Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. (English) Zbl 1102.93004
Summary: In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter \(h\) goes to zero [see J.-A. Infante and E. Zuazua, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 6, 713–718 (1998; Zbl 0910.65051)].
Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in \(L^{2}(0, T)\) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal \(L^{2}\)-norm control. We illustrate the mathematical results with several numerical experiments.
Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in \(L^{2}(0, T)\) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal \(L^{2}\)-norm control. We illustrate the mathematical results with several numerical experiments.
MSC:
| 93B05 | Controllability |
| 93C20 | Control/observation systems governed by partial differential equations |
Keywords:
boundary controllability; linear semi-discrete 1-D wave equation; mixed finite element methodCitations:
Zbl 0910.65051References:
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