Impulsive differential equations

by Samoilenko, Anatolii Mikhailovich

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Full title :Impulsive differential equations / A.M. Samoilenko & N.A. Perestyuk ; translated from the Russian by Yury Chapovsky

Summary

1. General description of impulsive differential systems. 1.1. Description of mathematical model. 1.2. Systems with impulses at fixed times. 1.3. Systems with impulses at variable times. 1.4. Discontinuous dynamical systems. 1.5. Motion of an impulsive oscillator under the effect of an impulsive force -- 2. Linear systems. 2.1. General properties of solutions of linear systems. 2.2. Linear systems with constant coefficients. 2.3. Stability of solutions of linear impulsive systems. 2.4. Characteristic exponents of functions and matrices of functions. 2.5. Adjoint systems. Perron theorem. 2.6. Reducible systems. 2.7. Linear periodic impulsive systems. 2.8. Linear Hamiltonian systems of impulsive differential equations. 2.9. Periodic solutions of a certain second order equation -- 3. Stability of solutions. 3.1. Linear systems with constant and almost constant matrices. 3.2. Stability criterion based on first order approximation. 3.3. Stability in systems with variable times of impulsive effect. 3.4. Direct Lyapunov method for studying stability of solutions of impulsive systems -- 4. Periodic and almost periodic impulsive systems. 4.1. Nonhomogeneous linear periodic systems. 4.2. Nonlinear periodic systems. 4.3. Numerical-analytical method for finding periodic solutions. 4.4. Almost periodic sequences. 4.5. Almost periodic functions. 4.6. Almost periodic differential systems. 4.7. Homogeneous linear periodic systems -- 5. Integral sets of impulsive systems. 5.1. Bounded solutions of nonhomogeneous linear systems. 5.2. Existence of bounded solutions of nonlinear systems. 5.3. Integral sets of systems with hyperbolic linear part. 5.4. Integral sets of a certain class of discontinuous dynamical systems -- 6. Optimum control in impulsive systems. 6.1. Formulation of the problem. Auxiliary results. 6.2. Necessary conditions for optimum. 6.3. Impulsive control with fixed times. 6.4. Necessary and sufficient conditions for optimum -- 7. Asymptotic study of oscillations in impulsive systems. 7.1. Formulation of the problem. 7.2. Formulas for an approximate solution of a non-resonance system. 7.3. Substantiation of the averaging method for a non-resonance system. 7.4. Averaging in a resonance system and its substantiation. 7.5. Formulas for approximate solutions for an impulsive effect occuring at fixed positions. 7.6. Substantiation of the averaging method for systems with impulses occurring at fixed positions. 7.7. General averaging scheme for impulsive systems. 7.8. On correspondence between exact and approximate solutions over an infinite time interval -- A. Periodic and almost periodic impulsive system / S.I. Trofimchuk

This monograph, written by the representatives of the Kiev school of Nonlinear Mechanics, contains a sufficiently complete study of systems of impulsive linear differential equations, as well as worked out nontrivial examples which could serve as guidelines for solving other particular applied problems