timeoptimal

Timeoptimal (or time-optimal) refers to problems and strategies that aim to reach a specified target state in the minimum possible time under given dynamics and constraints. In control theory, the time-optimal problem is a classical optimal control problem in which the objective is to minimize the final time T subject to state dynamics x'(t) = f(x(t), u(t)) and control constraints u(t) in U, with specified initial and terminal states.

Formulation: minimize T subject to x(0)=x0, x(T)=xf, x'(t)=f(x(t), u(t)) for t in [0,T], u(t) in U. The problem can be approached analytically for simple models, often using Pontryagin's Maximum Principle, which provides necessary conditions for optimality and frequently yields bang-bang controls when U is a convex polytope and f is affine in u. Singular arcs and chattering can occur in more complex cases.

Numerical methods: direct methods transcribe the problem into a finite-dimensional optimization (e.g., direct collocation, multiple shooting) with T as an additional decision variable, or indirect methods solve the associated two-point boundary-value problem. Robustness and model accuracy remain critical concerns in practical applications.

Applications: planning the fastest trajectory for robotics and unmanned vehicles, aerospace flight path optimization, manufacturing process control, and any system where time-to-target is critical.

History: time-optimal control grew from variational problems and the calculus of variations, with foundational results by Pontryagin and his school in the 1950s, and classical problems such as the brachistochrone dating to the 17th century.