关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Laplace Transform

Stability and stabilization of infinite dimensional systems with applications

Thank you very much for downloading introduction to infinite dimensional linear systems theory. As you may know, people have search numerous times for their favorite novels like this introduction to infinite dimensional linear systems theory, but end up in malicious downloads. Rather than enjoying a good book with a cup of tea in the afternoon, instead they juggled with some harmful bugs inside their laptop.

Introduction To Infinite Dimensional Linear Systems Theory

We present a new approach to mixed 2 /H ∞ output feedback control synthesis. Our method uses nonsmooth mathematical programming techniques to compute locally optimal 2 /H ∞ -controllers, which may have a predefined structure. We prove global convergence of our method and present numerical tests to validate it numerically.

/pdf/mixed-h-2-h-control-via-nonsmooth-optimization-2my5vbmaa6.pdf

Mixed H 2 /H ∞ control via nonsmooth optimization

We present a new approach to mixed $H_2/H_\infty$ output feedback control synthesis. Our method uses nonsmooth mathematical programming techniques to compute locally optimal $H_2/H_\infty$-controllers, which may have a predefined structure. We prove global convergence of our method and present tests to validate it numerically.

Mixed H2/H infinity control via nonsmooth optimization.

The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets $$A,B$$A,B under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is $${\mathcal {O}}(k^{-\rho })$$O(k-?) for some $$\rho \in (0,\infty )$$??(0,?).

/pdf/on-local-convergence-of-the-method-of-alternating-wkol6jprkn.pdf

On Local Convergence of the Method of Alternating Projections

We present a new proximity control bundle algorithm to minimize nonsmooth and nonconvex locally Lipschitz functions. In contrast with the traditional oracle-based methods in nonsmooth program- ming, our method is model-based and can accommodate cases where several Clarke subgradients can be computed at reasonable cost. We propose a new way to manage the proximity control parameter, which allows us to handle nonconvex objectives. We prove global convergence of our method in the sense that every accumulation point of the sequence of serious steps is critical. Our method is tested on a variety of examples in H∞-controller synthesis.

https://hal-unilim.archives-ouvertes.fr/hal-00464695/document

A Proximity Control Algorithm to Minimize Nonsmooth and Nonconvex Functions

In this paper, we study the behavior of solutions of the ODE associated to Nesterov acceleration. It is well-known since the pioneering work of Nesterov that the rate of convergence O(t 2) is optimal for the class of convex functions. In this work, we show that better convergence rates can be obtained with some additional geometrical conditions, such as Lojasiewicz property. More precisely, we prove the optimal convergence rates that can be obtained depending on the geometry of the function F to minimize. The convergence rates are new, and they shed new light on the behavior of Nesterov acceleration schemes.

https://hal.archives-ouvertes.fr/hal-01786117/document

Aude Rondepierre

Papers

Mixed H 2 /H ∞ control via nonsmooth optimization

Mixed H2/H infinity control via nonsmooth optimization.

On Local Convergence of the Method of Alternating Projections

A Proximity Control Algorithm to Minimize Nonsmooth and Nonconvex Functions

Optimal Convergence Rates for Nesterov Acceleration