Time Decay Rates for Undamped Constant Coefficients Linear Partial Differential Equations

✍ Scribed by Matei Kelemen

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 444 KB
• Volume: 125
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0030

✦ Synopsis

this work is dedicated to professor isaac horowitz, on his 75th birthday

The zeros of the characteristic polynomial of many important equations in mathematical physics (e.g. the wave equation, the Schro dinger equation) are situated on the imaginary axis. This causes a very slow decay in the time variable of the solution driven by initial conditions of such equations. In this article we show that by displacing (by feedback) the zeros to the left of the imaginary axis so that they approach this axis asymptotically, one can change drastically the above situation. Indeed, one can achieve a polynomial decay of arbitrary degree in the time variable of the absolute value of the solution, uniformly in the space variable, provided the initial conditions are smooth enough. For such equations ``smoothness in space implies decay in time.'' The relation between smoothness and decay is established in a quantitative way. The systems under investigation are linear undamped partial differential equations with constant coefficients, in multidimensional space. We provide also natural conditions for the exponential decay of the absolute value of the solution.

1996 Academic Press, Inc.

Introduction

1.1. Motivation

This work was motivated by a real problem: the feedback control of flexible structures and robots. These are modeled mathematically by partial differential equations (PDE). The results we have obtained in this article, which show that polynomial enhancement of stability is achievable by feedback, represent a significant step toward the solution of this difficult and important problem.

We present now the problem in some detail. Many physical systems have valuable engineering properties when they have very low stability margin, article no. 0030