Stochastic algebraic Riccati equations are almost as easy as deterministic ones theoretically

TL;DR

This paper introduces a new theoretical framework showing that stochastic algebraic Riccati equations are fundamentally as simple to solve as deterministic ones, enabling improved analysis and algorithms.

Contribution

The paper reveals the intrinsic algebraic structure of stochastic algebraic Riccati equations, demonstrating their solvability is comparable to deterministic cases.

Findings

Established a novel theoretical framework for stochastic Riccati equations

Proved the intrinsic algebraic structure simplifies solving these equations

Showed that solving stochastic Riccati equations is nearly as easy as deterministic ones

Abstract

Stochastic algebraic Riccati equations, also known as rational algebraic Riccati equations, arising in linear-quadratic optimal control for stochastic linear time-invariant systems, were considered to be not easy to solve. The-state-of-art numerical methods most rely on differentiability or continuity, such as Newton-type method, LMI method, or homotopy method. In this paper, we will build a novel theoretical framework and reveal the intrinsic algebraic structure appearing in this kind of algebraic Riccati equations. This structure guarantees that to solve them is almost as easy as to solve deterministic/classical ones, which will shed light on the theoretical analysis and numerical algorithm design for this topic.