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An effective form of the Variation Evolving Method (VEM), which originatesfrom the continuous-time dynamics stability theory, is developed for theclassic time-optimal control problem with control constraint. Within themathematic derivation, the Pontryagin's Minimum Principle (PMP) optimalityconditions are used. Techniques including limited integrator and corner pointsare introduced to capture the right solution. The variation dynamic evolvingequation may be reformulated as the Partial Differential Equation (PDE), andthen discretized as finite-dimensional Initial-value Problem (IVP) to be solvedwith common Ordinary Differential Equation (ODE) integration methods. Anillustrative example is solved to show the effectiveness of the method. Inparticular, the VEM is further developed to be more flexible in treating theboundary conditions of the Optimal Control Problem (OCP), by initializing thetransformed IVP with arbitrary initial values of variables.

Computation of time-optimal control problem with variation evolution principle