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

TL;DR

This work develops a complex-spatial-frequency approach to optimal control of finite-interval linear evolution PDEs by leveraging the unified transform (Fokas method) to decouple PDEs into frequency-parametrized ODEs. It derives real-line integral representations of LQR-optimal controls, then uses contour deformation in the complex plane to remove dependence on unknown boundary data, yielding a causal, frequency-domain solution that can be rewritten as a state-feedback form with a Toeplitz plus Hankel kernel. For the reaction-diffusion equation, the authors show how the complex-contour representation reduces to a convergent series and a convolution-based feedback law, illustrating both numerical advantages and structural kernel properties. The results demonstrate computational benefits of the integral representation over series representations and reveal how boundary data enter as additive boundary terms in the feedback form. The methodology provides a framework for extending LQR control to a broad class of linear evolution PDEs with general boundary conditions, with potential applications in diffusion, transport, and flow-control problems.

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.

Figures (9)

• Figure 1: Branch cuts for $\sqrt{\upkappa^{2n} + 1}$ with $n=4$
• Figure 2: Consider the PDE $\phi_t=\phi_{xxx}$ with $w_{\text{Re}}(\upkappa)=0, w_{\text{Im}}(\upkappa)=\mathbf{i} \upkappa^3$. $\upomega(\upkappa)\sim \mathbf{i} \upkappa^3$ and $\bar{\upomega}(\upkappa)\sim -\mathbf{i} \upkappa^3$ for large $\upkappa$. The blue regions represent positive real parts.
• Figure 3: Contour deformation from $\mathbb{R}$ to $\partial \mathcal{D}^+$ and $\partial \mathcal{D}^-$
• Figure 4: Classes of PDEs satisfying the requirements of the general procedure
• Figure 5: Contour deformation for \ref{['eq:state-before-deform']}. The red lines are branch cuts of $\upomega(\upkappa)$.

Theorems & Definitions (30)

• Remark 1
• Remark 2
• Example 1
• Example 2
• Remark 3
• Remark 4
• Example 3
• Example 4
• Example 5
• Theorem 1