Choice of a control input to minimize an objective subject to differential equations for a state and constraints on the control, state or terminal time. Necessary conditions such as the Pontryagin maximum principle require regularity hypotheses and do not by themselves prove global optimality.
A control input taking only extreme admissible values except possibly on a switching or singular set. If the Hamiltonian of an optimal-control problem is affine in a scalar control , a nonzero coefficient selects or ; a zero coefficient does not determine the control.
An adjoint variable enforcing the state dynamics in optimal control. In the normal minimization convention it satisfies for the Hamiltonian of an optimal-control problem, with terminal conditions determined by the endpoint constraints.
In a normal minimization convention for optimal control, the running cost and dynamics define . The multiplier is a costate; this Hamiltonian is a device for variational necessary conditions rather than necessarily a physical energy.
For a regular finite-horizon optimal control problem, an optimal trajectory admits nontrivial multipliers and a costate obeying adjoint equations, together with pointwise optimization of the Hamiltonian of an optimal-control problem. In the normal minimization convention , the conditions are , and minimization of over admissible controls; endpoint and state constraints change the terminal conditions. Abnormal extremals use a zero cost multiplier and must not be silently excluded.
For a smooth normal optimal control problem with freely variable terminal time, fixed terminal state independent of that time and terminal cost , the optimized Hamiltonian of an optimal-control problem at completion satisfies . It follows by varying the terminal time in the augmented objective; restrictions on terminal time or a moving endpoint introduce additional terms or inequalities.
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Optimal control refers to a mathematical and engineering discipline that deals with finding a control policy for a dynamic system to optimize a certain performance criterion. The goal is to determine the control inputs that will minimize (or maximize) a particular objective, which often involves the system's state over time. ### Key Concepts of Optimal Control: 1. **Dynamic Systems**: These are systems that evolve over time according to specific rules, often governed by differential or difference equations.