Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients

Abstract

Linear systems with partially uncertain time-dependent coefficients naturally arise in various areas of applied sciences. The Lyapunov function method became the prevailing approach to stability analysis of these systems, where it implies sufficient conditions for asymptotic stability. However, this method is not applicable to Hamiltonian systems, because they are at the utmost neutrally (but not asymptotically) stable. This paper describes stability analysis of linear Hamiltonian systems with partially uncertain periodic coefficients resulting in a generalization of a famous Yakubovich's stability theorem which significantly reduces its computational burden. We assume that the Hamiltonian is described as a sum of known and uncertain matrices, and for the latter only bilateral bounds are known. We propose a definition of such a system's stability and derive a necessary and sufficient stability criterion determined by positions of the multipliers of the corresponding marginal systems. Systems with a more general definition of uncertain matrices are also considered.