By Triggiani R., Imanuvilov O., Leugering G.

In those fabrics from the convention of May-June 2003, members of those 23 papers describe their paintings on the theoretical and alertness degrees during this self-discipline that has lately elevated significantly by means of extra practical versions. basic subject matters comprise elasticity, thermo-elasticity, aero-elasticity, interactions among fluids and elastic constructions, and fluid dynamics, parabolic platforms, dynamical Lame platforms, linear and non-linear hyperbolic equations, and pseudo-differential operators on a manifold

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It is independent of the Carleman inequality for the Stokes equation, which is necessary for the proof of local controllability. There is a large body of literature on the stabilization problem of steady-state solutions to NavierStokes equations. Here we confine ourselves to mention only a few of the papers (References 2 and 7), which are more related to this present work. We also refer to the recent paper of Fursikov [8] for a study of a boundary—rather than interior—problem for the Navier-Stokes equations, which, however, does not pertain to the topic of feedback stabilization in the established sense, as in the present paper.

G. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. , in press. [12] A. Bensoussan, G. Da Prato, M. Delfour, and S. Mitter, Representation and Control of Infinte Dimensional Systems, Vol. II, Birkh¨auser, Boston, (1993). [13] J. Cea, Lectures on Optimization—Theory and Algorithms Published for the Tata Institute of Fundamental Research, Springer-Verlag, New York, (1978). [14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, New York, New York, (1992).

43) a. By the regularity posted in Eq. 1, T 0 h [(G 1 − λG 2 )γ0 ϑ, φ] L 2 ( ) |h | T dt ≤ C t 0 1 4 ϑ 1− 1 2k L2( ) eA 2 x 0 t 1+ 1k 2 H dt. k Applying H¨older’s inequality to the right-hand side, with H¨older conjugates (2 2k2−1 , 1 ) now yields 1 +2−k−1 2 T 0 T h [(G 1 − λG 2 )γ0 ϑ, φ] L 2 ( ) dt ≤ C × ϑ 1− 1 2k L2( ) e A 2t 1+ 1k x0 H 2 dt ≤ C T 0 |h | h(t) 1 t 4 h(t) T k−1 −4 3 5×2k 2 −1 0 ϑ 1 −k−1 2 +2 2 L2( ) T + h(t)E 0 t dt. 1(1) 19 Because for any δ > 0, we can choose integer k large enough so that 3 5×22k −1−4 ≥ we then get k−1 T 0 h [(G 1 − λG 2 )γ0 ϑ, φ] L 2 ( ) dt ≤ C T T 15 2 − ϑ 2 L2( ) 0 T + h(t)E 0 15 2 − δ, t dt.

### Control Theory of Partial Differential Equations by Triggiani R., Imanuvilov O., Leugering G.

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