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Optimal Control Theory is a powerful mathematical tool, which has known arapid development since the 1950s, mainly for engineering applications. Morerecently, it has become a widely used method to improve process performance inquantum technologies by means of highly efficient control of quantum dynamics.This tutorial aims at providing an introduction to key concepts of optimalcontrol theory which is accessible to physicists and engineers working inquantum control or in related fields. The different mathematical results areintroduced intuitively, before being rigorously stated. This tutorial describesmodern aspects of optimal control theory, with a particular focus on thePontryagin Maximum Principle, which is the main tool for determining open-loopcontrol laws without experimental feedback. The different steps to solve anoptimal control problem are discussed, before moving on to more advanced topicssuch as the existence of optimal solutions or the definition of the differenttypes of extremals, namely normal, abnormal, and singular. The tutorial coversvarious quantum control issues and describes their mathematical formulationsuitable for optimal control. The connection between the Pontryagin MaximumPrinciple and gradient-based optimization algorithms used for high-dimensionalquantum systems is described. The optimal solution of different low-dimensionalquantum systems is presented in detail, illustrating how the mathematical toolsare applied in a practical way.

Introduction to the Pontryagin Maximum Principle for Quantum Optimal Control