We consider a stochastic process $X_t^x$ which solves an equation $$\mathrm{d}{X}_t^x=A{X}_t^x\mathrm{d}t+\Phi \mathrm{d}{B}_t^H,X_0^x=x$$ where $A$ and $\Phi$ are real matrices and $B^H$ is a fractional Brownian motion with Hurst parameter $H\in (1/2,1)$. The Kolmogorov backward equation for the function $u(t,x)=𝔼f\left(X_t^x\right)$ is derived and exponential convergence of probability distributions of solutions to the limit measure is established.