H-theorem states that the entropy production is nonnegative and, therefore, the entropy of a closed system should monotonically change in time. In information processing, the entropy production is positive for random transformation of signals (the information processing lemma). Originally, the H-theorem and the information processing lemma were proved for the classical Boltzmann-Gibbs-Shannon e...

The general method of Lyapunov functionals construction which was developed during the last decade for stability investigation of stochastic differential equations with aftereffect and stochastic difference equations is considered. It is shown that after some modification of the basic Lyapunov-type theorem, this method can be successfully used also for stochastic difference Volterra equations w...

We present a nonlinear Lyapunov function based small-gain theorem for analyzing input-to-state stability of discrete-time large-scale systems. Motivated by the fact that many feedback control laws lead to discontinuous closed loop systems, we pose no continuity assumptions on the system dynamics. For characterizing input-to-state stability in this discontinuous setting, we utilize a recently in...

The objective of this note is to prove, under certain technical conditions, that the top-Lyapunov exponent of a strictly stationary random sequence of block-triangular matrices is equal to the maximum of the top-Lyapunov exponents of its diagonal blocks. This study is partially motivated by a basic technical problem in the identification of GARCH processes. A recent extension of the above inher...

In this paper Kharitonov's theorem for the robust stability'.df kterval , polynomials is proved using the second method of Lyapunov. The Hermi+' matfi is taken as the matrix of the form which is used as a Lyapunov funct i i tq prove B m i B stability. It is shorn that if the four Hermite matrices eorrespqndmg; to the four Xharitonov extreme pobmomials are positive definite, the Hermitemhtriw of...

In this paper Kharitonov's theorem for the robust stability of interval polynomials is proved using the second method of Lyapunov. The Hermite matrix is taken as the matrix of the quadratic form which is used as a Lyapunov function to prove Hurwitz stability. I t is shown that if the four Hermite matrices corresponding to the four Kharitonov extreme polynomials are positive definite, the Hermit...

The general method of Lyapunov functionals construction has been developed during the last decade for stability investigations of stochastic differential equations with aftereffect and stochastic difference equations. After some modification of the basic Lyapunov type theorem this method was successfully used also for difference Volterra equations with continuous time. The latter often appear a...

We consider compact invariant sets Λ for C maps in arbitrary dimension. We prove that if Λ contains no critical points then there exists an invariant probability measure with a Lyapunov exponent λ which is the minimum of all Lyapunov exponents for all invariant measures supported on Λ. We apply this result to prove that Λ is uniformly expanding if every invariant probability measure supported o...

This paper generalizes the Lasalle-Yoshizawa Theorem to switched nonsmooth systems. It is established that a common candidate Lyapunov function with a negative semidefinite derivative is sufficient for boundedness of the system state and convergence of a positive semidefinite function of the system state to zero. The developed generalization is motivated by adaptive control of switched systems ...

in this paper, stabilization conditions and controller design for a class of nonlinear systems are proposed. the proposed method is based on the nonlinear feedback, quadratic lyapunov function and heuristic slack matrices definition. these slack matrices in null products are derived using the properties of the system dynamics. based on the lyapunov stability theorem and sum of squares (sos) dec...