Exponential decay for distributed bilinear control systems with damping
The system is \[ \begin{aligned}{ &y''(t) + Ay(t) + Dy'(t) + u(t)By(t) = 0, \\ &y(0) = y_0, \quad y'(0) = y_1 }\end{aligned} \] in a Hilbert space \(H.\) All operators are linear, \(D\) and \(B\) are bounded and \(A : V \to E,\) where \(V \hookrightarrow E\) with dense, continuous and compact imbedding. Under suitable assumptions, the author shows uniform exponential decay of the energy \[ E(t) = \tfrac 12\big( \| y(t)\| _V^2 + \| y'(t)\| _E^2 \big) \] by means of a scalar nonlinear feedback \(u(t) = f(\langle By(t), y(t) \rangle).\) This is related to results of Ball and Slemrod, where there is no damping \((D = 0)\) and only weak stabilization is obtained.
scientific article; zbMATH DE number 1279155