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47], the orthogonal space to H being made up of which are the gradients of by (5). Moreover, for (1) implies Interior and Boundary Stabilization of Navier-Stokes Equations 2. 45 The Main Results Assumptions. (i) The boundary of is a finite union of dimensional manifolds. 3, p. 11, p. 30] on for sufficiently smooth ] Preliminaries. The translated problem. By the substitutions we are readily led via (1), (2) to the study of null stabilization of the equation By use of (5) on H, (6), (7) on A and B, we see that (10), after application of P, can be rewritten abstractly as (compare with (9) again now introduced the operator since by (10)), where we have The operator in (12) is well-defined follows from the estimate This or equivalently, recalling (7), which is obtained directly by use of the definition (12b).

C. Delfour, N. -P. Zolésio, Extension of the uniform cusp property in shape optimization, in “Control of Partial Differential Equations”, G. Leugering, O. Imanuvilov, R. Triggiani, and B. Zhang, eds. Lectures Notes in Pure and Applied Mathematics, Marcel Dekker, May 2003, accepted. C. Delfour, N. -P. Zolésio, The uniform fat segment and cusp properties in shape optimization, in “Control and Boundary analysis”, J. -P. ), pp. 85–96, Marcel Dekker 2004. [6] E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, Boston, Basel, Stuttgart, 1984.

It was shown in [1] that, when is finite, The compactness result of [1] can be revisited and established in the from which convergence in all other topologies of Theorem 1 follows. 1 of [4] we get the analogues of the above Theorems 12 and 15. Recall the definition of the orthogonal subgroup of N × N matrices where *A is the transposed matrix of A. A direction can be specified either by a matrix (of rotation) or the corresponding unit vector THEOREM 16 Let ‚ borhood of 0 such that be given and assume that U is a bounded neigh- Let R > 0 be such that subset D of consider a family Given a bounded nonempty of subsets of with SYSTEM MODELING AND OPTIMIZATION 40 the following properties: for each there exist and a and and each where such that where (i) Assume that there exists and such that where Each of the uniform cusp property for the parameters Hence (from Theorem 12) the family is compact in (ii) Given in place of satisfies and the results of part (i) remain true with References [1] D.

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