Functional Differential Equations

This monograph presents the basics of i-smooth calculus, a new differential calculus of nonlinear functionals based on the notion of invariant derivative, and its application to some problems of the qualitative theory of functional differential equations. It separates finite and infinite dimensional components in the structures of functional differential equations and functionals, and uses conditional representation of FDEs, which is expedient for the application of methods and constructions of i-smooth calculus. Part I contains a foundation of i-smooth calculus. Part II is an introduction to FDEs based on i-smooth calculus. Part III describes the direct Lyapunov method for systems with delays in terms of i-smooth functionals. Part IV considers an approach to the development of a dynamical programming method for systems with delays in terms of i-smooth Bellman's functionals.