Optimal control of PDEs in a complex space setting; application to the Schr\"odinger equation

TL;DR

This paper develops optimality conditions for control problems in complex spaces and applies them to the Schrödinger equation, deriving first and second order conditions for bilinear controls with linear control inputs.

Contribution

It introduces a framework for optimal control in complex spaces and applies it specifically to quantum systems governed by the Schrödinger equation, including bilinear control analysis.

Findings

Derived first and second order optimality conditions for Schrödinger control problems.

Addressed control constraints and linear control input cases.

Provided theoretical foundation for quantum control optimization.

Abstract

In this paper we discuss optimality conditions for abstract optimization problems over complex spaces. We then apply these results to optimal control problems with a semigroup structure. As an application we detail the case when the state equation is the Schr\"{o}dinger one, with pointwise constraints on the "bilinear" control. We derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly.