Optimal control

Optimal control studies how to choose time-varying inputs to a dynamic system so a specified objective is minimized or maximized, subject to system dynamics and constraints.

Pontryagin's maximum principle: A set of necessary conditions for optimal trajectories and controls in constrained dynamic optimization. It derives the costate equations and pointwise control rule central to many optimal-control solutions.
State-space representation: A mathematical description of a system using state variables, inputs, and differential or difference equations. It supplies the state dynamics that an optimal control problem seeks to steer.
Linear–quadratic regulator: An optimal feedback controller for linear systems with quadratic state and control costs. It is a tractable, widely used special case with a closed-form Riccati solution.
Classical feedback control: Control design based mainly on input-output behavior, stability, and frequency-domain methods. It often prioritizes stability and response shaping rather than optimizing an explicit trajectory objective.
Richard Bellman: An American mathematician who developed dynamic programming and the principle of optimality. His recursive framework established a major foundation for feedback-based optimal control.
Hamilton–Jacobi–Bellman equation: A nonlinear partial differential equation characterizing the value function of a dynamic optimization problem. It gives a global, feedback-oriented route to optimal control through the value function.
Objective function: A function or functional that assigns a numerical value to a candidate decision or trajectory. It specifies which system outcomes count as better and what the controller optimizes.
Model predictive control: A feedback strategy that repeatedly optimizes a finite-horizon model and applies only the first control action. It implements constrained optimal control online as new measurements arrive.
Robust control: Control design that guarantees specified behavior despite bounded uncertainty or disturbances. It emphasizes worst-case guarantees, while nominal optimal control can be sensitive to model errors.
Lev Pontryagin: A Soviet mathematician who led the development of the maximum principle for optimal processes. His principle supplied necessary conditions suited to constrained control trajectories.

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