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The main aim of the book is to present new constructive methods of delay differential equation (DDE) theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays. Referred to as “systems with delays” in this volume, this class of differential equations is also called delay differential equations (DDE), time-delay systems, hereditary systems, and functional differential equations. Delay differential equations are widely used for describing and modeling various processes and systems in different applied problems At present there are effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). There are many applications for these types of equations, because of this progress, but there are not as many methodologies in systems with delays that are easily applicable for the engineer or applied mathematician. there are no methods of finding solutions in explicit forms, and there is an absence of generally available general-purpose software packages for simulating such systems. Systems with Delays fills this void and provides easily applicable methods for engineers, mathematicians, and scientists to work with delay differential equations in their operations and research.

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The edition introduces a new class of invariant derivatives and shows their relationships with other derivatives, such as the Sobolev generalized derivative and the generalized derivative of the distribution theory. This is a new direction in mathematics. i-Smooth analysis is the branch of functional analysis that considers the theory and applications of the invariant derivatives of functions and functionals. The important direction of i-smooth analysis is the investigation of the relation of invariant derivatives with the Sobolev generalized derivative and the generalized derivative of distribution theory. Until now, i-smooth analysis has been developed mainly to apply to the theory of functional differential equations, and the goal of this book is to present i-smooth analysis as a branch of functional analysis. The notion of the invariant derivative (i-derivative) of nonlinear functionals has been introduced in mathematics, and this in turn developed the corresponding i-smooth calculus of functionals and showed that for linear continuous functionals the invariant derivative coincides with the generalized derivative of the distribution theory. This book intends to introduce this theory to the general mathematics, engineering, and physicist communities.

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