Algebraic criteria for absolute (relative to nonlinearity) stability of stochastic automatic control systems with nonlinear feedback (Q1264128)

A necessary and sufficient condition for absolute stability of the trivial solution of Ito's differential equations under the Wiener disturbance w(t): \(dx(t)=[A x(t)+g \phi (\sigma)]dt+B x(t) dw(t)\), \(\sigma =\ell 'x\), \(\phi (\sigma)\in R^ 1\), \(0=\phi (\sigma)/\sigma =h\), \(\ell\), \(g\in R^ n\), is established by Lyapunov's direct method. The condition consists of some matrix inequalities for A, h, g, \(\ell\), H where H is a positive definite solution of the matrix equation \(A'H+HA+B'HB=-G\) and G is a positive definite matrix. The paper is a continuation of the author's previous ones [e.g., Sov. Math., Dokl. 34, 377-380 (1987; Zbl 0623.60074); translation from Dokl. Akad. Nauk SSSR 290, 1041-1044 (1986)].