In a technical lemma the authors give a sufficient condition for the exponential stability of the \(k\)th moment of the solution to a stochastic differential equation in \(\mathbb{R}^n\) whose drift and diffusion coefficients are depending on time and on a finite state Markov process \(\theta(t)\).
This condition is shown to be satisfied (with \(k=2\)) for the linear SDE \[ dX(t)= [A+\Delta A_{\theta(t)}] X(t) dt+\sum^J_{j=1} F_{\theta(t),j}dW^j(t), \] if the matrix \(A\) is such that for some positive definite matrices \(Q_i\) the system of associated Lyapunov-Itô equations is solvable. In that case the linear system is mean-square stable.
It is further shown that the stability condition holds, if the linear system is closed by means of a feedback control \(Bu(t) dt\), \(u(t)= M_{\theta(t)}X(t),M_{\theta(t)}\) a suitable matrix, where \((A,B)\) are controllable and such that for suitable positive definite matrices \(Q_i\) and \(R_i\) the associated system of Riccati-Itô equations can be solved. It follows that in that case the linear system is mean-square stabilizable.