CDS 140a Winter 2011 Homework 2 - Revision history

Murray at 00:03, 20 January 2011

2011-01-20T00:03:42Z

← Older revision		Revision as of 00:03, 20 January 2011
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	<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
	* 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 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.
	* 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>;		* 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>;
	** <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.)		** <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.)
	** <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.
	* 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>;		* 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>

Murray at 23:47, 19 January 2011

2011-01-19T23:47:18Z

← Older revision		Revision as of 23:47, 19 January 2011
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	* 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>;		* 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>;
	** <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.)		** <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.)
	** <font color=blue>Note:</font> ~~The notation~~ <amsmath>S \sim \{0\}</amsmath> ~~means~~ the set <amsmath>S</amsmath> minus the ~~origin~~.		** <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.
	* 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>;		* 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>

Murray at 23:44, 19 January 2011

2011-01-19T23:44:27Z

← Older revision		Revision as of 23:44, 19 January 2011
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	* 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 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>;		* 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>;
	** <font color=blue>Note: this problem should be modified to state that the projection of ''A'' onto the center subspace is not semisimple</font>		** <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.)
			** <font color=blue>Note:</font> The notation <amsmath>S \sim \{0\}</amsmath> means the set <amsmath>S</amsmath> minus the origin.
	* 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>;		* 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>

Murray at 02:53, 17 January 2011

2011-01-17T02:53:25Z

@@ Line 11: / Line 11: @@
 this homework set (including reading).
-# [Perko, Section 1.7, problem 3, parts (a) and (d)]
+<ol>
-# Perko, Section 1.8, problem 10
+<li> '''Perko, Section 1.7, problem 3, parts (a) and (d)''': Solve the initial value problem (1) with the matrix
-# Perko, Section 1.9, problem 3
+<center><amsmath>
-# '''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
+\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
-#* 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>;
+\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
-#* 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>;
+</amsmath></center>
-#** <font color=blue>Note: this problem should be modified to state that the projection of ''A'' onto the center subspace is not semisimple</font>
+</li>
 <hr>
 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.

Murray at 02:28, 17 January 2011

2011-01-17T02:28:07Z

← Older revision		Revision as of 02:28, 17 January 2011
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	this homework set (including reading).		this homework set (including reading).

	# Perko, Section 1.7, problem 3, parts (a) and (d)		# [Perko, Section 1.7, problem 3, parts (a) and (d)]
	# Perko, Section 1.8, problem 10		# Perko, Section 1.8, problem 10
	# Perko, Section 1.9, problem 3		# Perko, Section 1.9, problem 3
	# Perko, Section 1.9, problem 5, parts (c), (d1) and (d2)		# '''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
			#* 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>;
			#** <font color=blue>Note: this problem should be modified to state that the projection of ''A'' onto the center subspace is not semisimple</font>
			#* 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>;
	# Perko, Section 1.10, problem 2		# Perko, Section 1.10, problem 2
	# Perko, Section 1.10, problem 3		# Perko, Section 1.10, problem 3

	<hr>		<hr>
	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.		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.

Murray at 06:53, 12 January 2011

2011-01-12T06:53:07Z

← Older revision		Revision as of 06:53, 12 January 2011
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	~~{{HW draft}}~~

	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, G. Buzi		\| instructor = R. Murray, G. Buzi

Murray at 15:38, 11 January 2011

2011-01-11T15:38:55Z

← Older revision		Revision as of 15:38, 11 January 2011
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	<hr>		<hr>
	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 ~~text book~~. It is fine to check everything with MATLAB.		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.

Murray: Created page with '{{HW draft}} {{CDS homework | instructor = R. Murray, G. Buzi | course = ACM 101/AM 125a/CDS 140a | semester = Winter 2011 | title = Problem Set #2 | issued = 11 Jan 2011 (…'

2011-01-11T05:38:01Z

Created page with '{{HW draft}} {{CDS homework | instructor = R. Murray, G. Buzi | course = ACM 101/AM 125a/CDS 140a | semester = Winter 2011 | title = Problem Set #2 | issued = 11 Jan 2011 (…'

New page

{{HW draft}}

{{CDS homework
| instructor = R. Murray, G. Buzi
| course = ACM 101/AM 125a/CDS 140a
| semester = Winter 2011
| title = Problem Set #2
| issued = 11 Jan 2011 (Tue)
| due = 20 Jan 2011 (Thu)
}}

'''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).

# Perko, Section 1.7, problem 3, parts (a) and (d)
# Perko, Section 1.8, problem 10
# Perko, Section 1.9, problem 3
# Perko, Section 1.9, problem 5, parts (c), (d1) and (d2)
# Perko, Section 1.10, problem 2
# Perko, Section 1.10, problem 3

<hr>
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 text book. It is fine to check everything with MATLAB.