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The stability of equilibrium points plays a fundamental role in dynamical systems. For nonlinear dynamical systems, which represent the majority of real plants, an investigation of stability requires the characterization of the domain of attraction (DA) of an equilibrium point, i.e., the set of initial conditions from which the trajectory of the system converges to such a point. It is well-known that estimating the DA, or even more attempting to control it, are very difficult problems because of the complex relationship of this set with the model of the system. The book also offers a concise and simple description of the main features of SOS programming which can be used in research and teaching. In particular, it introduces various classes of SOS polynomials and their characterization via LMIs and addresses typical problems such as establishment of positivity or non-positivity of polynomials and matrix polynomials, determining the minimum of rational functions, and solving systems of polynomial equations, in cases of both unconstrained and constrained variables. The techniques presented in this book are available in the MATLAB® toolbox SMRSOFT, which can be downloaded from http://www.eee.hku.hk/~chesi. Domain of Attraction addresses the estimation and control of the DA of equilibrium points using the novel SOS programming scheme, i.e., optimization techniques that have been recently developed based on polynomials that are sums of squares of polynomials (SOS polynomials) and that amount to solving convex optimization problems with linear matrix inequality (LMI) constraints, also known as semidefinite programs (SDPs). For the first time in the literature, a means of dealing with these issues is presented in a unified framework for various cases depending on the nature of the nonlinear systems considered, including the cases of polynomial systems, uncertain polynomial systems, and nonlinear (possibly uncertain) non-polynomial systems. The methods proposed in this book are illustrated in a variety of real systems and simulated systems with randomly chosen structures and/or coefficients which includes chemical reactors, electric circuits, mechanical devices, and social models. The book also offers a concise and simple description of the main features of SOS programming which can be used in research and teaching. In particular, it introduces various classes of SOS polynomials and their characterization via LMIs and addresses typical problems such as establishment of positivity or non-positivity of polynomials and matrix polynomials, determining the minimum of rational functions, and solving systems of polynomial equations, in cases of both unconstrained and constrained variables. The techniques presented in this book are available in the MATLAB® toolbox SMRSOFT, which can be downloaded from http://www.eee.hku.hk/~chesi.

Polynomials --- Mathematical optimization --- Mechanical Engineering --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Algebra --- Mechanical Engineering - General --- Systems engineering. --- Equilibrium. --- Balance --- Balance (Physics) --- Balancing (Physics) --- Engineering systems --- System engineering --- Design and construction --- Engineering. --- System theory. --- Statistical physics. --- Complexity, Computational. --- Control engineering. --- Control. --- Systems Theory, Control. --- Nonlinear Dynamics. --- Complexity. --- Stability --- Statics --- Engineering --- Industrial engineering --- System analysis --- Systems theory. --- Control and Systems Theory. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Construction --- Industrial arts --- Technology --- Computational complexity. --- Complexity, Computational --- Electronic data processing --- Machine theory --- Physics --- Mathematical statistics --- Systems, Theory of --- Systems science --- Science --- Control engineering --- Control equipment --- Control theory --- Engineering instruments --- Automation --- Programmable controllers --- Statistical methods --- Philosophy --- sum of squares of polynominals --- domain of attraction --- robust domain of attraction

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We use addition on a daily basis-yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition's most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research.Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series-long sums that can only be defined by the concept of limit, as in the example of 1+1/2+1/4+. . .=? With the help of some group theory and geometry, the third section ties together the first two parts of the book through a discussion of modular forms-the analytic functions on the upper half-plane of the complex numbers that have growth and transformation properties. Ash and Gross show how modular forms are indispensable in modern number theory, for example in the proof of Fermat's Last Theorem.Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.

Number theory. --- Mathematics --- Number study --- Numbers, Theory of --- Algebra --- Absolute value. --- Addition. --- Analytic continuation. --- Analytic function. --- Automorphic form. --- Axiom. --- Bernoulli number. --- Big O notation. --- Binomial coefficient. --- Binomial theorem. --- Book. --- Calculation. --- Chain rule. --- Coefficient. --- Complex analysis. --- Complex number. --- Complex plane. --- Computation. --- Congruence subgroup. --- Conjecture. --- Constant function. --- Constant term. --- Convergent series. --- Coprime integers. --- Counting. --- Cusp form. --- Determinant. --- Diagram (category theory). --- Dirichlet series. --- Division by zero. --- Divisor. --- Elementary proof. --- Elliptic curve. --- Equation. --- Euclidean geometry. --- Existential quantification. --- Exponential function. --- Factorization. --- Fourier series. --- Function composition. --- Fundamental domain. --- Gaussian integer. --- Generating function. --- Geometric series. --- Geometry. --- Group theory. --- Hecke operator. --- Hexagonal number. --- Hyperbolic geometry. --- Integer factorization. --- Integer. --- Line segment. --- Linear combination. --- Logarithm. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Matrix group. --- Modular form. --- Modular group. --- Natural number. --- Non-Euclidean geometry. --- Parity (mathematics). --- Pentagonal number. --- Periodic function. --- Polynomial. --- Power series. --- Prime factor. --- Prime number theorem. --- Prime number. --- Pythagorean theorem. --- Quadratic residue. --- Quantity. --- Radius of convergence. --- Rational number. --- Real number. --- Remainder. --- Riemann surface. --- Root of unity. --- Scientific notation. --- Semicircle. --- Series (mathematics). --- Sign (mathematics). --- Square number. --- Square root. --- Subgroup. --- Subset. --- Sum of squares. --- Summation. --- Taylor series. --- Theorem. --- Theory. --- Transfinite number. --- Triangular number. --- Two-dimensional space. --- Unique factorization domain. --- Upper half-plane. --- Variable (mathematics). --- Vector space.