On a class of nonlinear distributed systems with linear finite dimensional feedback controls
The author studies certain asymptotic properties of nonlinear distributed parameter systems in the presence of linear finite dimensional feedback controls. He considers dynamical systems described by a class of nonlinear evolution equations in a Hilbert space, which includes the Navier-Stokes equations and the magnetohydrodynamic equations with homogeneous boundary conditions. He shows that linear feedback controls depending on a finite number of modes assure that for bounded solutions of the system, their asymptotic behavior as \(t\to \infty\) is completely determined by that of a corresponding finite number of modes, that is, the bounded solutions of the system tend to an invariant set having a finite dimensional structure as \(t\to \infty\).
linear finite dimensional feedback controls
scientific article; zbMATH DE number 3956923