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Register a new userNonuniform Mean-square Exponential Dichotomies and Mean-square Exponential Stability
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In this paper, the existence conditions of nonuniform mean-square exponential dichotomy (NMS-ED) for a linear stochastic differential equation (SDE) are established. The difference of the conditions for the existence of a nonuniform dichotomy between an SDE and an ordinary differential equation (ODE) is that the first one needs an additional assumption, nonuniform Lyapunov matrix, to guarantee that the linear SDE can be transformed into a decoupled one, while the second does not. Therefore, the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity. This paper is also concerned with the mean-square exponential stability of nonlinear perturbation of a linear SDE under the condition of nonuniform mean-square exponential contraction (NMS-EC). For this purpose, the concept of second-moment regularity coefficient is introduced. This concept is essential in determining the stability of the perturbed equation, and hence we deduce the lower and upper bounds of this coefficient. Our results imply that the lower and upper bounds of the second-moment regularity coefficient can be expressed solely by the drift term of the linear SDE.
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For linear stochastic differential equations (SDEs) with bounded coefficients, we establish the robustness of nonuniform mean-square exponential dichotomy (NMS-ED) on $[t_{0},+oo)$, $(-oo,t_{0}]$ and the whole $R$ separately, in the sense that such an NMS-ED persists under a sufficiently small linear perturbation. The result for the nonuniform mean-square exponential contraction (NMS-EC) is also discussed. Moreover, in the process of proving the existence of NMS-ED, we use the observation that the projections of the exponential growing solutions and the exponential decaying solutions on $[t_{0},+oo)$, $(-oo,t_{0}]$ and $R$ are different but related. Thus, the relations of three types of projections on $[t_{0},+oo)$, $(-oo,t_{0}]$ and $R$ are discussed.
We investigate how the following properties are related to each other: i)-A manifold is transversally exponentially stable; ii)-The transverse linearization along any solution in the manifold is exponentially stable; iii)-There exists a field of positive definite quadratic forms whose restrictions to the directions transversal to the manifold are decreasing along the flow. We illustrate their relevance with the study of exponential incremental stability. Finally, we apply these results to two control design problems, nonlinear observer design and synchronization. In particular, we provide necessary and sufficient conditions for the design of nonlinear observer and of nonlinear synchronizer with exponential convergence property.
The concept of square-mean almost automorphy for stochastic processes is introduced. The existence and uniqueness of square-mean almost automorphic solutions to some linear and non-linear stochastic differential equations are established provided the coefficients satisfy some conditions. The asymptotic stability of the unique square-mean almost automorphic solution in square-mean sense is discussed.
Detailed theoretical study of the mean square radius of extensive air shower electrons has been made in connection with further development of scaling formalism for electron lateral distribution function. A very simple approximation formula, which allows joint description of all our results obtained in wide primary energy range and for different observation depths is presented. The sensitivity of the mean square radius to variations of basic parameters of hadronic interaction model is discussed.
In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if $F$ is a finite collection of hyperbolic automorphisms of a CAT(0) square complex $X$, then either there exists a pair of words of length at most 10 in $F$ which freely generate a free semigroup, or all elements of $F$ stabilize a flat (of dimension 1 or 2 in $X$). As a corollary, we obtain a lower bound for the growth constant, $sqrt[10]{2}$, which is uniform not just for a given group acting freely on a given CAT(0) cube complex, but for all groups which are not virtually abelian and have a free action on a CAT(0) square complex.
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