Asymptotic properties of solutions such as stability/ instability,oscillation/ nonoscillation, existence of solutions with specific asymptotic, maximum principles present a classical part in the theory of higher order functional differential equations. The use of these equations in applications is one of the main reasons for the developments in this field. The control in the mechanical processes leads to mathematical models with second order delay differential equations. Stability and stabilization of second order delay equations are one of the main goals of this book. The book is based on the authors’ results in the last decade.
In this book, various types of functional differential equations are considered: second and higher orders delay differential equations with measurable coefficients and delays, integro-differential equations, neutral equations, operator equations. Oscillation/nonoscillation, existence of unbounded solutions, instability, special asymptotic behavior, positivity, exponential stability and stabilization of functional differential equations are studied. New methods for the study of exponential stability are proposed. Note among them such as the W-transform (right regularization), a priory estimation of solutions, maximum principles, differential and integral inequalities, matrix inequality method, reduction to a system of equations.
The book can be used by applied mathematicians and as a basis for a course on stability of functional differential equations for graduate students.
A Brief Biography. Preface. Introduction to Stability Methods. Stability: A priory Estimation Method. Stability: Reduction to a System of Equations. Stability: W-transform Method I. Stability: W-transform Method II. Exponential Stability for Equations with Positive and Negative Coeffcients. Connection Between Nonoscillation and Stability. Stabilization for Second Order Delay Models, Simple Delay Control. Stabilization by Delay Distributed Feedback Control. Wronskian of Neutral FDE and Sturm Separation Theorem. Vallee-Poussin Theorem for Delay and Neutral DE. Sturm Theorems and Distance between Adjacent Zeros. Unbounded Solutions and Instability of Second Order DDE. Upper and Lower Estimates of Distances Between Zeros and Floquet Theory for Second Order DDE. Distribution of Zeros and Unboundedness of Solutions to Partial DDE. Second Order Equations: Oscillation and Boundary Value Problems. Stability of Third Order DDE. Operator Differential Equations. Properties A and B of Equations with a Linear Minorant. On Kneser-Type Solutions. Monotonically Increasing Solutions. Specific Properties of FDE. A Useful Theorems from Analysis. B Functional-differential Equations. Bibliography. Index.