Methods for verified stabilizing solutions to continuous-time algebraic Riccati equations

TL;DR

This paper introduces improved verified numerical methods for solving continuous-time algebraic Riccati equations, combining the Krawczyk method with a new cubic-complexity verification approach, and demonstrates their effectiveness on benchmark problems.

Contribution

It enhances the Krawczyk method for Riccati equations and proposes a novel cubic-complexity verification technique based on a fixed-point formulation.

Findings

The improved Krawczyk method achieves results comparable to state-of-the-art algorithms.

The new verification method surpasses existing techniques in some examples.

Both methods are effective on standard benchmark problems.

Abstract

We describe a procedure based on the Krawczyk method to compute a verified enclosure for the stabilizing solution of a continuous-time algebraic Riccati equation $A^*X+XA+Q=XGX$ building on the work of [B.~Hashemi, \emph{SCAN} 2012] and adding several modifications to the Krawczyk procedure. We show that after these improvements the Krawczyk method reaches results comparable with the current state-of-the-art algorithm [Miyajima, \emph{Jpn. J. Ind. Appl. Math} 2015], and surpasses it in some examples. Moreover, we introduce a new direct method for verification which has a cubic complexity in term of the dimension of $X$, employing a fixed-point formulation of the equation inspired by the ADI procedure. The resulting methods are tested on a number of standard benchmark examples.