  Some sufficient conditions on the algebraic stability of non-homogeneous
regime-switching diffusion processes are established. In this work we focus on
determining the decay rate of a stochastic system which switches randomly
between different states, and owns different decay rates at various states. In
particular, we show that if a finite-state regime-switching diffusion process
is $p$-th moment exponentially stable in some states and is $p$-th moment
algebraically stable in other states, which are characterized by a common
Lyapunov function, then this process is ultimately exponentially stable
regardless of the jumping rate of the random switching between these states.
Moreover, these results can be applied to the stabilization of SDE by feedback
control to reduce the observation times.
