Algebraic Riccati Equations and the Distance to the Nearest Uncontrollable Pair
Pascal M. Gahinet, Alan J. Laub · SIAM Journal on Control and Optimization · 1992
A connection is established between nearness to unstabilizability of a stabilizable pair $(A,B)$ of matrices and nearness to singularity of the symmetric positive definite solution to an associated algebraic Riccati equation. From this result, computable upper and lower bounds are derived for the distance of $(A,B)$ to the nearest uncontrollable pair. Numerical tests confirm the validity of the method and potential applications are discussed.