Differential Operators and Related Topics pp 141-154 | Cite as

# Oscillations in Systems with Periodic Coefficients and Sector-restricted Nonlinearities

## Abstract

One of the fields of Applied Mathematics that grew up from the pioneering papers of A.M. Liapunov and M.G. Krein is the theory of dynamical systems with periodic coefficients. The results of M.G. Krein and V.A. Yakubovich concerning stability of linear periodic Hamiltonian systems turned out to have applications in the so-called linear-quadratic theory of the controlled systems. The present paper deals with a “subset” of this theory: existence of nonlinear oscillations (periodic and almost periodic solutions) in systems with sector-restricted nonlinearities (the so-called absolutely stable systems). Since almost all results obtained for differential equations have their discrete-time (more or less) counterpart, both continuous time and discrete time periodic cases are presented here. The existence conditions are expressed in terms of an associated periodic Hamiltonian system that is required to be dichotomic and strongly disconjugate. This property may be checked in terms of the properties of an associated matrix Riccati equation or of some Linear Matrix Inequalities.

## Keywords

Hamiltonian System Linear Matrix Inequality Riccati Equation Invariant Manifold Exponential Dichotomy## Preview

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