Bézout identities and control of the heat equation
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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)^1/2θ(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)=∑_k=1^n c_kcosh(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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