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Download Differential Equations And Control Theory by Sergiu Aizicovici; N H Pavel PDF

By Sergiu Aizicovici; N H Pavel

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Let us check that B is Gateaux differentiate. Indeed, it is seen that - f ' ( y + u)(z ^ \ \ f ( y + u + t(z + w))-f(y \(L 2 fl Lf(y +U + t T ( Z w) + Jo dr (z f'(y tr(z + w)} - f(y + u))dr \ 2 dx dx for all ( y , u ) , ( z , w } € ^ 2 (0) x L2(tt) and t e ffi \ {0}. The Lebesgue Dominated Convergence Theorem and assumption (i) ensure that the right hand side of the above equality converges to 0 as t —> 0. Moreover, the map (z,w) <= L2(ft] x L 2 (fi) h-» f ' ( y + u)(z + w) e L2(tt) is linear and continuous.

The following simple known lemma will be useful in the next section. 1. Let X be a real Hilbert space of inner product {•, •} and let (/> : X -^ R be a Frechet differentiable functional. 12) Flow-Invariant Sets for Navier-Stokes Equation 43 Then the tangent cone T/<-(/) to K at f £ K is given by: ( X, TM( f;) = < ~ I v£X;'f,v(f) < 0 = M, if / = 0 . (213); ' ' V Proof: The inclusion T/<-(/) C My is immediate. Viceversa. the interior M9 of My is: My0 = {v 6 X; {<£'(/), V ) < 0} C T K ( f ) .

By (Hi), Xi C T(y]U)M, so (9) follows from Lemma 1 with G - the null functional on X\. ,Y = 0,Vze D(A) with Az = By(y,u)z, which can be expressed as teN(A-By(y,u))^. , Brezis [3], p. 29). Then (10) and (11) imply teR((A-By(y,u))*). , w}x-,x = 0, Vw e D(B] with Cw + Bu(y, u)w = 0. Reasoning as above we see that r}£R((C + B u ( y , u ) ) * ) . (13) By (12) and (13) there exist p G D(A*} C E* and p e D(C)* C £* such that (A-By(y,u))*p = t, (C + 5 u (y, U ))*p = 7/, (14) so (9) can be rewritten as (p, (A - By(y, U))Z)E;E + (P, (C + Bu(y, U))W}E*,E = 0 (15) for all (z,w) e D(A) x D(C) satisfying (3).

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