Compact Formulation of the First Evolution Equation for Optimal Control Computation

TL;DR

This paper introduces a compact, dimension-reduced evolution equation for optimal control that enhances computational efficiency and accuracy, connecting classic iteration methods with continuous evolution equations.

Contribution

It develops a compact form of the first evolution equation in VEM, reducing problem scale and improving performance over the primary form.

Findings

The compact evolution equation outperforms the primary form in precision.

It reduces the scale of the initial-value problem significantly.

The scheme links gradient methods to discrete evolution equations.

Abstract

The first evolution equation is derived under the Variation Evolving Method (VEM) that seeks optimal solutions with the variation evolution principle. To improve the performance, its compact form is developed. By replacing the states and costates variation evolution with that of the controls, the dimension-reduced Evolution Partial Differential Equation (EPDE) only solves the control variables along the variation time to get the optimal solution, and its definite conditions may be arbitrary. With this equation, the scale of the resulting Initial-value Problem (IVP), transformed via the semi-discrete method, is significantly reduced. Illustrative examples are solved and it is shown that the compact form evolution equation outperforms the primary form in the precision, and the efficiency may be higher for the dense discretization. Moreover, in discussing the connections to the classic iteration methods, it is uncovered that the computation scheme of the gradient method is the discrete implementation of the third evolution equation, and the compact form of the first evolution equation is a continuous realization of the Newton type iteration mechanism.