Given the planar linear system $ x' = Ax $ with $ A $ as below, find a symmetric matrix $ P $ such that $ V(x) = x^T P x $ is a Lyapunov function whose time derivative along solutions satisfies $ \dot{V} = -x^T Q x $ with $ Q $ the identity matrix.\[A = \begin{pmatrix} -1 & 0 \\ 2 & -1 \end{pmatrix}\]Explanation: - The problem involves determining a symmetric matrix $ P $ for the Lyapunov function $ V(x) $.- The matrix $ A $ is given as $ \begin{pmatrix} -1 & 0 \\ 2 & -1 \end{pmatrix} $.- $ Q $ is specified as the identity matrix.- The objective is to ensure that the derivative $ \dot{V} $ is expressed in the form $ -x^T Q x $.

Answered: Image /qna-images/answer/f49e914e-9610-4058-be67-286993be7cfd.jpg

Given the planar linear system ' = A.x with A as below, find a symmetric matrix P such that V (x) = x" P is a Lyapunov function whose time derivative along solutions satisfies V = –xTQx with Q the identity matri -1 A = 2.

Given the planar linear system ' = A.x with A as below, find a symmetric matrix P such that V (x) = x" P is a Lyapunov function whose time derivative along solutions satisfies V = –xTQx with Q the identity matri -1 A = 2.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: The function of 2 variables f(x,y) has a stationary point with a Hessian Matrix: H = [1 -2]…

Q: Write the given linear system without the use of matrices. Assume X = dx dt dy dt dz dt || = = X' =…

Q: Solve the given (matrix) linear system: (3cos(t) X + 2et 2. X' 14

Q: Consider the linear system Find the eigenvalues and eigenvectors for the coefficient matrix. A₁ = 15…

A: For given system, the coefficient matrix is To find eigenvalues of ,…

Q: The general solution of the linear system x = Ax is et/4 X(t) = | C1 et/3 C2 Determine the constant…

A: We need to find the constant coefficient matrix A.

Q: y = [ 0 1] (2x2 matrix) + [-1 0] d/dx y [1] (1x2 matrix) y (0) [1] [1] %3D [2]

Q: First verify that the given vectors are solutions of the given system. Then use the Wronskian to…

A: Here first we have to verify that x1, x2, x3 are the solutions of the given equation. Then using…

Q: Verify that the vector Xp is a particular solution of the given nonhomogeneous linear system. dx dt…

Q: -I₁+I₂-3x4 = 4 1+ 3x3 + 24 = 0 12 13 -4 = 3 24 3x1 + x3 + 24 = 1 olve this system using Gaussian…

Q: Let k be a positive integer and y be a column vector of functions with length 2k. We have the system…

Q: By using the method of least squares, find the best line through the points: (-1,1), (1,0). (2,2).…

Q: Let f(x) = (x – b) Ax + c, where A is symmetric and positive definite matrix, b is a vector and c is…

A: Given that f(X)=X-bTAX+c, where A is symmetric and positive definite matrix, b is a vector and cis…

Q: Write in its matrix form X´=AX the following system of homogeneous linear equations:

A: Given x'1 = x1 + x2 + x3 x'2 = 2x1 - x2 -x3 x'3 = -x2 We need to write the above system in Matrix…

Q: For a system of two linear differential equations, the only equilibrium point is (0,0). This is…

Q: 4 Find the least-squares line given the matrix products Aª A = 37 A¹b = 590.4 6895.2 y = Ex: 1.2 +…

Q: Compute the matrix exponential eAt for the system x' = Ax given below. x₁ = 10x₁ + 2x₂, X²2 = 2x₁ +…

Q: Express the model Y₁ = Bo + Bixi + B₁r? + &i where the ɛ; have mean zero, variance o² and are…

A: Given Yi=β0+β1xi+β11xi2+εi ,i=1,2,......5

Q: (BH) Recall that a matrix Q is orthogonal if Q' Q = I.

A: Given, A matrix Q is orthogonal if QTQ=I Proof: (a) The one way to write the norm for vector x is…

Q: 1) A =| °, find fundamental matrix eta

Q: 1.1 s Find the chawantristice equiation and the eigonalas Cand a bais for each of the matrix. cores…

A: we have given A matrix as follows A = 0-39-44-18004 Hence now the characteristic equation is given…

Q: (a) Find the equation of the straight line y = ao + a₁t which best fits the following points: t Y 1…

A: Since you have posted multiple questions, we will provide the solution only to the first question as…

Question

Given the planar linear system $ x' = Ax $ with $ A $ as below, find a symmetric matrix $ P $ such that $ V(x) = x^T P x $ is a Lyapunov function whose time derivative along solutions satisfies $ \dot{V} = -x^T Q x $ with $ Q $ the identity matrix.

\[
A = \begin{pmatrix} -1 & 0 \\ 2 & -1 \end{pmatrix}
\]

Explanation:
- The problem involves determining a symmetric matrix $ P $ for the Lyapunov function $ V(x) $.
- The matrix $ A $ is given as $ \begin{pmatrix} -1 & 0 \\ 2 & -1 \end{pmatrix} $.
- $ Q $ is specified as the identity matrix.
- The objective is to ensure that the derivative $ \dot{V} $ is expressed in the form $ -x^T Q x $.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Complexity$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

First verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system
[] Solve the following system of equations. 3x - 4y = 3 7x+6y=7 X J = 1 + Continue 4 ابن W= Integrating Evide....docx ( 3 X WRITING-CENTER_.
Consider the following interpolating points for the function f(x) = x² – 2x: Xi 01 23 f(xi) 0 -1 03 (a) Use monomial basis to find P₁(x) that interpolates f(x) at x; for i = 0,1. Con- struct the Vandermonde matrix and then use the backslash \ in matlab to solve the linear system. (b) Repeat (a) to find P₂(x) that interpolates f(x) at x; for i = 0, 1, 2. (c) Repeat (a) to find P3(x) that interpolates f(x) at x, for i = 0, 1, 2, 3. (d) Compare P₂(x), P3(x) and f(x). What do you see? Can you explain it?
Find the first four iterations obtained by the Inverse Power method applied to the matrix A starting with iteration. x (0) = = (-_¹₁), and the linear functional ((*₂)) - =[17¹² 3] = = x₂. Normalize the vector for each
Use the Finite Difference method to write the equation x" + 2x' - 6x = 2, with the boundary conditions x(0) = 0 and x(9)-0 to a matrix form. Use the CD for the second order differences and the FW for the first order differences with a mesh h=3.
I tried -210e^-7t but it didn't work and idk what i did wrong...
The matrix that rotates by 3π 4 is X' -√2-√2 1 - 1/2 √2
Write the given linear system in matrix form. Assume X = X' = -7 3 dx dt dy dt dz dt = -7x + 8y - 9z = 3x - y = 10x + 8y + 7z 8 -9 0 -(:)) X + 0.00
X' = {(1, 2), (4, 3)} x X + {(8e3t), (12e2t)}. Where X1 = {(-e-t), (e-t)} and X2 = {(e5t), (2e5t)} are solutions of the corresponding homogeneous system. Show that the solutions of the corresponding homogeneous system are linearly independent. Find the fundamental matrix and its inverse. Solve the nonhomogeneous system given above. Show all of your work and multiply out all matrix calculations.
Let 0 = (B, o^2)^T denotes the unknown parameters for multiple linear regression. Find the Cramer-Rao lower bound matrix for the vector of the parameters.
10