Strong convergence and arbitrarily slow decay of energy for a class of bilinear control problems. (English) Zbl 0711.35017

Müller, Stefan

J. Differ. Equations 81, No. 1, 50-67 (1989).

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.

Reviewer: C.Simionescu-Badea

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35L70	Second-order nonlinear hyperbolic equations
35L15	Initial value problems for second-order hyperbolic equations
42B05	Fourier series and coefficients in several variables
35D05	Existence of generalized solutions of PDE (MSC2000)

Keywords:

nonlinear damping term; decay to zero; energy

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