Semidefinite approximations of the polynomial abscissa

Roxana Hess (1) , Didier Henrion (2, 1) , Jean-Bernard Lasserre (1) , Tien Son Pham (3)

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
2 CTU - Czech Technical University in Prague
3 DEPARTMENT OF MATHEMATICS - Department of Mathematics

Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is Hölder continuous, and not locally Lipschitz in general, which is a source of numerical difficulties for designing and optimizing control laws. In this paper we propose simple approximations of the abscissa given by polynomials of fixed degree, and hence controlled complexity. Our approximations are computed by a hierarchy of finite-dimensional convex semidefinite programming problems. When their degree tends to infinity, the polynomial approximations converge in norm to the abcissa, either from above or from below.

Mots clés: semialgebraic optimization; polynomial approximations; Linear systems control; non-convex non-smooth optimization; semidefinite programming

Domaines: Optimisation et contrôle [math.OC]

Roxana Hess, Didier Henrion, Jean-Bernard Lasserre, Tien Son Pham. Semidefinite approximations of the polynomial abscissa. SIAM Journal on Control and Optimization, 2016, 54 (3), pp. 1633-1656. ⟨hal-01181623⟩