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Stability of neutral functional differential equations. (English) Zbl 1315.34002

Atlantis Studies in Differential Equations 3. Amsterdam: Atlantis Press (ISBN 978-94-6239-090-4/hbk; 978-94-6239-091-1/ebook). xiii, 304 p. (2014).
The aim of the monograph is to provide last achievements for specialists in the stability theory of neutral type functional differential equations (NDEs), control system theory and mechanics.
Chapter 1 contains mainly results from the theory of Banach and ordered spaces and some norm estimates for the integral operators \(f \mapsto \int_0^\eta {dR_0(s)} f(t-s)\) and \(f \mapsto \int_0^\eta {d_s R(t,s)} f(t-s)\), \(t\geq0\). Here, \(R_0(s)\) is a real matrix-function of bounded variation on a finite segment \([0,\eta]\) and \(R(t,s)\) is a real matrix-function with bounded variation in \(s\).
Chapter 2 contains norm estimates for matrix-valued functions and estimates for eigenvalues of matrices, particularly for resolvent, powers of matrices, exponential, spectral radius, connections of the numerical range with the convex hull of the spectrum, perturbation theory results for matrix exponentials and determinants, Gerschgorin’s bounds for the eigenvalues.
In Chapters 3, 4 there are derived for linear autonomous and time-variant vector difference-delay equations with continuous time norm estimates for the resolvent operators, which are used in Chapter 5 and 6 for the stability investigations for the NDEs, linear autonomous NDEs in Chapter 5 and linear vector non-autonomous NDEs in Chapter 6. In particular, in Chapter 6 the Bohl-Perron principle is extended to a class of neutral type functional DEs together with its integral version for NDEs, on the base of which the stability conditions for time-variant NDEs close to autonomous systems and for time-variant systems with small principal operators, stability conditions independent of delay in the non-autonomous case are obtained.
Chapter 7 is devoted to nonlinear vector NDEs having linear autonomous neutral parts and nonlinear causal mappings, where the Lyapounov functional method is presented, formulated in terms of the determinants and eigenvalues of auxiliary matrices dependent on a parameter. Explicit conditions for the exponential, absolute and input-to state stabilities are obtained.
Chapter 8 is devoted to scalar nonlinear functional differential NDEs with autonomous linear parts. Here, explicit absolute \(L^2 \)-stability conditions are derived in terms of the norms of the Cauchy operators to the linear parts and the Lipschitz constants of the nonlinearities. Also the following generalized Aizerman problem is solved: for scalar NDEs of \(n\)-th order with special form nonlinearities to separate a class of linear parts of NDEs such that the exponential stability of the linear equation provides the \(L^2 \)-absolute stability for the nonlinear equation.
In Chapter 9 some bounds for the characteristic values of autonomous vector NDEs are obtained.
Thus, the reviewed monograph is the first book, which
1.
gives a systematic approach to the stability analysis of vector NDEs which is based on estimates for matrix-valued functions allowing to investigate various classes of equations from a unified viewpoint;
2.
presents the generalized Bohl-Perron principle for neutral type systems and its integral version;
3.
contains the solutions of the generalized Aizerman problem for NDEs;
4.
suggests explicit stability conditions for semilinear equations with linear neutral type parts and nonlinear causal mappings.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34K20 Stability theory of functional-differential equations
34K40 Neutral functional-differential equations
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