Lyapunov’s coefficient of regularity and nonuniform hyperbolicity

For a nonautonomous dynamics with discrete time defined by a sequence of matrices, we obtain sharp lower and upper bounds for the Lyapunov coefficient of regularity. This has the advantage of avoiding considering the adjoint dynamics (in contrast to what happens with the regularity coefficients considered by Perron and Grobman). We also show that the dynamics can always be reduced to one defined by upper triangular matrices with the additional properties that the canonical basis is normal and ordered. Moreover, we show in a simpler manner that the Lyapunov coefficient of regularity is related to the notion of nonuniform hyperbolicity, more precisely to the nonuniform part of a nonuniform exponential contraction or a nonuniform exponential dichotomy. Finally, as an application of this relation, we show that from the point of view of ergodic theory, for almost all trajectories with negative Lyapunov exponents the nonuniformity can be made arbitrarily small.     

Homogenization and corrector for the wave equation with discontinuous coefficients in time

In this paper we analyze the homogenization of the wave equation with bounded variation coefficients in time, generalizing the classical result, which assumes Lipschitz-continuity. We start showing a general existence and uniqueness result for a general sort of hyperbolic equations. Then, we obtain our homogenization result comparing the solution of a sequence of wave equations to the solution of a sequence of elliptic ones. We conclude the paper making an analysis of the corrector. Firstly, we obtain a corrector result assuming that the derivative of the coefficients in the time variable is equicontinuous. This result was known for non-time dependent coefficients. After, we show, with a counterexample, that the regularity hypothesis for the corrector theorem is optimal in the sense that it does not hold if the time derivative of the coefficients is just bounded.     

Lyapunov regularity of impulsive differential equations

For linear impulsive differential equations, we give a simple criterion for the existence of a nonuniform exponential dichotomy, which includes uniform exponential dichotomies as a very special case. For this we introduce the notion of Lyapunov regularity for a linear impulsive differential equation, in terms of the so-called regularity coefficient. The theory is then used to show that if the Lyapunov exponents are nonzero, then there is a nonuniform exponential behavior, which can be expressed in terms of the Lyapunov exponents of the differential equation and of the regularity coefficient. We also consider the particular case of nonuniform exponential contractions when there are only negative Lyapunov exponents. Having this relation in mind, it is also of interest to provide alternative characterizations of Lyapunov regularity, and particularly to obtain sharp lower and upper bound for the regularity coefficient. In particular, we obtain bounds expressed in terms of the matrices defining the impulsive linear system, and we obtain characterizations in terms of the exponential growth rate of volumes. In addition we establish the persistence of the stability of a linear impulsive differential equation under sufficiently small nonlinear perturbations.     

On the class of proper linear differential systems with unbounded coefficients

Proper linear differential systems (whose coefficients are not necessarily bounded on the half-line) are defined as systems for which there exists a generalized Lyapunov transformation reducing them to a diagonal system with constant coefficients (Basov). We prove that Lyapunov’s original definition of a proper system and the Perron and Vinograd criteria hold for the class of proper systems as well as for the class of proper systems with uniformly bounded coefficients. We show that the Lyapunov properness criterion for a triangular system fails for systems with unbounded coefficients; namely, we construct an improper system with the following properties: the Lyapunov exponents of all nonzero solutions of that system are finite and exact, and for an arbitrary reduction of this system by a generalized Lyapunov transformation to triangular form, its diagonal coefficients have finite exact mean values, whose set with regard of multiplicities is independent of the choice of the transformation. In addition, we show that the main property of proper systems with uniformly bounded coefficients (preservation of conditional exponential stability as well as the dimension of the exponentially stable manifold and the exponent of the asymptotic behavior of solutions under perturbations of higher-order smallness) holds for proper systems with unbounded coefficients as well.     

Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization

The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property. Jameson’s duality for two cones, which relates bounded linear regularity to property (G), is re-derived and refined. For polyhedral cones, a statement dual to Hoffman’s error bound result is obtained. A sharpening of a result on error bounds for convex inequalities by Auslender and Crouzeix is presented. Finally, for two subspaces, property (G) is quantified by the angle between the subspaces. Received October 1, 1997 / Revised version received July 21, 1998? Published online June 11, 1999     

The width and integer optimization on simplices with bounded minors of the constraint matrices

In this paper, we will show that the width of simplices defined by systems of linear inequalities can be computed in polynomial time if some minors of their constraint matrices are bounded. Additionally, we present some quasi-polynomial-time and polynomial-time algorithms to solve the integer linear optimization problem defined on simplices minus all their integer vertices assuming that some minors of the constraint matrices of the simplices are bounded.     

k-途径正则有向图

In this paper, we define a class of strongly connected digraph, called the k-walk- regular digraph, study some properties of it, provide its some algebraic characterization and point out that the 0-walk-regular digraph is the same as the walk-regular digraph discussed by Liu and Lin in 2010 and the D-walk-regular digraph is identical with the weakly distance-regular digraph defined by Comellas et al in 2004.     

Periodic segment implies infinitely many periodic solutions

In this note we show that the existence of a periodic segment for a non-autonomous ODE with periodic coefficients implies the existence of infinitely many periodic solutions inside this segment provided that a sequence of Lefschetz numbers of iterations of an associated map is not constant. In the case when this sequence is bounded we have to impose a geometric condition on the segment to get solutions by use of symbolic dynamics.

     

Improved delay-dependent stability criteria for continuous system with two additive time-varying delay components

This paper investigated the problem of improved delay-dependent stability criteria for continuous system with two additive time-varying delay components. Free weighting matrices and convex combination method are not involved, which achieves much less numbers of linear matrix inequalities (LMIs) and LMIs scalar decision variables. By taking advantage of integral inequality and new Lyapunov–Krasovskii functional, new less conservative delay-dependent stability criterion is derived. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.     

Emptiness of set of points of lower semicontinuity of Lyapunov exponents

We consider parametric families of differential systems with coefficients that are bounded and continuous on the half-line and uniformly in time continuously depend on a real parameter. For each Lyapunov exponent, we construct a family such that the Lyapunov exponent of its systems treated as a function of the parameter is not a lower semicontinuous function for any value of the parameter.     

Lagrange multipliers for regular and singular problems in the calculus of variations

A Lagrange multiplier rule is presented for a variational problem of Bolza type under hypotheses that allow certain components of the coefficient matrices involved in the functional being minimized to fail to be integrable near an endpoint of the interval on which the relevant functions are defined. The problem is also addressed when all coefficients are of classL ², but not necessarily bounded. Applications are made to ascertain properties of functions providing equality to certain singular and regular integral inequalities appearing in the literature.     

Periodic and almost periodic solution for a non-autonomous epidemic predator-prey system with time-delay

In this paper, we studied a non-autonomous predator-prey system with discrete time-delay, where there is epidemic disease in the predator. By using some techniques of the differential inequalities and delay differential inequalities, we proved that the system is permanent under some appropriate conditions. When all the coefficients of the system is periodic, we obtained the existence and global attractivity of the positive periodic solution by Mawhin’s continuation theorem and constructing a suitable Lyapunov functional. Furthermore, when the coefficients of the system are not absolutely periodic but almost periodic, sufficient conditions are also derived for the existence and asymptotic stability of the almost periodic solution.     

Lyapunov exponents,symmetric spaces,and a multiplicative ergodic theorem for semisimple Lie groups

The classical theory of Lyapunov characteristic exponents is reformulated in invariant geometric terms and carried over to arbitrary noncompact semisimple Lie groups with finite center. A multiplicative ergodic theorem (a generalization of a theorem of Oseledets) and the global law of large numbers are proved for semisimple Lie groups, as well as a criterion for Lyapunov regularity of linear systems of ordinary differential equations with subexponential growth of coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 30–46, 1987.     

Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations

In this note we state weighted Poincaré inequalities associated with a family of vector fields satisfying Hörmander rank condition. Then, applications are given to relative isoperimetric inequalities and to local regularity (Harnack's inequality) for a class of degenerate elliptic equations with measurable coefficients.     

Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients

In this article, we study the problem of estimating the pathwise Lyapunov exponent for linear stochastic systems with multiplicative noise and constant coefficients. We present a Lyapunov type matrix inequality that is closely related to this problem, and show under what conditions we can solve the matrix inequality. From this we can deduce an upper bound for the Lyapunov exponent. In the converse direction, it is shown that a necessary condition for the stochastic system to be pathwise asymptotically stable can be formulated in terms of controllability properties of the matrices involved.     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.