B\'ezout identities and control of the heat equation

Computing analytic B\'ezout 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 B\'ezout identity $f(x)\cosh(ax)+g(x)\cosh(bx)=1$, the omputation of which boils down to a B\'ezout 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 B\'ezout relation seems to converge.