem in a space of measures. Consequently, the problem was transformed into an infinite linear one, whose solution is guaranteed. At this stage, by applying two subsequent approximation steps, the optimal solution was identified through an optimization search technique. This method effectively transforms a potentially complex, nonlinear problem into a linear one by utilizing mathematical tools and approximations. It simplifies the problem-solving process, making it more tractable and amenable to computational methods. The steps involved mathematical abstraction, transformation into a suitable space, expansion to ensure solvability, reduction to a finite-dimensional problem, and finally, determination and visualization of the optimal solutions.</p>

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A linearization technique for solving some quantum optimal control problems governed by the Schrödinger equation

  • 期刊论文

  • 发表日期：

    2025年8月18日

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  • Hajar Alimorad

摘要

In this paper, we attempted to synthesize a nonlinear optimal control procedure for the quantum equation. To this end, to analyze the quantum optimal control system derived from the time-dependent Schrödinger equation, we developed a linearization method. Using a new approach based on the embedding process, we first formulated the system in variational form. Then, by defining a positive Radon measure, we represented the problem in a space of measures. Consequently, the problem was transformed into an infinite linear one, whose solution is guaranteed. At this stage, by applying two subsequent approximation steps, the optimal solution was identified through an optimization search technique. This method effectively transforms a potentially complex, nonlinear problem into a linear one by utilizing mathematical tools and approximations. It simplifies the problem-solving process, making it more tractable and amenable to computational methods. The steps involved mathematical abstraction, transformation into a suitable space, expansion to ensure solvability, reduction to a finite-dimensional problem, and finally, determination and visualization of the optimal solutions.