Computation of Optimal Control Problems with Terminal Constraint via Variation Evolution

TL;DR

This paper develops a variation evolution method for solving optimal control problems with terminal constraints, deriving explicit conditions and discretizing the problem for numerical solution.

Contribution

It introduces a compact variation evolution approach with explicit costate and Lagrange multiplier expressions for constrained optimal control problems.

Findings

Derivation of the evolution PDE for optimal control with terminal constraints

Explicit expressions for costates and Lagrange multipliers

Discretization method using PDE numerical techniques for solution

Abstract

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.