CDS 140a Winter 2011 Homework 2

R. Murray, G. Buzi	Issued: 11 Jan 2011 (Tue)
ACM 101/AM 125a/CDS 140a, Winter 2011	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).

1. [Perko, Section 1.7, problem 3, parts (a) and (d)]
2. Perko, Section 1.8, problem 10
3. Perko, Section 1.9, problem 3
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
   • 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: this problem should be modified to state that the projection of A onto the center subspace is not semisimple
   • 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
6. Perko, Section 1.10, problem 3

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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.