Asymptotic 2p-Moment Stability of Stochastic Linear Systems

Abstract

The exponential p-moment stability of dynamical systems governed by a system of linear Itô stochastic differential equations is revisited. It is well-known that the system of equations governing the evolution of these p-moments is linear and, therefore, available results for asymptotic stability of the linear systems of deterministic first-order homogeneous differential equations are applicable [2,4,7,10,11]. Specifically, the necessary and sufficient conditions for asymptotic stability of a system of deterministic linear equations is that the real parts of all the eigenvalues of the system matrix are negative. The search for stability boundaries involves repeated solutions of eigenvalue problems of dimension equal to the dimension of the system of moments. Alternatively, the well-known Routh-Hurwitz procedure provides conditions for stability in the form of inequalities which involve the system and parametric excitation characteristics. However, these conditions are quite cumbersome since they involve computation of a large number of determinants of orders up to the order of the system of moments. Therefore, Routh-Hurwitz conditions are practical for obtaining stability boundaries only for low order systems. For the special case of an n-th order linear Ito stochastic differential equation, Khasminskii [9] has derived simplified conditions for exponential mean-square (p = 2) stability which mainly depend on the conditions of stability of the first moments, supplemented by only an extra condition involving the evaluation of an n-th order determinant. In this study, new simplified 2p-moment stability conditions are developed which provide significant advantages for the analytical and numerical estimation of the 2p-moment stability border and stability regions. Specifically, it is shown that there exists a real eigenvalue of the system of 2p-moments which is an upper bound of the real parts of all other eigenvalues. Thus, the stability of the system of moments can be examined by computing only the maximum real eigenvalue of the state matrix describing the evolution of the 2p moments. In particular, at the border of stability, one eigenvalue of the system of 2p-moments is equal to zero. Thus, a necessary condition for the system configuration to correspond to a point on the 2p-moment stability boundary is that the determinant of the matrix describing the system of 2p-moments be zero. This condition is a generalization of the mean-square stability boundary condition obtained in [8]. It provides a single algebraic expression for computing all candidate stability boundaries. It is also shown in this study that all candidate 2p-moment stability boundaries can be obtained by computing the real eigenvalues of a matrix of dimension equal to the dimension describing the system of 2p-moments.

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