Strong convergence and arbitrarily slow decay of energy for a class of bilinear control problems
The paper deals with a differential equation of hyperbolic type, having a weak nonlinear damping term. The equation arises from the study of stabilizability for a bilinear control problem, \(\ddot u+Au+p(t)Bu=0\), where p(t) is a control function. If one asks to determine p(t) in such a way that all solutions of this equation decay to zero as \(t\to \infty\), then a choice would be \(p(t)=(Bu(t),\dot u(t))\). The main result of the paper shows that the initial value problem \(u(0)=u_ 0\), \(\dot u(0)=u_ 1\), \(u_ 0\in D(A^{1/2})\), for the above equation, under certain conditions for the operators A and B, possess a unique weak solution and \(\{u(t),\dot u(t)\}\to \{0,0\}\) in \(D(A'''\times H\), H a Hilbert space, the convergence being strongly. The proof is based on estimates for the natural energy associated with the unperturbed equation \(\ddot u+Au=0\) and results from the theory of nonharmonic Fourier series.
scientific article; zbMATH DE number 4169188