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By Weijiu Liu
From the reviews:
“This e-book is an creation to the mathematical keep an eye on thought of a few utilized partial differential equations. … the fabric during this ebook is a pleasant simplification of that from the prevailing complex monographs on infinite-dimensional regulate conception. … this article can be utilized as a textbook for a one-semester graduate path on keep watch over idea for the structures ruled by means of partial differential equations.” (Xu Zhang, Mathematical experiences, factor 2010 m)
“This booklet is an introductory textual content up to speed conception of partial differential equations (PDEs) meant for first-year graduate scholars in arithmetic or engineering … . e-book is definitely idea out and the subjects move jointly properly. … This textbook will be very appropriate as a prime textual content for a path on keep watch over thought of PDEs that emphasizes software of suggestions stabilization thought to concrete PDEs. … additionally function a precious aspect reference in a extra normal or summary path with its many fantastic examples.” (Scott W. Hansen, SIAM assessment, Vol. fifty three (2), 2011)
Read Online or Download Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation PDF
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Extra info for Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation
Sample text
Also c2 ddxu2 = λ 2 u − f2 − λ f1 belongs to L2 (0, L). 49) have a unique solution. This proves that R(λ I − A) = H . It remains to show that A is dissipative. We note that the inner product in H01 (0, L) is defined by L d f d g¯ dx ( f , g)H 1 = c2 0 dx dx 0 and the inner product in L2 (0, L) is defined by ( f , g) = L 0 f (x)g(x)dx. 6 Semigroups 37 For any ( f1 , f2 ) ∈ D(A) = (H 2 (0, L) ∩ H01 (0, L)) × H01 (0, L), integration by parts gives (the inner product here is the one in H01 (0, L) × L2 (0, L)) Re A f f1 , 1 f2 f2 f2 d 2 f1 2 c dx2 = Re , H01 ×L2 f1 f2 H01 ×L2 d f2 d f¯1 d 2 f1 + c2 2 f¯2 dx dx dx dx 0 L ¯ d f2 d f1 d f1 ¯ L dx + Re c2 − Re c2 = Re f2 dx dx dx 0 0 = 0.
If v = 0, the eigenvalue problem is complicated and not all properties in the above theorem are true. 20) are not orthogonal in L2 ((0, L)× (0, H)). 21) then they are orthogonal. Unfortunately, for a general v, we do not know whether there exists a weighted inner product to make the eigenfunctions orthogonal. To state a result in this case, we introduce the concept of direct sum. Let X1 and X2 be two subspaces of a vector space X . The sum of these two subspaces, denoted by X1 + X2 , is the set of all the sums x1 + x2 , where x1 ∈ X1 and x2 ∈ X2 .
Also c2 ddxu2 = λ 2 u − f2 − λ f1 belongs to L2 (0, L). 49) have a unique solution. This proves that R(λ I − A) = H . It remains to show that A is dissipative. We note that the inner product in H01 (0, L) is defined by L d f d g¯ dx ( f , g)H 1 = c2 0 dx dx 0 and the inner product in L2 (0, L) is defined by ( f , g) = L 0 f (x)g(x)dx. 6 Semigroups 37 For any ( f1 , f2 ) ∈ D(A) = (H 2 (0, L) ∩ H01 (0, L)) × H01 (0, L), integration by parts gives (the inner product here is the one in H01 (0, L) × L2 (0, L)) Re A f f1 , 1 f2 f2 f2 d 2 f1 2 c dx2 = Re , H01 ×L2 f1 f2 H01 ×L2 d f2 d f¯1 d 2 f1 + c2 2 f¯2 dx dx dx dx 0 L ¯ d f2 d f1 d f1 ¯ L dx + Re c2 − Re c2 = Re f2 dx dx dx 0 0 = 0.