Elementary Feedback Stabilization of the Linear by Weijiu Liu

By Weijiu Liu

From the reviews:

“This ebook is an creation to the mathematical keep watch over conception of a few utilized partial differential equations. … the fabric during this booklet is a pleasant simplification of that from the prevailing complicated monographs on infinite-dimensional keep watch over concept. … this article can be utilized as a textbook for a one-semester graduate direction on keep an eye on thought for the structures ruled by means of partial differential equations.” (Xu Zhang, Mathematical studies, factor 2010 m)

“This e-book is an introductory textual content on top of things thought of partial differential equations (PDEs) meant for first-year graduate scholars in arithmetic or engineering … . booklet is definitely suggestion out and the subjects circulation jointly properly. … This textbook will be very appropriate as a prime textual content for a path on keep an eye on conception of PDEs that emphasizes software of suggestions stabilization idea to concrete PDEs. … additionally function a useful aspect reference in a extra basic or summary path with its many fantastic examples.” (Scott W. Hansen, SIAM evaluation, Vol. fifty three (2), 2011)

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Additional resources for Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation

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Then the growth bound ω0 of the semigroup satisfies ω0 ≥ sup{Re(λ ), λ ∈ σ (A)}. 58) Proof. We argue by contradiction. If ω0 < sup{Re(λ ), λ ∈ σ (A)}, then there exists a constant ω1 such that ω0 < ω1 < sup{Re(λ ), λ ∈ σ (A)}. By the definition of the growth bound, we deduce that there exists a constant M(ω1 ) such that T (t) ≤ M(ω1 )eω1t . Moreover, there exists a λ1 ∈ σ (A) such that Re(λ1 ) > ω1 . 10 that λ1 ∈ ρ (A). This is a contradiction. 58) may not hold. 6]. Fortunately, the equality does hold for a large class of semigroups such as analytic semigroups.

0 0 0 · · · λi 0 0 0 ··· 0 ⎡ λi 0 0 · · · 0 ⎢ 0 λi 0 · · · 0 ⎢ ⎢ = ⎢ ... ... · · · ... ⎢ ⎣ 0 0 0 · · · λi 0 0 0 ··· 0 = Λi + N. ⎤ 0 0⎥ ⎥ .. ⎥ . ⎥ ⎥ 1⎦ λi ⎤ ⎡ 01 0 ⎢0 0 0⎥ ⎥ ⎢ .. ⎥ + ⎢ .. ⎢ . ⎥ ⎥ ⎢. ⎣0 0 ⎦ 0 00 λi ⎤ 00 0 0⎥ ⎥ .. ⎥ . ⎥ ⎥ 0 ··· 0 1 ⎦ 0 ··· 0 0 0 ··· 1 ··· .. 9) 54 3 Finite Dimensional Systems Since exp(Λit) = exp(λit)I and Nk = 0 for k ≥ mi , we deduce that exp(Jit) = exp(Λi t) exp(Nt) = exp(λit) ⎡ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ mi −1 k k t N ∑ k! k=0 eλit teλit 0 .. 0 0 λi t 2!

Suppose that the linear, closed operator A has eigenvalues {λn }∞ n=1 and the corresponding eigenvectors {φn }∞ n=1 form an orthogonal basis on a Hilbert space H. Then (1) A has the representation Ax = ∞ ∑ λn(x, φn )φn , x ∈ D(A) = ∞ x∈H| n=1 ∑ |λn|2 |(x, φn )|2 < ∞ . 61) n=1 (2) If sup Re(λn ) < ∞, then A is the infinitesimal generator of a C0 semigroup T (t) n≥1 given by ∞ ∑ (x, φn )eλnt φn . 62) n=1 (3) The growth bound of the semigroup is given by ω0 = sup Re(λn ). n≥1 Proof. For x ∈ D(A), let ∞ ∑ (x, φn )φn .

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