On the Dirichlet Boundary Controllability of the 1 D-Heat Equation . Semi-Analytical Calculations and Ill-posedness Degree

The ill-posedness degree for the controlability of the one-dimensional heat equation by a Dirichlet boundary control is the purpose of this work. This problem is severely (or exponentially) ill-posed. We intend to shed more light on this assertion and the underlying mathematics. We start by discussing the framework liable to fit an efficient numerical implementation without introducing further complications in the theoretical analysis. We expose afterward the Fourier calculations that transform the ill-posedness issue to a one related to the linear algebra. This consists in investigating the singular values of some infinite structured matrices that are obtained as sums of Cauchy matrices. Calling for the Gershgorin-Hadamard theorem and the Collatz-Wielandt formula, we are able to provide lower and upper bounds for the largest singular value of these matrices. After checking out that they are also solutions of some symmetric Lyapunov (or Sylvester) equations with a very low displacement rank, we use an estimate that improves former Penzl’s result to bound the ratio smaller/largest singular values of these matrices. Accordingly, the controllability problem is confirmed to be severely ill-posed. The bounds proved here will be supported by computations made by means of Matlab procedures. At last, the well known explicit inverse of Cauchy’s type matrices allows to provide a formal exponential series representation of the Dirichlet control in a long horizon controllability. That series has to be checked afterward whether it is convergent or not to find out whether the desired state is reachable or not. Here again, some examples run within Matlab will be discussed and commented. keywords: Controllability, HUM control, Dirichlet control, Cauchy matrices, Löewner matrices, Lyapunov equation, Penzl’s type bounds, ill-posedness degree.

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