Stability and stabilizability of mixed retarded-neutral type systems

Autor: Grigory M. Sklyar, Pavel Yu. Barkhayev, Rabah Rabah
Přispěvatelé: Institut de Recherche en Communications et en Cybernétique de Nantes (IRCCyN), Mines Nantes (Mines Nantes)-École Centrale de Nantes (ECN)-Ecole Polytechnique de l'Université de Nantes (EPUN), Université de Nantes (UN)-Université de Nantes (UN)-PRES Université Nantes Angers Le Mans (UNAM)-Centre National de la Recherche Scientifique (CNRS), Institute of Mathematics, University of Szczecin, University of Szczecin, Department of Differential Equations and Control, Kharkov National University, Kharkov National University, Institute of Mathematics, Department of Differential Equations and Control, The work was partially supported by the Polish Ministry of Science and High Education grant No. N514 238438 and École Centrale de Nantes, France.
Jazyk: angličtina
Rok vydání: 2009
Předmět:
0209 industrial biotechnology
Pure mathematics
Control and Optimization
Differential equation
infinite dimensional systems
[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]
Dynamical Systems (math.DS)
02 engineering and technology
01 natural sciences
[SPI.AUTO]Engineering Sciences [physics]/Automatic
93C23
34K06
34K20
34K40
49K25
symbols.namesake
020901 industrial engineering & automation
Retarded-neutral type systems
Exponential stability
FOS: Mathematics
State space
MSC. 93C23
34K06
34K20
34K40
49K25
Mathematics - Dynamical Systems
0101 mathematics
Invariant (mathematics)
Mathematics - Optimization and Control
Eigenvalues and eigenvectors
Mathematics
Resolvent
asymptotic non-exponential stability
010102 general mathematics
Mathematical analysis
Hilbert space
93C23
93D15
34K40
34K20
stabilizability
Linear subspace
Computational Mathematics
Optimization and Control (math.OC)
Control and Systems Engineering
symbols
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Zdroj: ESAIM: Control, Optimisation and Calculus of Variations
ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2012, 18 (3), pp.656-692. ⟨10.1051/cocv/2011166⟩
ISSN: 1292-8119
1262-3377
DOI: 10.1051/cocv/2011166⟩
Popis: We analyze the stability and stabilizability properties of mixed retarded-neutral type systems when the neutral term is allowed to be singular. Considering an operator model of the system in a Hilbert space we are interesting in the critical case when 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 asymptotic stability property. The behavior of spectra of mixed retarded-neutral type systems does not allow to apply directly neither methods of retarded system nor the approach of neutral type systems for analysis of stability. In this paper two technics are combined to get 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 technics introduced allow to analyze the concept of regular strong stabilizability for mixed retarded-neutral type systems. The present paper extends the results by R. Rabah, G.M. Sklyar, A.V. Rezounenko on stability obtained in [J. Diff. Equat., 214(2005), No. 2, 391-428] and on stabilizability from [J. Diff. Equat., 245(2008), No. 3, 569-593].
46 pages
Databáze: OpenAIRE
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