By Weijiu Liu
From the reviews:
“This ebook is an advent to the mathematical regulate thought of a few utilized partial differential equations. … the fabric during this e-book is a pleasant simplification of that from the prevailing complex 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 concept for the structures ruled through partial differential equations.” (Xu Zhang, Mathematical studies, factor 2010 m)
“This booklet is an introductory textual content up to speed thought of partial differential equations (PDEs) meant for first-year graduate scholars in arithmetic or engineering … . publication is definitely concept out and the themes stream jointly properly. … This textbook will be very appropriate as a first-rate textual content for a direction on keep an eye on thought of PDEs that emphasizes program of suggestions stabilization concept to concrete PDEs. … additionally function a worthwhile part reference in a extra normal or summary path with its many glorious examples.” (Scott W. Hansen, SIAM evaluation, Vol. fifty three (2), 2011)
Read or Download Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation PDF
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Additional info for Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation
T t0 Since t0 is arbitrary, we can let t0 go to ∞ to obtain lim sup t→∞ Hence lim t→∞ ln T (t) ln T (t) ≤ lim inf . t→∞ t t ln T (t) ln T (t) ln T (t) = lim sup = lim inf < ∞. t→∞ t t t t→∞ For any ω > ω0 , there exists M(ω ) such that T (t) ≤ M(ω )eω t . Consequently, ln T (t) M ≤ ω + lim = ω. t→∞ t→∞ t t Since ω is arbitrary, we have lim ln T (t) ≤ ω0 . t→∞ t lim If ln T (t) < ω0 , t→∞ t lim then there exists ω such that 40 2 Elementary Functional Analysis lim t→∞ ln T (t) < ω < ω0 . t This implies that there exists a t0 such that and then ln T (t) ≤ω t for t ≥ t0 T (t) ≤ eω t for t ≥ t0 .
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! e eλit teλit ..
7) is exponentially stable if and only if all eigenvalues of A satisfy Reλi < 0. 2. A matrix A is said to be Hurwitz if all eigenvalues of A have negative real parts. 1. 11) where A= 0 1 . −1 0 The characteristic equation of A is det(λ I − A) = λ 2 + 1 = 0. Solving the equation, we obtain the eigenvalues λ = ±i. It is clear that the algebraic multiplicity of both i and −i is 1 and the ranks of iI − A and −iI − A are equal to 1. 1 are satisfied and then the equilibrium 0 is stable. In fact, the solution of the system is given by x1 x2 = c1 cost sint + c2 , − sint cost which shows that the equilibrium point 0 is stable.
Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation by Weijiu Liu