Computation of Optimal Control Problems with Terminal Constraint via Variation Evolution
For researchers in optimal control, this provides a new computational approach for problems with terminal constraints, but the paper lacks experimental validation or comparison to existing methods.
This paper extends the Variation Evolving Method (VEM) to solve optimal control problems with terminal constraints, deriving evolution PDEs and costate-free optimality conditions. The method is discretized into initial-value problems solved with standard ODE integrators.
Enlightened from the inverse consideration of the stable continuous-time dynamics evolution, the Variation Evolving Method (VEM) analogizes the optimal solution to the equilibrium point of an infinite-dimensional dynamic system and solves it in an asymptotically evolving way. In this paper, the compact version of the VEM is further developed for the computation of Optimal Control Problems (OCPs) with terminal constraint. The corresponding Evolution Partial Differential Equation (EPDE), which describes the variation motion towards the optimal solution, is derived, and the costate-free optimality conditions are established. The explicit analytic expressions of the costates and the Lagrange multipliers adjoining the terminal constraint, related to the states and the control variables, are presented. With the semi-discrete method in the field of PDE numerical calculation, the EPDE is discretized as finite-dimensional Initial-value Problems (IVPs) to be solved, with common Ordinary Differential Equation (ODE) numerical integration methods.