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Additional resources for Optimal Control of Partial Differential Equations: International Conference in Chemnitz, Germany, April 20-25, 1998

Example text

For the same reason as above we can assume that h ::; l2. Following in a similar way to the case p(x) = 1 and using controllability results proved in Section 2, one can prove that bijectivity of the map (13) is equivalent to the bijectivity of the map (14). From (21) it follows that u(L, t) = v: L L L 40000 1 k r=Ok=Oq=O A(r, k, q)f(t - w(r, k, q) - L). (22) Let us suppose now that the gap between the intervals hand 12 is not large, more exactly, G := a2 - b1 ::; h. In this case the restriction of the right-hand side of (22) to the interval [T - L, T - L + hl contains only three nonzero terms.

The key ingredients are the following: The uniqueness of the corresponding static problem (due to the structural Assumption 1), and the compactness of the embeddings which are inherent in the analytic estimate (31). The explicit details of this argument are given here. If the G. Avalos, I. Lasiecka and R. ) = dt OOj E(n)(o) < constant, "In. 1, then the sequence {E(n)(t)} is then also bounded uniformly in n for t ::::; T. ) ] dt::::; C, t (43) where the dependence of the constant C on T is not noted here.

32 G . Avalos, 1. Lasiecka and R. Triggiani [34] T. Wah, Large amplitude flexural vibrations of rectangular plates, Int. J. Meeh. SeL, Vol. 5 (1963), pp. 425-438. [35] D. Tataru, On the regularity of boundary traces for the wave equation, Annali di Seuola Normale di Pisa, to appear. George Avalos Department of Mathematics and Statistics Texas Tech University Lubbock Texas 79409 (USA). Irena Lasiecka and Roberto Triggiani Department of Mathematics University of Virginia Kerchof Hall Charlottesville Virginia 22903 (USA).

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