Bézout identities and control of the heat equation
Identités de Bézout et contrôle de l'équation de la chaleur
Résumé
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 θ(b, t) being then a flat output, with parametrization θ(x, t) = cosh((b − x)(∂/∂t) 1/2 θ(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 computation 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) = ∑ n k=1 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.
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