Feedback control of chaotic systems using multiple shooting shadowing and application to Kuramoto–Sivashinsky equation

We propose an iterative method to evaluate the feedback control kernel of a chaotic system directly from the system’s attractor. Such kernels are currently computed using standard linear optimal control theory, known as linear quadratic regulator theory. This is however applicable only to linear systems, which are obtained by linearizing the system governing equations around a target state. In the present paper, we employ the preconditioned multiple shooting shadowing (PMSS) algorithm to compute the kernel directly from the nonlinear dynamics, thereby bypassing the linear approximation. Using the adjoint version of the PMSS algorithm, we show that we can compute the kernel at any point of the domain in a single computation. The algorithm replaces the standard adjoint equation (that is ill-conditioned for chaotic systems) with a well-conditioned adjoint, producing reliable sensitivities which are used to evaluate the feedback matrix elements. We apply the idea to the Kuramoto–Sivashinsky equation. We compare the computed kernel with that produced by the standard linear quadratic regulator algorithm and note similarities and differences. Both kernels are stabilizing, have compact support and similar shape. We explain the shape using two-point spatial correlations that capture the streaky structure of the solution of the uncontrolled system.