Dissipative Linear Stochastic Hamiltonian Systems

TL;DR

This paper analyzes linear stochastic Hamiltonian systems, focusing on their energy balance, stability, invariant measures, and dissipation properties under random external forces, with applications to feedback control.

Contribution

It provides a comprehensive study of energy relations, stability criteria, and dissipation in linear stochastic Hamiltonian systems, including new insights into their invariant measures and feedback applications.

Findings

Derived energy balance relations for LSH systems.

Established stability conditions for linear stochastic Hamiltonian systems.

Demonstrated dissipation properties using Lyapunov functions.

Abstract

This paper is concerned with stochastic Hamiltonian systems which model a class of open dynamical systems subject to random external forces. Their dynamics are governed by Ito stochastic differential equations whose structure is specified by a Hamiltonian, viscous damping parameters and system-environment coupling functions. We consider energy balance relations for such systems with an emphasis on linear stochastic Hamiltonian (LSH) systems with quadratic Hamiltonians and linear coupling. For LSH systems, we also discuss stability conditions, the structure of the invariant measure and its relation with stochastic versions of the virial theorem. Using Lyapunov functions, organised as deformed Hamiltonians, dissipation relations are also considered for LSH systems driven by statistically uncertain external forces. An application of these results to feedback connections of LSH systems is outlined.