On the data-sparsity of the solution of Riccati equations with applications to feedback control

TL;DR

This paper investigates the structure of solutions to large-scale Riccati equations with quasiseparable coefficients, introducing two efficient solvers that leverage this property for improved computational performance.

Contribution

It establishes that solutions inherit quasiseparability from coefficients and develops two specialized algorithms for these cases, enhancing solution efficiency.

Findings

Algorithms are effective on synthetic and real-world examples.

Solutions maintain quasiseparability, enabling faster computations.

Applicable to control problems involving PDEs and agent models.

Abstract

Solving large-scale continuous-time algebraic Riccati equations is a significant challenge in various control theory applications. This work demonstrates that when the matrix coefficients of the equation are quasiseparable, the solution also exhibits numerical quasiseparability. This property enables us to develop two efficient Riccati solvers. The first solver is applicable to the general quasiseparable case, while the second is tailored to the particular case of banded coefficients. Numerical experiments confirm the effectiveness of the proposed algorithms on both synthetic examples and case studies from the control of partial differential equations and agent-based models.