Difference between revisions of "CDS 140a Winter 2013 Homework 2"

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D = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.
D = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.
</amsmath></center>
</amsmath></center>
<ol type="a">
(a) Show that if all of the eigenvalues of <amsmath>A</amsmath> have nonpositive real parts, then for each <amsmath>x_0 \in {\mathbb R}^n</amsmath> there is a positive constant <amsmath>M</amsmath> such that <amsmath>|x(t)| \leq M</amsmath> for all <amsmath>t \geq 0></amsmath> where <amsmath>x(t)</amsmath> is the solution of the initial value problem.
<li> Show that if all of the eigenvalues of <amsmath>A</amsmath> have nonpositive real parts, then for each <amsmath>x_0 \in {\mathbb R}^n</amsmath> there is a positive constant <amsmath>M</amsmath> such that <amsmath>|x(t)| \leq M</amsmath> for all <amsmath>t \geq 0></amsmath> where <amsmath>x(t)</amsmath> is the solution of the initial value problem. </li>
 
<li> Show via a simple counterexample that this is not true if the Jordan blocks have non-zero off diagonal entries (with the same constraint on the eigenvalues).</li>
(b) Show via a simple counterexample that this is not true if the Jordan blocks have non-zero off diagonal entries (with the same constraint on the eigenvalues).
</ol>
</li>
</li>


<li> '''Perko, Section 1.9, problem 3''': Solve the system
<li> '''Perko, Section 1.9, problem 3''' (modified): Consider the linear system
<center><amsmath>
<center><amsmath>
\dot x = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{bmatrix} x.
\dot x = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{bmatrix} x.
</amsmath></center>
</amsmath></center>
Find the stable, unstable and center subspaces for this system and sketch the phase portrait.  Show that the flow of initial points in each of the subspaces remains in that subspace.
(a) Compute the solutions to the differential equation.  You should provide the matrices used to transform the system to Jordan form along with the appropriate matrix exponential of the relevant Jordan form matrix (you don't need to multiply everything out to get the solution in the original basis).
:* Note: you should show the various (regular and generalized) eigenvectors associated with each eigenvalue.  OK to check your answer with MATLAB, but be sure to show that you know how to solve it by hand.
 
(b) Find the stable, unstable and center subspaces for this system.
</li>
</li>


<li> '''Perko, Section 1.9, problem 5, parts (c), (d1) and (d2)''': Let <amsmath>A</amsmath> be an <amsmath>n \times n</amsmath> nonsingular matrix and let <amsmath>x(t)</amsmath> be the solution of the initial value problem (1) with <amsmath>x(0) = x_0</amsmath>.  Show that
<li> '''Perko, Section 1.9, problem 5, parts (c), (d1) and (d2)''': Let <amsmath>A</amsmath> be an <amsmath>n \times n</amsmath> nonsingular matrix and let <amsmath>x(t)</amsmath> be the solution of the initial value problem (1) with <amsmath>x(0) = x_0</amsmath>.  Show that<br>
* If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq |x(t)| \leq M</amsmath>;
 
** <font color=blue>Note:</font> Perko defines <amsmath>\sim</amsmath> to mean "set subtraction".  So <amsmath>E \sim \{0\}</amsmath>is the set <amsmath>E</amsmath> minus the point 0.
(c) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq |x(t)| \leq M</amsmath>;
* If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and <amsmath>A</amsmath> is not semisimple, then there is an <amsmath>x_0 \in {\mathbb R}^n</amsmath> such that <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty</amsmath>;
:* Note: Perko defines <amsmath>\sim</amsmath> to mean "set subtraction".  So <amsmath>E \sim \{0\}</amsmath>is the set <amsmath>E</amsmath> minus the point 0.
** <font color=blue>Note: this problem should be modified to state that the projection of ''A'' onto the center subspace is not semisimple</font>.  (Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.)
 
* If <amsmath>E^s \neq \{0\}</amsmath>, <amsmath>E^u \neq \{0\}</amsmath>, and <amsmath>x_0 \in E^s \oplus E^u \sim (E^s \cup E^u)</amsmath>, then <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty</amsmath>;
(d1) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and the projection of <amsmath>A</amsmath> onto the center subspace is not semisimple, then there is an <amsmath>x_0 \in {\mathbb R}^n</amsmath> such that <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty;</amsmath>
:* Note: there is a bug in the statement in Perko; use the modified version here
:* Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.
 
(d2) If <amsmath>E^s \neq \{0\}</amsmath>, <amsmath>E^u \neq \{0\}</amsmath>, and <amsmath>x_0 \in E^s \oplus E^u \sim (E^s \cup E^u)</amsmath>, then <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty</amsmath>;
</li>
</li>


<li> '''Perko, Section 1.10, problem 2''': Use Theorem 1 to solve the nonhomogeneous linear system
<li> '''Perko, Section 1.10, problem 2''': Use Theorem 1 in Section 1.10 to solve the nonhomogeneous linear system
<center><amsmath>
<center><amsmath>
\dot x = \begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix} x + \begin{bmatrix} t \\ 1 \end{bmatrix}
\dot x = \begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix} x + \begin{bmatrix} t \\ 1 \end{bmatrix}
Line 69: Line 75:
x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.
x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.
</amsmath></center>
</amsmath></center>
</li>
<li> '''Perko, Section 1.10, problem 3''': Show that
<center><amsmath>
\Phi(t) = \begin{bmatrix} e^{-2t} \cos t & -\sin t \\ e^{-2t} \sin t & \cos t \end{bmatrix}
</amsmath></center>
is a fundamental matrix of the nonautonomous linear system
<center><amsmath>
\dot x = A(t) x
</amsmath></center>
with
<center><amsmath>
A(t) = \begin{bmatrix} -2 \cos^2 t & -1 - \sin 2t \\ 1 - \sin 2t & -2 \sin^2 t \end{bmatrix}
</amsmath></center>
Find the inverse of <amsmath>\Phi(t)</amsmath> and use Theorem 1 and Remark 1 to solve the nonhomogeneous linear system
<center><amsmath>
\dot x = A(t) x + b(t)
</amsmath></center>
with <amsmath>A(t)</amsmath> given above and <amsmath>b(t) = (1, e^{-2t})^T</amsmath>.  Note that, in general, if <amsmath>A(t)</amsmath> is a periodic matrix of period <amsmath>T</amsmath>, then corresponding to any fundamental matrix <amsmath>\Phi(t)</amsmath> there exists a periodic matrix <amsmath>P(t)</amsmath> of period <amsmath>T</amsmath> and a constant matrix <amsmath>B</amsmath> such that
<center><amsmath>
\Phi(t) = P(t) e^{B t}
</amsmath></center>
Show that <amsmath>P(t)</amsmath> is a rotation matrix and <amsmath>B = \text{diag}[-2, 0]</amsmath> in this problem.
* <font color=blue>Note:</font> A rotation matrix is defined as a matrix <amsmath>R</amsmath> that satisfies <amsmath>R R^T = I</amsmath> and <amsmath>\det R = 1</amsmath>.
</li>
</li>
</ol>
</ol>

Latest revision as of 18:47, 15 January 2013

R. Murray, D. MacMartin Issued: 12 Jan 2011 (Tue)
ACM 101/AM 125b/CDS 140a, Winter 2013 Due: 19 Jan 2011 (Tue)

Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading).

  1. Perko, Section 1.7, problem 3, parts (a) and (d): Solve the initial value problem (1) with the matrix
    <amsmath>

    \text{(a)}\quad A = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & -1 & 1 & 0 \end{bmatrix}, \qquad \text{(d)}\quad A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 1 & 1 \end{bmatrix}, \qquad

    </amsmath>
  2. Perko, Section 1.8, problem 10 Suppose that the elementary blocks <amsmath>B</amsmath> in the Jordan form of the matrix <amsmath>A</amsmath>, have no ones or <amsmath>I_2</amsmath> blocks off the diagonal, so that they are of the form
    <amsmath>

    B = \begin{bmatrix}

     \lambda & 0 & 0 & \dots & 0 \\
     0 & \lambda & 0 & \dots & 0 \\
     \dots & & & & & \\
     0 & \dots & & \lambda & 0 \\
     0 & \dots & & 0 & \lambda 
    

    \end{bmatrix} \qquad\text{or}\qquad B = \begin{bmatrix}

     D & 0 & 0 & \dots & 0 \\
     0 & D & 0 & \dots & 0 \\
     \dots & & & & & \\
     0 & \dots & & D & 0 \\
     0 & \dots & & 0 & D
    

    \end{bmatrix}, \quad D = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.

    </amsmath>

    (a) Show that if all of the eigenvalues of <amsmath>A</amsmath> have nonpositive real parts, then for each <amsmath>x_0 \in {\mathbb R}^n</amsmath> there is a positive constant <amsmath>M</amsmath> such that <amsmath>|x(t)| \leq M</amsmath> for all <amsmath>t \geq 0></amsmath> where <amsmath>x(t)</amsmath> is the solution of the initial value problem.

    (b) Show via a simple counterexample that this is not true if the Jordan blocks have non-zero off diagonal entries (with the same constraint on the eigenvalues).

  3. Perko, Section 1.9, problem 3 (modified): Consider the linear system
    <amsmath>

    \dot x = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 2 & 0 & 6 \end{bmatrix} x.

    </amsmath>

    (a) Compute the solutions to the differential equation. You should provide the matrices used to transform the system to Jordan form along with the appropriate matrix exponential of the relevant Jordan form matrix (you don't need to multiply everything out to get the solution in the original basis).

    • Note: you should show the various (regular and generalized) eigenvectors associated with each eigenvalue. OK to check your answer with MATLAB, but be sure to show that you know how to solve it by hand.

    (b) Find the stable, unstable and center subspaces for this system.

  4. Perko, Section 1.9, problem 5, parts (c), (d1) and (d2): Let <amsmath>A</amsmath> be an <amsmath>n \times n</amsmath> nonsingular matrix and let <amsmath>x(t)</amsmath> be the solution of the initial value problem (1) with <amsmath>x(0) = x_0</amsmath>. Show that
    (c) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and <amsmath>A</amsmath> is semisimple, then there are postive constants <amsmath>m</amsmath> and <amsmath>M</amsmath> such that for all <amsmath>t \in R</amsmath>, <amsmath>m \leq |x(t)| \leq M</amsmath>;
    • Note: Perko defines <amsmath>\sim</amsmath> to mean "set subtraction". So <amsmath>E \sim \{0\}</amsmath>is the set <amsmath>E</amsmath> minus the point 0.
    (d1) If <amsmath>x_0 \in E^c \sim \{0\}</amsmath> and the projection of <amsmath>A</amsmath> onto the center subspace is not semisimple, then there is an <amsmath>x_0 \in {\mathbb R}^n</amsmath> such that <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty;</amsmath>
    • Note: there is a bug in the statement in Perko; use the modified version here
    • Hint: put the system into Jordan form; the center subspace is now the set of coordinate directions corresponding to eigenvalues with zero real part.
    (d2) If <amsmath>E^s \neq \{0\}</amsmath>, <amsmath>E^u \neq \{0\}</amsmath>, and <amsmath>x_0 \in E^s \oplus E^u \sim (E^s \cup E^u)</amsmath>, then <amsmath>\lim_{t \to \pm \infty} |x(t)| = \infty</amsmath>;
  5. Perko, Section 1.10, problem 2: Use Theorem 1 in Section 1.10 to solve the nonhomogeneous linear system
    <amsmath>

    \dot x = \begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix} x + \begin{bmatrix} t \\ 1 \end{bmatrix}

    </amsmath>

    with the initial condition

    <amsmath>

    x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.

    </amsmath>

Notes:

  • The problems are transcribed above in case you don't have access to Perko. However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.
  • There are a number of problems that can be solved using MATLAB. If you just give the answer with no explanation (or say "via MATLAB"), the TAs will take off points. Instead, you should show how the solutions can be worked out by hand, along the lines of what is done in the textbook. It is fine to check everything with MATLAB.