Existence of Solutions for the Continuous Algebraic Riccati Equation via Polynomial Optimization

TL;DR

This paper presents a novel polynomial optimization approach using Lasserre's hierarchy to determine the existence of solutions for the continuous algebraic Riccati equation, including cases with nonexistence detection.

Contribution

It introduces a new method that formulates CARE as polynomial optimization problems and employs semi-definite relaxations to find or verify the absence of solutions.

Findings

Can obtain exact positive semi-definite solutions for symmetric coefficient matrices

Can detect nonexistence of solutions effectively

Numerical examples demonstrate improved effectiveness over existing methods

Abstract

This paper studies the solution existence of the continuous-time algebraic Riccati equation (CARE). We formulate the CARE as two constrained polynomial optimization problems, and then use Lasserre's hierarchy of semi-definite relaxations to solve them. Compared to the existing work, our approaches can obtain the exact positive semi-definite solution of the CARE when the coefficient matrices are symmetric. Moreover, our methods can detect the nonexistence of the positive semi-definite solution for the CARE. Numerical examples show that our approaches are effective compared to the existing methods.