Preface

1. The suggested book deals with the stability of linear and nonlinear vector neutral type functional differential equations. Equations with neutral type linear parts and nonlinear causal mappings are also considered. Explicit conditions for the exponential, absolute and input-to-state stabilities are derived. Moreover, solution estimates for the considered equations are established. These estimates provide the bounds for regions of attraction of steady states. The main methodology presented in the book is based on a combined usage of the recent norm estimates for matrix-valued functions with the following methods and results: the generalized Bohl-Perron principle, the integral version of the generalized Bohl-Perron principle and the positivity conditions for fundamental solutions to scalar neutral equations. We also apply the so-called generalized norm. A significant part of the book is devoted to the generalized Aizerman problem. 2. Neutral type functional differential equations (NDEs) naturally arise in various applications, such as control systems, mechanics, nuclear reactors, distributed networks, heat flows, neural networks, combustion, interaction of species, microbiology, learning models, epidemiology, physiology, and many others. The theory of functional differential equations has been developed in the works of V. Volterra, A.D. Myshkis, N.N. Krasovskii, B. Razumikhin, N. Minorsky, R. Bellman, A. Halanay, J. Hale and other mathematicians. The problem of stability analysis of various neutral type equations continues to attract the attention of many specialists despite its long history. It is still one of the most burning problems because of the absence of its complete solution. The basic method for the stability analysis is the method of the Lyapunov type functionals. By that method many very strong results are obtained. We do not consider the Lyapunov functionals method because several excellent books cover this topic. It should be noted that finding the Lyapunov type functionals for vector neutral type equations is often connected with serious mathematical difficulties, especially in regard to nonautonomous and nonlinear equations. On the contrary, the stability conditions presented in the suggested book are mainly formulated in terms of the determinants and eigenvalues of auxiliary matrices