This tutorial paper introduces the fundamental aspects of Quantum Optimal Control (QOC), a method for designing time-varying electromagnetic fields to perform operations in quantum technologies. The paper is structured to be accessible to physicists, using an analogy with classical Lagrangian and Hamiltonian mechanics to explain key concepts. It emphasizes numerical algorithms for solving QOC problems, providing detailed examples from two-level quantum systems to Bose-Einstein Condensates (BEC) in one-dimensional optical lattices. The paper discusses the connection between optimal processes and the quantum speed limit in two-level systems and describes experimental implementations of optimal control protocols for BEC, highlighting current constraints and limitations. The tutorial covers the mathematical framework of QOC, including the Pontryagin Maximum Principle (PMP), and provides numerical codes for shooting and gradient-based methods. It also addresses the challenges of translating theoretical results into experimental applications, emphasizing the importance of accurate modeling and experimental constraints. The paper concludes with a discussion on future perspectives and a list of mathematical symbols and acronyms used throughout.This tutorial paper introduces the fundamental aspects of Quantum Optimal Control (QOC), a method for designing time-varying electromagnetic fields to perform operations in quantum technologies. The paper is structured to be accessible to physicists, using an analogy with classical Lagrangian and Hamiltonian mechanics to explain key concepts. It emphasizes numerical algorithms for solving QOC problems, providing detailed examples from two-level quantum systems to Bose-Einstein Condensates (BEC) in one-dimensional optical lattices. The paper discusses the connection between optimal processes and the quantum speed limit in two-level systems and describes experimental implementations of optimal control protocols for BEC, highlighting current constraints and limitations. The tutorial covers the mathematical framework of QOC, including the Pontryagin Maximum Principle (PMP), and provides numerical codes for shooting and gradient-based methods. It also addresses the challenges of translating theoretical results into experimental applications, emphasizing the importance of accurate modeling and experimental constraints. The paper concludes with a discussion on future perspectives and a list of mathematical symbols and acronyms used throughout.