1. Find the energy E(t) of the system. 2. Define the appropriate Hilbert energy space H and its inner product. 3. Find the linear unbounded operator A and its domain D(A), and write the system in the form U₁ = AU \ U(0) = Uo € H 4. Show that A is m-dissipative (for maximality: prove that 0 € p(A)). 5. Deduce the existence and uniqueness of system (1). 6. Prove that the energy of this system tends to zero as t goes to infinity (non-compact resolvent). 7. Replace the the Kelvin-voigt dampings by frictional dampings (compact resolvent), then prove that the energy of this system tends to zero as t goes to infinity.
1. Find the energy E(t) of the system. 2. Define the appropriate Hilbert energy space H and its inner product. 3. Find the linear unbounded operator A and its domain D(A), and write the system in the form U₁ = AU \ U(0) = Uo € H 4. Show that A is m-dissipative (for maximality: prove that 0 € p(A)). 5. Deduce the existence and uniqueness of system (1). 6. Prove that the energy of this system tends to zero as t goes to infinity (non-compact resolvent). 7. Replace the the Kelvin-voigt dampings by frictional dampings (compact resolvent), then prove that the energy of this system tends to zero as t goes to infinity.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Utt- (rux + X(a,b) Utx)x - Byt = 0, x ]0, 1[, t > 0,
Ytt - (Yx + X(a,b) Ytx) x + But = 0, x ]0, 1[, t > 0,
u(0, t) = u(1,t) = y(0, t) = y(1, t) = 0, t > 0,
u(x,0) = u(x), y(x,0) = yo(x), x ≤]0, 1[
ut (x, 0) = u₁(x), yt(x,0) = y₁(x), x ≤]0, 1[,
where r, ß are positive constants and 0 < a <b<1
1. Find the energy E(t) of the system.
2. Define the appropriate Hilbert energy space H and its inner product.
3. Find the linear unbounded operator A and its domain D(A), and write the system
in the form
U₁ = AU
\ U(0)=Uo € H
4. Show that A is m-dissipative (for maximality: prove that 0 € p(A)).
(1)
5. Deduce the existence and uniqueness of system (1).
6. Prove that the energy of this system tends to zero as t goes to infinity (non-compact
resolvent).
7. Replace the the Kelvin-voigt dampings by frictional dampings (compact resolvent),
then prove that the energy of this system tends to zero as t goes to infinity.
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9612a6a5-e19c-4346-ae6e-6941d4a9f0e7%2F1e89619f-bf86-4c58-835d-fe4c9fca878e%2Fh33rfh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Utt- (rux + X(a,b) Utx)x - Byt = 0, x ]0, 1[, t > 0,
Ytt - (Yx + X(a,b) Ytx) x + But = 0, x ]0, 1[, t > 0,
u(0, t) = u(1,t) = y(0, t) = y(1, t) = 0, t > 0,
u(x,0) = u(x), y(x,0) = yo(x), x ≤]0, 1[
ut (x, 0) = u₁(x), yt(x,0) = y₁(x), x ≤]0, 1[,
where r, ß are positive constants and 0 < a <b<1
1. Find the energy E(t) of the system.
2. Define the appropriate Hilbert energy space H and its inner product.
3. Find the linear unbounded operator A and its domain D(A), and write the system
in the form
U₁ = AU
\ U(0)=Uo € H
4. Show that A is m-dissipative (for maximality: prove that 0 € p(A)).
(1)
5. Deduce the existence and uniqueness of system (1).
6. Prove that the energy of this system tends to zero as t goes to infinity (non-compact
resolvent).
7. Replace the the Kelvin-voigt dampings by frictional dampings (compact resolvent),
then prove that the energy of this system tends to zero as t goes to infinity.
1
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Step 1: Introduction
VIEWStep 2: Finding the total energy E(t) of the system
VIEWStep 3: Defining the Hilbert space and the inner product
VIEWStep 4: Finding the linear unbounded operator A
VIEWStep 5: Checking the uniqueness of the system
VIEWStep 6: Proving E(t) → 0 as t → ∞
VIEWStep 7: Replacing the Kelvin-Voigt damping by frictional damping
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