We analyze the stability and stabilizability properties of mixed retarded-neutral type systems when the neutral term may be singular. We consider an operator differential equation model of the system in a Hilbert space, and we are interested in the critical case when there is a sequence of eigenvalues with real parts converging to zero. In this case, the system cannot be exponentially stable, and we study conditions under which it will be strongly stable. The behavior of spectra of mixed retarded-neutral type systems prevents the direct application of retarded system methods and the approach of pure neutral type systems for the analysis of stability. In this paper, two techniques are combined to obtain the conditions of asymptotic non-exponential stability: the existence of a Riesz basis of invariant finite-dimensional subspaces and the boundedness of the resolvent in some subspaces of a special decomposition of the state space. For unstable systems, the techniques introduced enable the concept of regular strong stabilizability for mixed retarded-neutral type systems to be analyzed.

Keywords: retarded-neutral type systems, asymptotic non-exponential stability, stabilizability, infinite dimensional systems

@article{COCV_2012__18_3_656_0, author = {Rabah, Rabah and Sklyar, Grigory Mikhailovitch and Barkhayev, Pavel Yurevitch}, title = {Stability and stabilizability of mixed retarded-neutral type systems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {656--692}, publisher = {EDP-Sciences}, volume = {18}, number = {3}, year = {2012}, doi = {10.1051/cocv/2011166}, mrnumber = {3041660}, zbl = {1263.34115}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2011166/} }

TY - JOUR AU - Rabah, Rabah AU - Sklyar, Grigory Mikhailovitch AU - Barkhayev, Pavel Yurevitch TI - Stability and stabilizability of mixed retarded-neutral type systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 656 EP - 692 VL - 18 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2011166/ DO - 10.1051/cocv/2011166 LA - en ID - COCV_2012__18_3_656_0 ER -

%0 Journal Article %A Rabah, Rabah %A Sklyar, Grigory Mikhailovitch %A Barkhayev, Pavel Yurevitch %T Stability and stabilizability of mixed retarded-neutral type systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 656-692 %V 18 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2011166/ %R 10.1051/cocv/2011166 %G en %F COCV_2012__18_3_656_0

Rabah, Rabah; Sklyar, Grigory Mikhailovitch; Barkhayev, Pavel Yurevitch. Stability and stabilizability of mixed retarded-neutral type systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 3, pp. 656-692. doi : 10.1051/cocv/2011166. http://archive.numdam.org/articles/10.1051/cocv/2011166/

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