Criteria for exponential stability of linear differential equations with positive evolution on ordered Banach spaces

In this paper, the problem of exponential stability of the zero state equilibrium of a linear differential equation defined by an operator-valued function with positive evolution on an infinite-dimensional ordered Banach space is investigated. The class of linear differential equations considered in this paper contains, as particular cases, linear differential equations described by Lyapunov operators. Such equations occur in connection with the problem of mean square exponential stability for a class of stochastic differential equations affected by multiplicative white noise perturbations and Markovian jumping. Both the case of the causal exponential stability and the case of the anticausal exponential stability are considered. The criteria for the two kinds of exponential stability are derived in terms of the existence of some globally defined and bounded solutions of some suitable forward and backward affine differential equations and in terms of solvability of some forward (backward) differential inequalities. We show that in the time-invariant case, our criteria recover the ones given in Damm (2004, Rational Matrix Equations in Stochastic Control. Lectures Notes in Control and Information Sciences, vol. 297. Berlin: Springer) and Damm & Hinrichsen (2003, Newton's method for concave operators with resolvent positive derivatives in ordered Banach spaces. Linear Algebra Appl., 363, 43–64). The problem of robustness of exponential stability with respect to some additive perturbations modelled by positive operator-valued functions is analysed in the case when the involved operator-valued functions are periodic functions. Finally, the criteria for exponential stability for linear differential equations with positive evolution are used to derive useful criteria for exponential stability in mean square in the case of systems of stochastic linear differential equations affected by multiplicative white noise perturbations and by a Markov process with a countable set of the states. The results derived in this paper can be used in the designing process of some stabilizing control laws for a wide class of processes modelled by systems of stochastic differential equations with state and control multiplicative white noise and coefficients affected by a Markov process with an infinite number of states. Also the results of this paper allow us to develop numerical procedures to compute the stabilizing solution for different kinds of Riccati equations involved in different control problems as: the linear quadratic optimal regulator or H2 optimal control for the aforementioned class of controlled stochastic systems.