On the uniform asymptotic stability in probability of systems with random disturbances (Q2740274)

This paper deals with the system \(\dot x=G(t,x,\xi(t))\) in \(\mathbb R^n\) where \(\xi(t)\) is an \(\mathbb R^k\)-valued stochastic process determined on a probability space \(({\mathbf P},\mathcal{F},\Omega)\). Natural conditions on \(G\) and \(\xi \) are imposed to guarantee the existence and trajectory-wise uniqueness of solutions.NEWLINENEWLINENEWLINELet \(x(t,t_0,x_0)\) stand for the solution satisfying \(x(t_0,t_0,x_0)=x_0\). It is assumed that the trivial solution \(x\equiv 0\) is uniformly stable in probability and that there exists \(r_0>0\) such that the following condition holds:NEWLINE\[NEWLINE\forall x_0\in \mathbb R^n:\|x_0\|\leq r_0 \Rightarrow \forall \epsilon >0:\lim_{t\to \infty }{\mathbf P}\{\|x(t,t_0,x_0)\|>\epsilon \}=0.\tag{1}NEWLINE\]NEWLINE The author proves that the limit in (1) is uniform in \(x_0\) for any \(\epsilon >0\).