A complex spatial frequency approach to optimal control of finite-extent linear evolution systems

TL;DR

This paper develops a novel complex spatial frequency method for deriving optimal boundary control laws for finite-extent linear PDEs, overcoming limitations of existing series-based approaches and enabling efficient computation.

Contribution

It introduces a unified transform approach to derive explicit, causal optimal control formulas for PDEs with general boundary conditions, extending beyond homogeneous cases.

Findings

Integral representation improves computational efficiency.

Series representation provides a convolution form for control.

Method handles general boundary conditions effectively.

Abstract

We consider the linear quadratic regulator (LQR) for one-dimensional linear evolution partial differential equations (PDEs) on a finite interval in space. The control is applied as an additive forcing term to PDEs. Existing methods for closed-form optimal control only apply to homogeneous (zero) boundary conditions, often resulting in series representations. In this paper, we consider general smooth boundary conditions. We use the unified transform, namely the Fourier transform restricted to the bounded spatial domain, to decouple PDEs into a family of ordinary differential equations (ODEs) parameterized by complex spatial frequency variables. Then, optimal control in the frequency domain is derived using LQR theory for ODEs. The inverse Fourier transform leads to non-causal terms in optimal control corresponding to integrals, over the real line, of future values of unspecified boundary conditions. To eliminate this non-causality, we deform the integrals to well-constructed contours in the complex plane along which the contribution of unknowns vanishes. For the reaction-diffusion equation, we show that the integral representation can be reformulated as a series representation, which leads to a state-feedback convolution form for optimal control, with the boundary conditions appearing as an additive term. In numerical experiments, we illustrate the computational advantages of the integral representation in comparison to the series representation and structural properties of the convolution kernel.