A singular perturbation problem in exact controllability of the Maxwell system

Abstract. This paper studies the exact controllability of the Maxwell system in a bounded domain, controlled by a current flowing tangentially in the boundary of the region, as well as the exact con-trollability the same problem but perturbed by a dissipative term multiplied by a small parameter in the boundary condition. This boundary perturbation improves the regularity of the problem and is therefore a singular perturbation of the original problem. The purpose of the paper is to examine the connection, for small values of the perturbation parameter, between observability estimates for the two systems, and between the optimality systems corresponding to the problem of norm minimum exact control of the solutions of the two systems from the rest state to a specied terminal state

A global attractor for a fluid--plate interaction model accounting only for longitudinal deformations of the plate

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. The main peculiarity of the model is the assumption that the transversal displacements of the plate are negligible relative to in-plane displacements. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. We also show that the corresponding linearized system generates exponentially stable $C_0$-semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system.Comment: 18 page

Dynamics of thermoelastic thin plates: A comparison of four theories

Four distinct theories describing the flexural motion of thermoelastic thin plates are compared. The theories are due to Chadwick, Lagnese and Lions, Simmonds, and Norris. Chadwick's theory requires a 3D spatial equation for the temperature but is considered the most accurate as the others are derivable from it by different approximations. Attention is given to the damping of flexural waves. Analytical and quantitative comparisons indicate that the Lagnese and Lions model with a 2D temperature equation captures the essential features of the thermoelastic damping, but contains systematic inaccuracies. These are attributable to the approximation for the first moment of the temperature used in deriving the Lagnese and Lions equation. Simmonds' model with an explicit formula for temperature in terms of plate deflection is the simplest of all but is accurate only at low frequency, where the damping is linearly proportional to the frequency. It is shown that the Norris model, which is almost as simple as Simmond's, is as accurate as the more precise but involved theory of Chadwick.Comment: 2 figures, 1 tabl

Detecting a long lived false vacuum with quantum quenches

Distinguishing whether a system supports alternate low-energy (locally stable) states -- stable (true vacuum) versus metastable (false vacuum) -- by direct observation can be difficult when the lifetime of the state is very long but otherwise unknown. Here we demonstrate, in a tractable model system, that there are physical phenomena on much shorter time scales that can diagnose the difference. Specifically, we study the spectral density following a quench in the tilted quantum Ising model, and show that the evolution of the spectral density is a powerful diagnostic. Small transition bubbles are more common than large ones, and we see characteristic differences in the size dependence of bubble lifetimes even well below the critical size for false vacuum decay. We expect this sort of behavior to be generic in systems of this kind. We identify a scaling limit that connects the discrete, simulatable model to continuum quantum field theory.Comment: 7+7 pages, 3+1 figure

COMMUNITY COLLEGE STUDENT MOTIVATION, RETENTION, AND ENGAGEMENT IN A CULTURALLY RELEVANT DEVELOPMENTAL WRITING CLASS

As an Assistant Professor at Northeast Riverside Community College, I planned this action research project in order to seek solutions for the persistent problems of low retention and achievement faced by students designated to multiple semesters of basic literacy coursework. Writing 60 is the first developmental writing course of a two-course sequence required for students with significant skill gaps. Students are identified as appropriate for Writing 60 based on their college placement test scores. In my redesigned Writing 60 course, I maintained adherence to the college identified course objectives listed on the master course syllabus, but altered my former approaches to lessons, materials, reading selections and writing tasks with careful consideration of culturally relevant pedagogy and the critical language approach. Twenty-seven students registered for two sections of the redesigned Writing 60 course that ran in the fall semester of 2016. In order to assess the influence of culturally relevant pedagogy and the critical language approach upon course outcomes, I utilized a mixed methods design, and collected both quantitative and qualitative data. Attendance data, student persistence, and passing rates, were collected from the two sections of the redesigned Writing 60 course and from 12 sections of the previous Writing 60 course that I also taught from spring of 2014 through spring of 2016. Additional data was gathered from students enrolled in the redesigned Writing 60 course that included pre- and post- student motivation surveys, pre- and post- assessments of academic writing, field notes I collected throughout the semester, student completed checklists of major assignments, and two focus groups that I audio-recorded, transcribed, and coded for themes. The implications of the data I analyzed suggested correlation between the modifications of the course and improved attendance, retention, and passing rates

Strichartz Estimates for the Vibrating Plate Equation

We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schr\"odinger-type equations we show its close relation with the Schr\"odinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary homogeneous vibrating plate equation.Comment: 18 pages, 4 figures, some misprints correcte

Exponential stabilization of well-posed systems by colocated feedback

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