On the closest stable/unstable nonnegative matrix and related stability radii

SIAM Journal on Matrix Analysis and Applications. 2018. Vol. 39. No. 4. P. 1642–1669.

Guglielmi N., Protasov V.

We consider the problem of computing the closest stable/unstable nonnegative matrix

to a given real matrix. The distance between matrices is measured in the Frobenius norm. The

problem is addressed for two types of stability: the Schur stability (the matrix is stable if its spectral

radius is smaller than one) and the Hurwitz stability (the matrix is stable if its spectral abscissa is

negative). We show that the closest unstable matrix can always be explicitly found. The problem

of computing the closest stable matrix to a nonnegative matrix is a hard problem even if the stable

matrix is not constrained to be nonnegative. Adding the nonnegativity constraint makes the problem

even more dicult. For the closest stable matrix, we present an iterative algorithm which converges

to a local minimum with a linear rate. It is shown that the total number of local minima can be

exponential in the dimension. Numerical results and the complexity estimates are presented.

Keywords: stability, dynamical systems, динамическая система, асимптотическая устойчивость, Stability analysis, Positive matrix, положительная матрица, gradient relaxation, градиентный метод