Pontryagin's maximum principle

Pontryagin's maximum principle gives necessary conditions for an optimal control trajectory. It requires the trajectory and an associated costate to satisfy differential equations while the control maximizes a Hamiltonian.

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Optimal control: Optimal control selects time-varying inputs to steer a dynamical system while optimizing an objective. The principle supplies necessary conditions for solving this broader class of control problems.
State-space representation: A mathematical description of a system using state variables and equations governing their evolution. The principle applies its state equation along the candidate optimal trajectory.
Lev Pontryagin: A Soviet mathematician who led the team that developed the maximum principle for optimal control. The principle bears his name and emerged from his research group's work.
Spacecraft trajectory optimization: The design of spacecraft paths and control inputs to optimize objectives such as fuel use or travel time. The principle can derive necessary conditions for fuel-efficient or time-optimal trajectories.
Bellman equation: A functional equation expressing the recursive value of an optimization problem across states and time. It characterizes optimal value functions, unlike the principle's trajectory-based necessary conditions.
Hamiltonian (optimal control): A function combining a system's dynamics, control, and costate to encode the instantaneous contribution to an optimal-control objective. Its maximization with respect to control is the principle's central condition.
Control variable: A decision variable that an optimizer can choose to influence a system's behavior. The Hamiltonian condition selects among the controls allowed by the problem.
Vladimir Boltyansky: A Soviet mathematician who contributed to the development and exposition of the maximum principle. He was a member of Pontryagin's research group that established the result.
Time-optimal control: Control that steers a system to a target in the least possible time. Maximizing the Hamiltonian yields necessary conditions for minimum-time policies.
Hamilton–Jacobi–Bellman equation: A partial differential equation whose solution gives the value function for a class of optimal-control problems. It can provide sufficient global solutions where the maximum principle typically gives necessary conditions.