Positive semigroups and algebraic Riccati equations in Banach spaces

TL;DR

This paper extends Wonham's theorem to Banach spaces, establishing conditions for the unique stabilizing solution of algebraic Riccati equations using positive semigroup theory and a novel iterative approach.

Contribution

It introduces a new method treating the linear part as a positive semigroup generator and the quadratic part as an order concave map, generalizing Riccati equation solvability to Banach spaces.

Findings

Proves existence and uniqueness of stabilizing solutions in Banach spaces.

Develops a monotone convergence method for approximating solutions.

Generalizes Lyapunov equations and stability criteria to Banach space setting.

Abstract

We generalize Wonham's theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to A*P+PA-PBB*P+C*C=0 when (A,B) is exponentially stabilizable and (C,A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton's iteration for concave functions is then used to approximate the solution, the approximations converge in the strong operator topology, and the convergence is monotone. The linearized equations are the well-known Lyapunov equations of the form A*P+PA=-Q, and semigroup stability criterion in terms of them is also generalized.