Contents -- Preface -- List of Notations -- 1. Foundations of Theory of Differential Equations with Discontinuous Right-Hand Sides -- 1.1 Notion of Solution to Differential Equation with Discontinuous Right-Hand Side -- 1.1.1 Difficulties encountered in the definition of a solution. Sliding modes -- 1.1.2 The concept of a solution of a system with discontinuous nonlinearities accepted in this book. Connection with the theory of differential equations with multiple-valued right-hand sides -- 1.1.3 Relation to some other definitions of a solution to a system with discontinuous right-hand side -- 1.1.4 Sliding modes. Extended nonlinearity. Example -- 1.2 Systems of Differential Equations with Multiple-Valued Right-Hand Sides (Differential Inclusions) -- 1.2.1 Concept of a solution of a system of differential equations with a multivalued right-hand side the local existence theorem the theorems on continuation of solutions and continuous dependence on initial values -- 1.2.2 "Extended" nonlinearities -- 1.2.3 Sliding modes -- 1.3 Dichotomy and Stability -- 1.3.1 Basic definitions -- 1.3.2 Lyapunov-type lemmas -- 2. Auxiliary Algebraic Statements on Solutions of Matrix Inequalities of a Special Type -- 2.1 Algebraic Problems that Occur when Finding Conditions for the Existence of Lyapunov Functions from Some Multiparameter Functional Class. Circle Criterion. Popov Criterion -- 2.1.1 Equations of the system. Linear and nonlinear partsof the system. Transfer function and frequency response -- 2.1.2 Existence of a Lyapunov function from the class of quadratic forms. S-procedure -- 2.1.3 Existence of a Lyapunov function in the class of quadratic forms (continued). Frequency-domain theorem -- 2.1.4 The circle criterion.

2.1.5 A system with a stationary nonlinearity. Existence of a Lyapunov function in the class "a quadratic form plus an integral of the nonlinearity" -- 2.1.6 Popov criterion -- 2.2 Relevant Algebraic Statements -- 2.2.1 Controllability observability and stabilizability -- 2.2.2 Frequency-domain theorem on solutions of some matrix inequalities -- 2.2.3 Additional auxiliary lemmas -- 2.2.4 The S-procedure theorem -- 2.2.5 On the method of linear matrix inequalities in control theory -- 3. Dichotomy and Stability of Nonlinear Systems with Multiple Equilibria -- 3.1 Systems with Piecewise Single-Valued Nonlinearities -- 3.1.1 Systems with several nonlinearities. Frequencydomain conditions for quasi-gradient-like behavior and pointwise global stability. Free gyroscope with dry friction -- 3.1.2 The case of a single nonlinearity and det P#0 . Theorem 3.4 on gradient-like behavior and pointwise global stability of the segment of rest. Examples -- 3.1.3 The case of a single nonlinearity and one zero pole of the transfer function. Theorem 3.6 on quasi-gradientlike behavior and pointwise global stability. The Bulgakov problem -- 3.1.4 The case of a single nonlinearity and double zero pole of the transfer function. Theorem 3.8 on global stability of the segment of rest. Gyroscopic roll equalizer. The problem of Lur'e and Postnikov. Control system for a turbine. Problem of an autopilot -- 3.2 Systems with Monotone Piecewise Single-Valued Nonlinearities -- 3.2.1 Systems with a single nonlinearity. Frequency-domain conditions for dichotomy and global stability. Corrected gyrostabilizer with dry friction. The problem of Vyshnegradskii.

3.2.2 Systems with several nonlinearities. Frequencydomain criteria for dichotomy. Noncorrectable gyrostabilizer with dry friction -- 3.3 Systems with Gradient Nonlinearities -- 3.3.1 Dichotomy and quasi-gradient-likeness of systems with gradient nonlinearities -- 3.3.2 Dichotomy and quasi-gradient-like behavior of nonlinear electrical circuits and of cellular neural networks -- 4. Stability of Equilibria Sets of Pendulum-Like Systems -- 4.1 Formulation of the Stability Problem for Equilibrium Sets of Pendulum-Like Systems -- 4.1.1 Special features of the dynamics of pendulum-like systems. The structure of their equilibria sets -- 4.1.2 Canonical forms of pendulum-like systems with a single scalar nonlinearity -- 4.1.3 Dichotomy. Gradient-like behavior in a class of nonlinearities with zero mean value -- 4.2 The Method of Periodic Lyapunov Functions -- 4.2.1 Theorem on gradient-like behavior -- 4.2.2 Phase-locked loops with first- and second-order lowpass filters -- 4.3 An Analogue of the Circle Criterion for Pendulum-Like Systems -- 4.3.1 Criterion for boundedness of solutions of pendulumlike systems -- 4.3.2 Lemma on pointwise dichotomy -- 4.3.3 Stability of two- and three-dimensional pendulum-like systems. Examples -- 4.3.4 Phase-locked loops with a band amplifier -- 4.4 The Method of Non-Local Reduction -- 4.4.1 The properties of separatrices of a two-dimensional dynamical system -- 4.4.2 The theorem on nonlocal reduction -- 4.4.3 Theorem on boundedness of solutions and on gradient-like behavior -- 4.4.4 Generalized Bohm-Hayes theorem -- 4.4.5 Approximation of the acquisition bands of phaselocked loops with various low-pass filters -- 4.5 Necessary Conditions for Gradient-Like Behavior of Pendulum-Like Systems.

4.5.1 Conditions for the existence of circular solutions and cycles of the second kind -- 4.5.2 Generalized Hayes theorem -- 4.5.3 Estimation of the instability regions in searching PLL systems and PLL systems with 1/2 filter -- 4.6 Stability of the Dynamical Systems Describing the Synchronous Machines -- 4.6.1 Formulation of the problem -- 4.6.2 The case of zero load -- 4.6.3 The case of a nonzero load -- 5. Appendix. Proofs of the Theorems of Chapter 2 -- 5.1 Proofs of Theorems on Controllability Observability Irreducibility and of Lemmas 2.4 and 2.7 -- 5.1.1 Proof of the equivalence of controllability to properties (i)-(iv) of Theorem 2.6 -- 5.1.2 Proof of the Theorem 2.7 -- 5.1.3 Completion of the proof of Theorem 2.6 -- 5.1.4 Proof of Theorem 2.8 -- 5.1.5 Proof of Theorem 2.9 in the scalar case m = l = 1 -- 5.1.6 Proof of Theorem 2.9 for the case when either m > 1 or l > 1 and proof of Theorem 2.10 -- 5.1.7 Proof of Lemma 2.4 -- 5.1.8 Proof of Lemma 2.7 -- 5.2 Proof of Theorem 2.13 (Nonsingular Case). Theorem on Solutions of Lur'e Equation (Algebraic Riccati Equation) -- 5.2.1 Two lemmas. A detailed version of frequency-domain theorem for the nonsingular case -- 5.2.2 Proof of Theorem 5.1. The theorem on solvability of the Lur'e equation -- 5.2.3 Lemma on J-orthogonality of the root subspaces of a Hamiltonian matrix -- 5.3 Proof of Theorem 2.13 (Completion) and Lemma 5.1 -- 5.3.1 Proof of Lemma 5.1 -- 5.3.2 Proof of Theorem 2.13 -- 5.4 Proofs of Theorems 2.12 and 2.14 (Singular Case) -- 5.4.1 Proof of Theorem 2.12 -- 5.4.2 Necessity of the hypotheses of Theorem 2.14 -- 5.4.3 Sufficiency of the hypotheses of Theorem 2.14 -- 5.5 Proofs of Theorems 2.17-2.19 on Losslessness of S-procedure -- 5.5.1 The Dines theorem.

5.5.2 Proofs of the theorems on the losslessness of the Sprocedure for quadratic forms and one constraint -- Bibliography -- Index.