Conditions For Exponential Decay Research Articles

Diffusing-Horizon Model Predictive Control

We analyze a time-coarsening strategy for model predictive control (MPC) that we call diffusing-horizon MPC. This strategy seeks to overcome the computational challenges associated with optimal control problems that span multiple timescales. The coarsening approach uses a time discretization grid that becomes exponentially more sparse as one moves forward in time. This design is motivated by a recently established property of optimal control problems that is known as exponential decay of sensitivity (EDS). This property states that the impact of a parametric perturbation at a future time decays exponentially as one moves backward in time. We establish conditions under which this property holds for a constrained MPC formulation with linear dynamics and costs. Moreover, we show that the proposed coarsening scheme can be cast as a parametric perturbation of the MPC problem and, thus, the exponential decay condition holds. We use a HVAC plant case study with real data to demonstrate the proposed approach. Specifically, we show that computational times can be reduced by two orders of magnitude while increasing the closed-loop cost by only 3%.

Exponential decay for negative feedback loop with distributed delay

We derive sufficient conditions for exponential decay of solutions of the delay negative feedback equation with distributed delay. The conditions are written in terms of exponential moments of the distribution. Our method only uses elementary tools of calculus and is robust towards possible extensions to more complex settings, in particular, systems of delay differential equations. We illustrate the applicability of the method to particular distributions — Dirac delta, Gamma distribution, uniform and truncated normal distributions.

Existence, uniqueness, and approximation solutions to linearized Chandrasekhar equation with sharp bounds

This article continues to study the linearized Chandrasekhar equation. We use the Hilbert-type inequalities to accurately calculate the norm of the Fredholm integral operator and obtain the exact range for the parameters of the linearized Chandrasekhar equation to ensure that there is a unique solution to the equation in $$L^p$$ space. A series of examples that can accurately calculate the norm of Fredholm integral operator shows that the Chandrasekhar kernel functions do not need to meet harsh conditions. As the symbolic part of the Chandrasekhar kernel function and the non-homogeneous terms satisfy the exponential decay condition, we yield a normed convergence rate of the approximation solution in $$L^p$$ sense, which adds new results to the theory of radiation transfer in astrophysics.

On a general nonlinear problem with distributed delays

The paper considers a general system of ordinary differential equations appearing in the neural network theory. The activation functions are assumed to be continuous and bounded by power type functions of the states and distributed delay terms. These activation functions are not necessarily Lipschitz continuous as it is commonly assumed in the literature. We obtain sufficient conditions for exponential decay of solutions.

Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators

We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrödinger operators. Let $Λ_L = (-L/2, L/2)^d$ and $H_L = -Δ_L + V_L$ be a Schrödinger operator on $L^2 (Λ_L)$ with a bounded potential $V_L : Λ_L \to \mathbb{R}^d$ and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the type $ \int_{Λ_L} \lvert φ \rvert^2 \leq C_{\rm {sfuc}} \int_{W_δ (L)} \lvert φ \rvert^2,$ where $φ$ is an infinite complex linear combination of eigenfunctions of $H_L$ with exponentially decaying coefficients, $W_δ (L)$ is some union of equidistributed $δ$-balls in $Λ_L$ and $C_{{\rm {sfuc}}} > 0$ an $L$-independent constant. The exponential decay condition on $φ$ can alternatively be formulated as an exponential decay condition of the map $λ \mapsto \lVert χ_{[λ, ∞)} (H_L) φ \rVert^2$. The novelty is that at the same time we allow the function $φ$ to be from an infinite dimensional spectral subspace and keep an explicit control over the constant $C_{{\rm {sfuc}}}$ in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.

Remark on a weakly coupled system of nonlinear damped wave equations

We study global existence of small solutions to the Cauchy problem for a weakly coupled nonlinear damped wave equation{(∂t2+∂t−Δ)u=N1(v),(∂t2+∂t−Δ)v=N2(u),x∈Rn,t>0u(0,x)=εu0(x),∂tu(0,x)=εu1(x),v(0,x)=εv0(x),∂tv(0,x)=εv1(x),x∈Rn, with super-critical nonlinearities Nk(ϕ)=|ϕ|ρk, k=1,2, where ε>0, the space dimension n≥4. Our purpose is to remove the exponential decay condition on the data and the lower bound for ρ1 which was assumed in [4] when proving the global existence of solutions in the case of higher space dimensions.

On large deviations for some sequences of weighted means of Gaussian processes

In this paper we study some sequences of weighted means of continuous real valued Gaussian processes. More precisely we consider suitable generalizations of both arithmetic and logarithmic means of a Gaussian process with covariance function which satisfies either an exponential decay condition or a power decay condition. Our aim is to provide limits of variances of functionals of such weighted means which allow the application of some large deviation results in the literature.

Large deviations for estimators of unknown probabilities, with applications in risk theory

We present large deviation results for estimators of unknown probabilities which satisfy a suitable exponential decay condition. These results provide some extensions of the large deviation estimates given in Macci and Petrella (2006). Furthermore we propose a classical approach which is different from the one presented in Ganesh et al. (1998) and we cannot say that the Bayesian approach is more conservative as in that paper.

Criteria of exponential decay for eigenfunctions of some classes of integral operators

We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ n . We consider integral operators K whose kernels have the form $$ k\left( {x,y} \right) = c\left( {x,y} \right)\frac{{{e^{ - \alpha \left| {x - y} \right|}}}}{{{{\left| {x - y} \right|}^\beta }}},\,\left( {x,y} \right) \in \Omega \times \Omega, $$ where α > 0, 0 ≤ β < n, and c(x, y) ∈ L ∞(Ω × Ω). It is shown that if the corresponding operator T = I − K is Fredholm, then all the solutions of the integral equation φ = Kφ exponentially decay at infinity. We consider applications to Wiener–Hopf operators with oscillating coefficient and to some classes of convolution operators with variable coefficients. Bibliography: 14 titles.

A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations

AbstractIn [6], Walter Philipp wrote that “… the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.” Many authors [1], [2], [4], [5], [9] have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables.We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn [8] for certain Gibbs measures. These do not fall into the ϕ-mixing or strong mixing cases established in the literature, but are needed for our investigations [7] into diffusions on configuration space.The proofs are all obtained by patching together standard results from [5], [9] while keeping a careful eye on the correlations.

An Inverse Scattering Problem for Dirac Equations with Time-Dependent Electromagnetic Potentials

We consider an inverse scattering problem, by using a time-dependent method, for the Dirac equation with a time-dependent electromagnetic field. The Fourier transform of the field is reconstructed from the scattering operator on a Lorents invariant set D: = \{(τ, ξ) \in \boldsymbol R \times \boldsymbol R^3;|τ|&lt;c|ξ|\} in the dual space of the space-time. As corollaries of this result, we can reconstruct the electromagnetic field completely if it is a finite sum of fields each of which is a time-independent one by a suitable Lorentz transform, and we can also determine the field uniquely if the fields satisfies some exponential decay condition. Our assumptions and results are independent of a choice of inertial frames.

Stability of uncertain systems with hysteresis nonlinearities

Stability of systems with hysteresis nonlinearities, parametric uncertainty and finite dimensional unmodelled dynamics is considered. Conditions for exponential decay of the signals in the system to an equilibrium position are given. The equilibrium is generally not unique. The stability condition is given in terms of a frequency domain inequality, which involves the transfer function of the nominal system and a multiplier, The search for a suitable multiplier can be performed by use of convex optimization in terms of linear matrix inequalities. A simple example is given that illustrates our result. © 1998 John Wiley & Sons, Ltd.

Time dependence in quantum mechanics—Study of a simple decaying system

A simple quantum-mechanical model consisting of a two-level system coupled to a continuum, where the interaction can be varied from perturbative to strong, is examined for what can be learned about quantum mechanical decay using elementary methods. The decay spectrum and time dependence of the amplitudes can be calculated in this model, without resorting to perturbation theory. The spectrum is not precisely Lorentzian nor is the decay precisely exponential, features that appear to be shared by all quantum-mechanical models of decaying systems. The conditions for exponential decay and some of the reasons for deviation from this at both short and long times are discussed.

Hitting-time and occupation-time bounds implied by drift analysis with applications

Bounds of exponential type are derived for the first-hitting time and occupation times of a real-valued random sequence which has a uniform negative drift whenever the sequence is above a fixed level. The only other assumption on the random sequence is that the increments satisfy a uniform exponential decay condition. The bounds provide a flexible technique for proving stability of processes frequently encountered in the control of queues.Two applications are given. First, exponential-type bounds are derived for a GI/G/1 queue when the service distribution is exponential type. Secondly, geometric ergodicity is established for a certain Markov chain in which arises in the decentralized control of a multi-access, packet-switched broadcast channel.

Hitting-time and occupation-time bounds implied by drift analysis with applications

Bounds of exponential type are derived for the first-hitting time and occupation times of a real-valued random sequence which has a uniform negative drift whenever the sequence is above a fixed level. The only other assumption on the random sequence is that the increments satisfy a uniform exponential decay condition. The bounds provide a flexible technique for proving stability of processes frequently encountered in the control of queues. Two applications are given. First, exponential-type bounds are derived for a GI/G/1 queue when the service distribution is exponential type. Secondly, geometric ergodicity is established for a certain Markov chain in which arises in the decentralized control of a multi-access, packet-switched broadcast channel.