On the Computation of Optimal Control Problems with Terminal Inequality Constraint via Variation Evolution

TL;DR

This paper develops a variation evolving method for solving optimal control problems with terminal inequality constraints, deriving new optimality conditions and introducing a numerical soft barrier to improve solution accuracy.

Contribution

It introduces a novel variation evolving framework for OCPs with terminal inequalities, including new costate-free optimality conditions and a numerical soft barrier technique.

Findings

Derived the right EPDE and costate-free optimality conditions.

Established analytic relations between states, controls, and multipliers.

Demonstrated the effectiveness of the numerical soft barrier in simulations.

Abstract

Studies regarding the computation of Optimal Control Problems (OCPs) with terminal inequality constraint, under the frame of the Variation Evolving Method (VEM), are carried out. The attributes of equality constraints and inequality constraints in the generalized optimization problem is traversed, and the intrinsic relations to the multipliers are uncovered. Upon these preliminaries, the right Evolution Partial Differential Equation (EPDE) is derived, and the costate-free optimality conditions are established. Besides the analytic expression for the costates in the classic treatment, they also reveal the analytic relations between the states, the controls and the (Lagrange and KKT) multipliers, which adjoin the terminal (equality and inequality) constraints. Moreover, in solving the transformed Initial-value Problems (IVPs) with common Ordinary Differential Equation (ODE) integration methods, the numerical soft barrier is proposed to eliminate the numerical error resulting from the suddenly triggered inequality constraint and it is shown to be effective.