The Variance Margin for the Stabilization of Stochastic Systems and Applications to the Solvability of GAREs

A variance margin problem for linear systems with multiplicative noise is considered. The variance margin <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$vm$</tex-math></inline-formula> is the maximal variance tolerance such that the systems can be robustly stabilized by a state-feedback control. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$vm$</tex-math></inline-formula> also provides a fundamental limit for the solvability of a generalized algebraic Riccati equation (GARE). Fundamental limits in literature are only valid for modified algebraic Riccati equations, which are special cases of GAREs. The purpose of this article is to derive new information on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$vm$</tex-math></inline-formula>. Main results are as follows: 1) The variance margin of a control is established in direct and analytical ways and a new characterization of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$vm$</tex-math></inline-formula> is proposed via the solution to a spectral optimization problem. 2) New analytical bounds of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$vm$</tex-math></inline-formula> are established. 3) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$vm$</tex-math></inline-formula> is completely determined by the optimal cost of a standard homogeneous linear quadratic regulation problem with a specified initial state if the systems have no state dependent noise and have a single input. If additionally, certain eigenvalue and controllability conditions are satisfied, an analytical expression of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$vm$</tex-math></inline-formula> is available.