Pathconstrained

Pathconstrained is a term used in optimization, control theory, and robotics to describe problems or systems in which the trajectory of the state and/or control variables must satisfy constraints at every point along a path, rather than only at initial or final points. In continuous-time formulations, path constraints take the form h_i(x(t), u(t), t) ≤ 0 for all t in [t0, tf], where x is the state, u is the control, and h_i are given constraint functions. Additional endpoint constraints on the initial or final state may be present, but the key feature is the per-time requirement.

This concept is central to optimal control and trajectory optimization. Examples include an autonomous vehicle required to stay within lane boundaries and avoid obstacles, a robotic arm that must maintain joint limits and safe torque during a move, and aerospace trajectory planning where environmental constraints depend on position along the flight path.

Common path-constrained problems are solved using direct methods, such as transcription to a nonlinear programming problem via direct collocation or multiple shooting, or indirect methods based on necessary conditions like the Pontryagin maximum principle. To improve feasibility and convergence, practitioners employ penalty methods, barrier methods, augmented Lagrangian techniques, or specialized constraint-preserving integrators. Problem structure, such as convexity of the path constraints or linearity of dynamics, influences tractability.

In summary, pathconstrained problems enforce constraints along the entire trajectory, shaping feasible solutions beyond endpoint conditions and enabling safer, compliant, and feasible motion and control.