Stability analysis of mixed retarded-neutral type systems in Hilbert space

36 pagesWe investigate the stability property of mixed retarded-neutral type systems. Considering an operator model of the system in Hilbert space we are interesting in the critical case when the spectrum of the operator belongs to the open left half-plane and there exists a sequence of eigenvalues with real parts approaching to zero. In this case the exponential stability is not possible and we are studying the strong asymp- totic stability property. The present paper extends the results obtained in [R. Rabah, G.M. Sklyar, A. V. Rezounenko, Stability analysis of neutral type systems in Hilbert space. J. of Dierential Equations, 214(2005), No. 2, 391{428] in which stability of systems of neutral type was studied using the existence of a Riesz basis of invariant finite-dimensional subspaces. However, for mixed retarded-neutral type systems such a basis may not exist for the whole state space. Though the main result on stability remains the same for mixed retarded-neutral type systems, the technic of its proof had to be changed and it involves a proof of resolvent boundedness on some invariant subspace. We show that the property of asymptotic stability is determinated not only by the spectrum of the system but also depends on geometrical characteristics of its main neutral term which in our situation may be singular. We also give an explicit example of two systems having the same spectrum in the open left half-plane, but one of them is asymptotically stable while the other one is unstable