Iteration Argument Research Articles

Observability of heat processes by transmutation without geometric restrictions

The goal of this note is to explain how transmutation techniques (originally introduced in [14] in the context of the control of the heat equation, inspired on the classical Kannai transform, and recently revisited in [4] and adapted to deal with observability problems) can be applied to derive observability results for the heat equation without any geometric restriction on the subset in which the control is being applied, from a good understanding of the wave equation. Our arguments are based on the recent results in [15] on the frequency depending observability inequalities for waves without geometric restrictions, an iteration argument recently developed in [13] and the new representation formulas in [4] allowing to make a link between heat and wave trajectories.

Decentralized control of discrete‐time linear time invariant systems with input saturation

AbstractWe study decentralized stabilization of discrete‐time linear time invariant (LTI) systems subject to actuator saturation using LTI controllers. The requirement of stabilization under both saturation constraints and decentralization imposes obvious necessary conditions on the open‐loop plant, namely that its eigenvalues are in the closed unit disc and further that the eigenvalues on the unit circle are not decentralized fixed modes. The key contribution of this work is to provide a broad sufficient condition for decentralized stabilization under saturation. Specifically, we show through an iterative argument that the stabilization is possible: whenever (1) the open‐loop eigenvalues are in the closed unit disc; (2) the eigenvalues on the unit circle are not decentralized fixed modes; and (3) these eigenvalues on the unit circle have algebraic multiplicity of 1. Copyright © 2009 John Wiley & Sons, Ltd.

On Gaussian decay estimates of solutions to some linear elliptic equations and their applications

In this paper we establish pointwise decay estimates of solutions to some linear elliptic equations by using the Nash–Moser iteration arguments and the ODE method. As applications we obtain sharp Gaussian decay estimates for solutions to nonlinear elliptic equations that are related with self-similar solutions to nonlinear heat equations and standing wave solutions to nonlinear Schrodinger equations with harmonic potential.

Local well-posedness for quadratic nonlinear Schrödinger equations and the ``good'' Boussinesq equation

The Cauchy problem for 1-D nonlinear Schrödinger equations with quadratic nonlinearities are considered in the spaces $H^{s,a}$ defined by $ \| f \|_{H^{s,a}}=\| (1+|\xi|)^{s-a} |\xi|^a \widehat{f} \|_{L^2}, $ and sharp local well-posedness and ill-posedness results are obtained in these spaces for nonlinearities including the term $u\bar{u}$. In particular, when $a=0$ the previous well-posedness result in $H^s$, $s>-1/4$, given by Kenig, Ponce and Vega (1996), is improved to $s\ge -1/4$. This also extends the result in $H^{s,a}$ by Otani (2004). The proof is based on an iteration argument similar to that of Kenig, Ponce and Vega, with a modification of the spaces of the Fourier restriction norm. Our result is also applied to the ``good'' Boussinesq equation and yields local well-posedness in $H^s\times H^{s-2}$ with $s>-1/2$, which is an improvement of the previous result given by Farah (2009).

Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case

Extending previous results of Oh–Zumbrun and Johnson–Zumbrun, we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type, removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions. Key to our analysis is a nonlinear cancellation estimate observed by Johnson and Zumbrun, along with a detailed understanding of the Whitham averaged system. The latter motivates a careful analysis of the Bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave, allowing us to derive the sharpened low-frequency estimates needed to close the nonlinear iteration arguments.

Counterexamples to Bilinear Estimates for the Korteweg-de Vries Equation in the Besov-Type Bourgain Space

Recently it was shown by Z. Guo and the author independently that the Cauchy problem for the nonperiodic Korteweg-de Vries equation is locally well-posed in the Sobolev space of the critical regularity. Their proofs were based on the iteration argument in the Besov endpoint of the Bourgain space with some additional modification in low frequency, and it was conjectured that the Bourgain space with the Besov-type modification only does not restore the bilinear estimate which is essential to the iteration. In the present article we give the answer to this conjecture by constructing an exact counterexample to the bilinear estimate.

ON THE CONDITIONAL LIKELIHOOD RATIO TEST FOR SEVERAL PARAMETERS IN IV REGRESSION

For the problem of testing the hypothesis that all m coefficients of the right-hand-side endogenous variables in an instrumental variables (IV) regression are zero, the likelihood ratio (LR) test can, if the reduced form covariance matrix is known, be rendered similar by a conditioning argument. To exploit this fact requires knowledge of the relevant conditional cumulative distribution function (c.d.f.) of the LR statistic, but the statistic is a function of the smallest characteristic root of an (m + 1)-square matrix and is therefore analytically difficult to deal with when m > 1. We show in this paper that an iterative conditioning argument used by Hillier (2009) and Andrews, Moreira, and Stock (2007 Journal of Econometrics 139, 116–132) to evaluate the c.d.f. in the case m = 1 can be generalized to the case of arbitrary m. This means that we can completely bypass the difficulty of dealing with the smallest characteristic root. Analytic results are obtained for the case m = 2, and a simple and efficient simulation approach to evaluating the c.d.f. is suggested for larger values of m.

Complete Stability in Multistable Delayed Neural Networks

We investigate the complete stability for multistable delayed neural networks. A new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property. An iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity. The existence of 3n equilibria and 2n positively invariant sets for the n-neuron system remains valid under the new formulation. The theory is demonstrated by a numerical illustration.

$${\mathcal{R}}$$ -boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class (\({\mathcal{HT}}\)) and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with \({\mathcal{R}}\) -bounded symbols, yielding by an iteration argument the \({\mathcal{R}}\) -boundedness of λ(A−λ)−1 in \(\Re(\lambda) \geq \gamma\) for some \(\gamma \in {\mathbb{R}}\) . To this end, elements of a symbolic and operator calculus of pseudodifferential operators with \({\mathcal{R}}\) -bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on \({\mathbb{R}}^{d}\) with operator valued coefficients.

DIRAC EQUATION WITH CERTAIN QUADRATIC NONLINEARITIES IN ONE SPACE DIMENSION

We discuss the time local existence of solutions to the Dirac equation for special types of quadratic nonlinearities in one space dimension. Solutions with more rough data than those of the previous work [15] are obtained. The Fourier transforms of solutions with respect to both variables x and t are investigated. Certain linear and bilinear estimates on solutions are derived, and a standard iteration argument gives the existence results.

Remarks on the uniqueness problem for the logistic equation on the entire space

We consider the logistic equation −Δu=a(x)u−b(x)uqon all ofRNwitha(x)/|x|γandb(x)/|x|τbounded away from 0 and infinity for all large |x|, where γ > −2, τ ∈ (−∞, ∞). We show that this problem has a unique positive solution. This considerably improves some earlier results. The main new technique here is a Safonov type iteration argument. The result can also be proved by a technique introduced by Marcus and Veron, and the two different techniques are compared.

Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum

In this paper, we study the Cauchy problem for the Boltzmann equation with an external force and the Vlasov-Poisson-Boltzmann system in infinite vacuum. The global existence of solutions is first proved for the Boltzmann equation with an external force which is integrable with respect to time in some sense under the smallness assumption on initial data in weighted norms. For the Vlasov-Poisson-Boltzmann system, the smallness assumption on initial data leads to the decay of the potential field which in turn gives the global existence of solutions by the result on the case with external forces and an iteration argument. The results obtained here generalize those previous works on these topics and they hold for a class of general cross sections including the hard-sphere model.

Pointwise curvature estimates for F-stable hypersurfaces

We consider immersed hypersurfaces in euclidean Rn+1 which are stable with respect to an elliptic parametric functional of the form F(X)=∫MF(N)dμ. We prove a pointwise curvature estimate provided that n⩽5 and F is sufficiently close to the area integrand. This extends the pointwise curvature estimates of Schoen, Simon and Yau [Acta Math. 134 (1975) 275] for stable minimal hypersurfaces in Rn+1 and of Simon [Math. Z. 154 (1977) 265] for minimizers of F. Our result follows from an integral curvature estimate and a generalized Simons inequality that were established recently [Calc. Var. Partial Differential Equations (2004), DOI: 10.1007/S00526-004-0306-5], together with a Moser type iteration argument.

Nucleation in the FitzHugh–Nagumo system: Interface–spike solutions

We find nucleation solutions of N interfaces and K spikes to the one-dimensional FitzhHugh–Nagumo system. Each spike sits asymptotically in the middle between two interfaces. We use the Lyapunov–Schmidt reduction method, in which the problem is split into a finite-dimensional problem related to the translation of the K spikes and an infinite-dimensional complement problem. However the complement problem remains near degenerate due to the translation of the N interfaces. To overcome this difficulty we move the interfaces by a small distance and solve the complement problem with the help of a Newton iteration argument.

Linear and quasilinear parabolic equations in Sobolev space

We consider linear parabolic equations of second order in a Sobolev space setting. We obtain existence and uniqueness results for such equations on a closed two-dimensional manifold, with minimal assumptions about the regularity of the coefficients of the elliptic operator. In particular, we derive a priori estimates relating the Sobolev regularity of the coefficients of the elliptic operator to that of the solution. The results obtained are used in conjunction with an iteration argument to yield existence results for quasilinear parabolic equations.

Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices

We present a new approach of asymptotic freeness in mean and almost everywhere for independent Gaussian or Wishart complex matrices together with some independent set {A i | i ∈ N*} of random matrices. This approach is based on the convergence of the generalized moments together with an iterative argument on the number of involved Gaussian or Wishart matrices. In the Gaussian case, a first proof is realized without any counting argument. Nevertheless in both cases, a precise computation of these moments leads asymptotically to the convolution relation set up by Speicher and Nica between the moments of free variables. We draw a sufficient condition on Hermitian matricial models for asymptotic freeness. The asymptotic freeness is then proved between independent Beta and Wishart matrices, and the limiting eigenvalues distribution of the last ones is determined.

Uniqueness results for some PDEs

Existence of solutions to many kinds of PDEs can be proved by using a fixed point argument or an iterative argument in some Banach space. This usually yields uniqueness in the same Banach space where the fixed point is performed. We give here two methods to prove uniqueness in a more natural class. The first one is based on proving some estimates in a less regular space. The second one is based on a duality argument. In this paper, we present some results obtained in collaboration with Pierre-Louis Lions, with Kenji Nakanishi and with Fabrice Planchon.

Resonances and time operator for the cusp map

A stochastic model for the spread of an SIS epidemic among a population consisting of N individuals, each having heterogeneous infectiousness and/or susceptibility, is considered and its behavior is analyzed under the practically relevant situation when N is small. The model is formulated as a finite time-homogeneous continuous-time Markov chain X. Based on an appropriate labeling of states, we first construct its infinitesimal rate matrix by using an iterative argument, and we then present an algorithmic procedure for computing steady-state measures, such as the number of infected individuals, the length of an outbreak, the maximum number of infectives, and the number of infections suffered by a marked individual during an outbreak. The time till the epidemic extinction is characterized as a phase-type random variable when there is no external source of infection, and its Laplace–Stieltjes transform and moments are derived in terms of a forward elimination backward substitution solution. The inverse iteration method is applied to the quasi-stationary distribution of X, which provides a good approximation of the process X at a certain time, conditional on non-extinction, after a suitable waiting time. The basic reproduction number R0 is defined here as a random variable, rather than an expected value.

Global Well-Posedness for Schrödinger Equations with Derivative

We prove that the one-dimensional Schrödinger equation with derivative in the nonlinear term is globally well-posed in Hs for s>2/3, for small L2 data. The result follows from an application of the "I-method." This method allows us to define a modification of the energy norm H1 that is "almost conserved" and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in Hs, for s>2/3.

Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations

In this paper we prove that the null controllability property of the heat equation may be obtained as limit of the exact controllability properties of singularly perturbed damped wave equations. We impose Dirichlet, homogeneous boundary conditions. The control is supported in a neighborhood of a subset of the boundary that satisfies the classical requirements to apply multiplier techniques. The proof needs an iterative argument that allows to treat separately the low and high frequencies and to make use of the dissipativity of the systems under consideration. This is combined with sharp observability estimates on the eigenfunctions of the Laplacian due to G. Lebeau and L. Robbiano, and global Carleman estimates. This proof applies in any space dimension. As a consequence of the uniform controllability we derive uniform observability estimates which can not be proved by classical methods due to the singular character of the perturbations we deal with.