Titlebook: Algebraic and Differential Methods for Nonlinear Control Theory; Elements of Commutat Rafael Martínez-Guerra,Oscar Martínez-Fuentes,Juan Bo

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Mathematical Background,This chapter focuses on the basic concepts and algebra of sets as well as a brief introduction to the theory of functions and the well known principle of mathematical induction. All this background will be needed as a tool to understand the theory of linear algebra and differential equations as set forth in the following chapters.

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Permutations and Determinants,This chapter introduces basic concepts about permutations and determinants of matrices, the first part contains permutation groups explained through various definitions and theorems, the second part is about determinants of matrices, a method to obtain the determinant of a matrix is given along various other useful results.

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Group Theory,nd its characteristics, followed by homomorphisms, the chapter concludes with various theorems about isomorphisms, along with the concepts, various examples are provided to facilitate the understanding of the theory. This material ca ben found in any introductory book on abstract algebra (also called modern algebra).

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Differential Equations,nomena that can be represented with differential equations, after that we introduce definitions and methods for first order and second order linear differential equations, various exercises are given to the reader to better illustrate the contents of this chapter.

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Differential Algebra for Nonlinear Control Theory,eory, it begins with necessary information of differential algebra, it continues with definitions of single-input single-output systems, invertible systems, realization and canonical forms, finally we present methods for stabilization of nonlinear systems throughout linearization by dynamical feedback and some examples of such processes.