By Jean-Paul Zolesio

In keeping with the operating convention on Boundary regulate and Boundary version held lately in Sophia Antipolis, France, this beneficial source presents vital examinations of form optimization and boundary keep watch over of hyperbolic platforms, together with unfastened boundary difficulties and stabilization.

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Let A : D(A) ~ X be t~O the infinitesimal generator of the Co-semigroup (T(t))t>o. We have to show A = S to complete the proof. 3 Generators of analytic semigroups and their powers 49 d dtT(t)x Hence T(t)x E D(A) for t > 0 with AT(t)x = T(t)Sx. Since T(t)x - t x and t-+O AT(t)x = T(t)Sx - t Sx, the closedness of A shows x E D(A) with Ax = Sx. t-+O Thus, SeA and AId - A extends AId - S for A > O. But now A = S is implied 0 by (0,00) c p(A) n p(S). In the sequel we will show that powers of sectorial operators are sectorial again.

B) A is closable if and only if D(A*) A = A**. (c) N(A*) = R(A)1- and R(A) c H densely. In this case we have = N(A*)1- . (d) If A is symmetric, then A is closable and IIAx - Axil 2:: 11m A E (: and x E D(A). Alllxll for We prove the closedness of A* by proving the closedness of the graph Q(A*) := {(x, A*x) : x E D(A*)} of A* in the Hilbert space H x H. Note that U : H x H :3 (x, y) 1--7 (-y, x) E H x H defines a unitary operator. Take y E D(A*) and x E D(A), then ((y, A'*y) , U(x, Ax)) = -(y, Ax)+(A*y, x) = 0, which proves Q(A*) C (U(Q(A)))1-.

9 Bibliographical remarks. 2 is a standard result in semigroup theory and goes back to Hille. It can be found in many textbooks on semigroup theory. 6 is due to de Laubenfels [38]. 1] and obtained a resolvent estimate as well. 13]. 3 we already introduced natural powers of densely-defined linear operators with non-void resolvent set. In this section we will extend this definition to fractional powers of densely-defined operators, if we assume in addition the resolvent set to include the negative real axis and certain resolvent estimates.