Absolute stability criteria for nonlinear operator equations

A. L. Likhtarnikov

Izvestiya. Mathematics , Tome 11 (1977) no. 5, pp. 1011-1029

Résumé

Conditions are obtained for the stability in the large of solutions of nonlinear equations of the form \begin{equation} \frac{dx}{dt}=Ax+bu+f,\qquad u=\varphi(y,t),\quad y=Cx. \end{equation} Here $A$ is the infinitesimal generator of a semigroup of class $C_0$, the maps $b\colon U\to X$ and $C\colon X\to Y$ are bounded linear operators, and $U,X$ and $Y$ are (generally different) Hilbert spaces. The equations (1) describe a wide class of distributed parameter control systems. The results obtained have the following features: a) The stability conditions pertain not to an individual system but to classes of systems; the stability holds uniformly in a certain sense for all systems of a particular class (“absolute stability in a given class of nonlinearities”). b) For some classes of nonlinearities, the conditions are not only sufficient but necessary. Bibliography: 15 titles.