Bézout identities and control of the heat equation

Abstract

Computing analytic Bézout identities remains a difficult task, which has many applications in control theory. Flat PDE systems have cast a new light on this problem. We consider here a simple case of special interest: a rod of length $a+b$, insulated at both ends and heated at point $x=a$. The case $a=0$ is classical, the temperature of the other end $\theta (b,t)$ being then a flat output, with parametrization $\theta (x,t)=\cosh ((b-x)(\partial /\partial t{)}^{1/2}\theta (b,t)$. When $a$ and $b$ are integers, with $a$ odd and $b$ even, the system is flat and the flat output is obtained from the Bézout identity $f(x)\cosh (ax)+g(x)\cosh (bx)=1$, the omputation of which boils down to a Bézout identity of Chebyshev polynomials. But this form is not the most efficient and a smaller expression $f(x)={\sum }_{k=1}^n{c}_k\cosh (kx)$ may be computed in linear time. These results are compared with an approximations by a finite system, using a classical discretization. We provide experimental computations, approximating a non rational value $r$ by a sequence of fractions $b/a$, showing that the power series for the Bézout relation seems to converge.

24 pages, 5 figures