For these systems in the book a special method of constructing of Lyapunov

# For these systems in the book a special method of

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have the cylindrical phase space. For these systems, in the book a special method of constructing of Lyapunov functions and Popov functionals is presented, the latter containing so- lutions of certain comparison systems. We shall note that continual stationary sets appear also in discontinuous relay systems and the systems with solid (Coulomb) friction. Frequency-domain investigation of these systems is expounded in the monograph [Gelig et al. 1978] where the theory of differential inclusions is systematically set forth. The limited capacity of this book made us confine ourselves only to smooth dynamical systems. In the third part of this book the problem of the existence of cycles and of the estimation of their frequency is considered. The powerful tool of investigation of cycles is the Poincare mapping. If one succeeds to prove that the Poincare mapping transfers the transversal cross-section, on which it is defined, into itself, then by a certain theorem about a fixed point one can deduce the existence of a cycle. That is why the problem of existence of a cycle often can be reduced to the estimation of the Poincare mapping. In the third part of the book various frequency-domain estimates of the Poincare mapping are presented. For the systems with the cylindrical phase space two kinds of the cycles are possible: the cycles of the first kind remain closed in the covering space, the cycles of the second kind lose the closure property there. Both for the cycles of the first kind and for the cycles of the second kind frequency-domain criteria of existence are obtained. Analogous tool is used in the forth part in order to establish the frequency-domain criterion of existence of homoclinic orbits. For
the systems with the cylindrical phase space. Such orbits appear when one transfers in the parameters space of the system from the region of gradient-like behavior to the region of existence of circular solutions and cycles of the second kind. We present in this book a frequency-domain criterion of the existence of a homoclinic orbit which can be regarded as a generalization of well-known Tricomi theorem about the existence of a separatrix loop. In the fifth part the short review of basic notions of the dimension theory is brought about. Principal attention is paid here to the Hausdorff dimension which is one of the basic characteristics of strange attractors. We present analytical methods of obtaining of upper estimates of Hausdorff measure and dimension of attractors. Then frequency-domain estimates of Hausdorff dimension of attractors are demonstrated. These results are applied to the well-known Lorenz system. Contents I Stability of Control Systems with the Unique Equilib- rium 1 1 Classical Theory of Absolute Stability 3 1.1 Feedback Control Equation and its Transfer Function 3 1.2 Controllability. Observability. Kalman Duality Principle 4 1.3 The Transfer Function of Controllable and Observable Linear Block . 12 1.4 Stable Linear Blocks 14 1.4.1 Hermite-Michailov Criterion ......................................................................................................

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• Spring '16
• Stability theory, Pendulum-Like Systems

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