Speaker
Description
This talk is devoted to the study of periodic solutions for a class of weakly nonlinear control systems in a Hilbert space. We assume that the linear part of the system generates a C₀-semigroup of bounded linear operators and that the nonlinearity is Lipschitz continuous. For an arbitrary bounded measurable input defined on the interval [0, T], we formulate the problem of finding a solution x(t) satisfying the periodic boundary condition x(0) = x(T). Under technical assumptions concerning the resolvent of the infinitesimal generator and the Lipschitz constant, we prove an existence and uniqueness result for such periodic solutions. The proof is based on the Banach contraction principle and is used to derive an iterative scheme for approximating the periodic solutions.
As an example, we consider a mathematical model of a dispersed-flow tubular reactor (DFTR) with boundary control under Danckwerts-type boundary conditions. This nonlinear parabolic model is represented as an abstract differential equation, for which the periodic solutions are constructed using the proposed iterative scheme.
