On exponential stability of second order delay differential equations

We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays.     

Portfolio Optimization with Stochastic Volatilities and Constraints: An Application in High Dimension

In this paper we are interested in an investment problem with stochastic volatilities and portfolio constraints on amounts. We model the risky assets by jump diffusion processes and we consider an exponential utility function. The objective is to maximize the expected utility from the investor terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB in short) equation which could not be solved when the risky assets number exceeds three. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semilinear equation. We prove the existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We present an example that shows the importance of this reduction for numerical study of the optimal portfolio. We then compute the optimal strategy of investment by solving the associated optimization problem.     

Stability conditions to Bresse systems with indefinite memory dissipation

ABSTRACT

We study the asymptotic behavior of Bresse system with non-dissipative kernel memory acting only in the equation of longitudinal displacement. We show that the exponential stability depends on conditions regarding the decay rate of the kernel and a new relationship between the coefficients of the system. Moreover, this new condition on the constants of the system is used to prove strong stability and exponential stability for the homogeneous case (frictional dissipation in the longitudinal equation).     

A representation theorem for smooth Brownian martingales

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a ‘frozen path’. The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.     

Existence and robustness of exponential dichotomy of linear skew-product semiflows over semiflows

In this paper we investigate the exponential dichotomy of linear skew-product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on Banach bundles. We characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy.     

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 spatially homogeneous landau equation for hard potentials part ii : h-theorem and applications

We find a lower bound for the entropy dissipation of the spatially homogeneous Landau equation with hard potentials in terms of the entropy itself. We deduce from this explicit estimates on the speed of convergence towards equilibrium for the solution of this equation. In the case of so-called overmaxwellian potentials, the convergence is exponential. We also compute a lower bound for the spectral gap of the associated linear operator in this setting.     

Exponential Attractors for the Generalized Ginzburg-Landau Equation

Abstract In this paper, we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions. We show the squeezing property and the existence of finite dimensional exponential attractors for this equation * The author is supported by the Postdoctoral Foundation of China     

Convergence of Exponential Attractors for a Finite Element Approximation of the Allen–Cahn Equation

Abstract

We consider a space semidiscretization of the Allen–Cahn equation by continuous piecewise linear finite elements. For every mesh parameter h, we build an exponential attractor of the dynamical system associated with the approximate equations. We prove that, as h tends to 0, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated with the Allen–Cahn equation. We also prove that the fractal dimension of the exponential attractor and of the global attractor is bounded by a constant independent of h. Our proof is adapted from the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semigroup. Here, the perturbation is a space discretization. The case of a time semidiscretization has been analyzed in a previous paper.     

Pullback exponential attractors for nonautonomous dynamical system in space of higher regularity

Under what condition, a process which exists a $(E,E)$-pullback exponential attractor implies the existence of $(E,V)$- pullback exponential attractor when $V$ embedded in $E$? We answer this question in this paper. As an application of this result, we prove the existence of pullback exponential attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity in $L^q(\Omega)(\forall q\geq 2)$ and $H_0^1(\Omega)$.     

The inversion problem and applications of the generalized radon transform

We prove that under certain conditions the inversion problem for the generalized Radon transform reduces to solving a Fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation. We consider applications of the generalized Radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential Radon transforms.     

Asymptotic dynamics for reaction diffusion equations in unbounded domain

In this paper we study the asymptotic dynamics for reaction diffusion equation defined in $\mathbb{R}^n$. We will prove that the equation possesses a fixed point when the nonlinearity satisfies some restrictive conditions and then we show that the fixed point is an exponential attractor.     

Lie symmetries and travelling wave solutions of the nonlinear waves in the inhomogeneous Fisher-Kolmogorov equation

In this work, we consider a Fisher-Kolmogorov equation depending on two exponential functions of the spatial variables. We study this equation from the point of view of symmetry reductions in partial differential equations. Through two-dimensional abelian subalgebras, the equation is reduced to ordinary differential equations. New solutions have been derived and interpreted.     

Finite-parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearities

We study the problem of global exponential stabilization of original Burgers' equations and the Burgers' equation with nonlocal nonlinearities by controllers depending on finitely many parameters. We investigate both equations by employing controllers based on finitely many Fourier modes and the latter equation by employing finitely many volume elements. To ensure global exponential stabilization, we have provided sufficient conditions on the control parameters for each problem. We also show that solutions of the controlled equations are steering a concrete solution of the non-controlled system as t → ∞ with an exponential decay rate.     

Exponential stability of a second-order integro-differential equation with delay

We consider a second order integro-differential equation with time-delay and show that this equation can be written as an evolutionary equation. Within the framework of evolutionary equations we show the well-posedness and the exponential stability of the equation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence

We consider the mean field equation arising in the high-energy scaling limit of point vortices with a general circulation constraint, when the circulation number density is subject to a probability measure. Mathematically, such an equation is a non-local elliptic equation containing an exponential nonlinearity which depends on this probability measure. We analyze the behavior of blow-up sequences of solutions in relation to the circulation numbers. As an application of our analysis we derive an improved Trudinger-Moser inequality for the associated variational functional.     

The theory of non-Abelian tensor fields: Gauge transformations and curvature

We take one more step in formulating the theory of non-Abelian two-tensor fields: we find gauge transformation rules and the curvature tensor for them. To define the theory, we use the surface exponential. We derive a differential equation for the exponential and attempt to formulate its definition as a matrix model. We discuss applications of our construction to the Yang-Baxter equation for integrable models and to string field theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 73–91, April, 2006.     

Generalized calculus and sdes with non regular drift

In this note we prove a precise asymptotic estimate for Laplace type functionals for a parabolic SPDE. We use a large deviation principle, the stochastic Taylor expansion, some exponential inequalities and support theorems for our stochastic partial differential equation     

A Refinement of the Discrete Wirtinger Inequality

In this paper, we obtain an improved discrete Wirtinger inequality associated with a nonlinear second order differential equation. We apply this result to prove a Bonnesen-style isoperimetric inequality for plane polygons and reinterpret the main theorem as a weighted exponential inequality.     

Periodic problem for an evolution equation with a quadratic and a cubic nonlinearity

We study conditions for the existence of a solution of a periodic problem for a model nonlinear equation in the spatially multidimensional case and consider various types of large time asymptotics (exponential and oscillating) for such solutions. The generalized Kolmogorov-Petrovskii-Piskunov equation, the nonlinear Schrödinger equation, and some other partial differential equations are special cases of this equation. We analyze the solution smoothing phenomenon under certain conditions on the linear part of the equation and study the case of nonsmall initial data for a nonlinearity of special form. The leading asymptotic term is presented, and the remainder in the asymptotics of the solution is estimated in a spatially uniform metric.