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.

翻译：解析贝祖恒等式的计算仍是一项难点，其在控制理论中具有广泛应用。平坦偏微分方程系统为这一问题提供了新视角。本文考虑一个具有特殊意义的简单案例：一根长度为$a+b$的杆，两端绝热，在$x=a$处加热。当$a=0$时，另一端温度$\theta(b,t)$成为经典平坦输出，其参数化形式为$\theta(x,t)=\cosh((b-x)(\partial/\partial t)^{1/2}\theta(b,t)$。当$a$和$b$为整数且$a$为奇数、$b$为偶数时，系统是平坦的，平坦输出可由贝祖恒等式$f(x)\cosh(ax)+g(x)\cosh(bx)=1$获得，该等式的计算归结为切比雪夫多项式的贝祖恒等式。但此形式并非最高效，可采用线性时间计算更紧凑的表达式$f(x)=\sum_{k=1}^{n} c_{k}\cosh(kx)$。我们将这些结果与通过经典离散化得到的有限系统近似值进行比较。通过将非有理数$r$近似为分数序列$b/a$进行实验计算，结果表明贝祖关系的幂级数似乎收敛。