Computation of time-optimal control problem with variation evolution principle
This work provides a new computational approach for solving classic optimal control problems, but it is incremental as it builds on existing PMP conditions and is demonstrated only on a single example.
The authors developed a Variation Evolving Method (VEM) for solving time-optimal control problems with control constraints, demonstrating its effectiveness on an illustrative example. The method reformulates the problem as a PDE and then as an IVP solvable by standard ODE integrators.
An effective form of the Variation Evolving Method (VEM), which originates from the continuous-time dynamics stability theory, is developed for the classic time-optimal control problem with control constraint. Within the mathematic derivation, the Pontryagin's Minimum Principle (PMP) optimality conditions are used. Techniques including limited integrator and corner points are introduced to capture the right solution. The variation dynamic evolving equation may be reformulated as the Partial Differential Equation (PDE), and then discretized as finite-dimensional Initial-value Problem (IVP) to be solved with common Ordinary Differential Equation (ODE) integration methods. An illustrative example is solved to show the effectiveness of the method. In particular, the VEM is further developed to be more flexible in treating the boundary conditions of the Optimal Control Problem (OCP), by initializing the transformed IVP with arbitrary initial values of variables.