Criteria for the asymptotic stability of solutions of dynamical systems (Q2722248)

Let \(M\) be a semiordered finite-dimensional Banach space in which any monotonically increasing bounded sequence has a limit and let \(L:B \to B\) be a linear monotone operator. The authors prove that the difference equation \(x(n+1)=Lx(n)\), \(n=0,1,\ldots \), is asymptotically stable iff for some \(b>0\), \(b\in M,\) the equation \(c=b+Lc\) has a solution \(c>0\). With the help of this result new proofs of stability criteria for certain classes of systems are presented. Namely, the following \(n\)-dimensional deterministic and stochastic systems are studied: NEWLINE\[NEWLINE\begin{gathered} x(n+1)=Ax(n),\\ dx(t)=Ax(t) dt+\sum_{s=1}^{N}A_sx(t) dw_s(t), \\ dx(t)=Ax(t) dt+\sum_{s=1}^{N}A_sx(t-\tau_s) dw_s(t),\quad \tau_s\geq 0. \end{gathered} NEWLINE\]NEWLINE Here \(w_s(t)\), \(s=1,\ldots,N,\) are independent Wiener processes with unit dispersion. For example, it is shown that the trivial solution of the second (or the third) of these systems is asymptotically stable in mean square iff for some matrix \(B>0\) the matrix equation \(B+AC+CA^*+\sum_{s=1}^{N}A_sCA_s^*=0\) has a solution \(C>0\).