Optimal control
Introduction
Optimal control is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in both science and engineering.
History
The field of optimal control originated in the 1950s and 1960s, due to the technological need of optimizing the trajectories of manned spacecraft and unmanned rockets. As such, the field is largely credited to the work of Lev Pontryagin and Richard Bellman.
Mathematical Formulation
The mathematical formulation of an optimal control problem involves the definition of a control system, a performance measure, and constraints on the controls or state variables. The control system is usually described by a set of ordinary differential equations (ODEs) known as the state equations. The performance measure, also known as the cost functional, is an integral over the time interval of interest. The constraints can be either equality constraints, given by the state equations, or inequality constraints on the state and control variables.
Pontryagin's Minimum Principle
One of the key results in optimal control theory is Pontryagin's minimum principle, which provides necessary conditions for optimality. The principle states that the optimal control minimizes the Hamiltonian, a function constructed from the cost functional and the state equations, along the optimal trajectory.
Bellman's Principle of Optimality
Another fundamental result in optimal control theory is Bellman's principle of optimality, which provides the theoretical foundation for dynamic programming, a method for solving complex problems by breaking them down into simpler subproblems. The principle states that an optimal policy has the property that, whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.
Applications
Optimal control theory has been applied in numerous fields, including aerospace engineering, operations research, and economics. In aerospace engineering, for example, it is used to determine the optimal path for a spacecraft to travel from one point to another. In operations research, it is used to optimize the operation of complex systems, such as supply chains or production lines. In economics, it is used to model and analyze dynamic economic systems, such as the growth of an economy over time.