Numerical approximation of null controls for the heat equation : Ill-posedness and remedies
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The numerical approximation of exact or trajectory controls for the wave equation is known to be a delicate issue, since the pioneering work of Glowinski-Lions in the nineties, because of the anomalous behavior of the high-frequency spurious numerical waves. Various efficient remedies have been developed and analyzed in the last decade to filter out these highfrequency components: Fourier filtering, Tychonoff's regularization, mixed finite-element methods, multi-grid strategies, etc. Recently convergence rate results have also been obtained. This work is devoted to analyzing this issue for the heat equation, which is the opposite paradigm because of its strong dissipativity and smoothing properties. The existing analytical results guarantee that, at least in some simple situations, as in the finite-difference scheme in 1 - d, the null or trajectory controls for numerical approximation schemes converge. This is due to the intrinsic high-frequency damping of the heat equation that is inherited by its numerical approximation schemes. But when developing numerical simulations the topic appears to be much more subtle and difficult. In fact, efficiently computing the null control for a numerical approximation scheme of the heat equation is a difficult problem in itself. The difficulty is strongly related to the regularizing effect of the heat kernel. The controls of minimal L2-norm are characterized as minima of quadratic functionals on the solutions of the adjoint heat equation, or its numerical versions. These functionals are shown to be coercive in very large spaces of solutions, sufficient to guarantee the L2 character of controls, but very far from being identifiable as energy spaces for the adjoint system. The very weak coercivity of the functionals under considerationmakes the approximation problem exponentially ill-posed and the functional framework far from being well adapted to standard techniques in numerical analysis. In practice, the controls of the minimal L2-norm exhibit a singular highly oscillatory behavior near the final controllability time, which cannot be captured numerically. Standard techniques, such as Tychonoff's regularization or quasi-reversibility methods, allow a slight smoothing of the singularities but significantly reduce the quality of the approximation. In this paper we develop some more involved and less-standard approaches which turn out to be more efficient. We first discuss the advantages of using controls with compact support with respect to the time variable or the effect of adding numerical dissipative singular terms. We also develop the numerical version of the so-called transmutation method that allows writing the control of a heat process in terms of the corresponding control of the associated wave process, by means of a 'time convolution' with a one-dimensional controlled fundamental heat solution. This method, although it can be proved to converge, is also subtle in its computational implementation. Indeed, it requires using convergent numerical schemes for the control of the wave equation, a problem that, as mentioned above, is delicate in itself. But we also need to compute an accurate approximation of a controlled fundamental heat solution, an issue that requires its own analysis and significant numerical and computational new developments. These methods are thoroughly illustrated and discussed in the paper, accompanied by some numerical experiments in one space dimension that show the subtlety of the issue. These experiments allow one to compare the efficiency of the various methods. This is done in the case where the control is distributed in some subdomain of the domain where the heat process evolves but similar results and numerical experiments could be derived for other cases, such as the one in which the control acts on the boundary. The techniques we employ here can also be adapted to the multi-dimensional case. © 2010 IOP Publishing Ltd.