Murray at 23:42, 7 February 2015

2015-02-07T23:42:57Z

← Older revision		Revision as of 23:42, 7 February 2015
Line 42:		Line 42:

	<!-- 2014 TA comments: Some of the students mentioned they couldn't use Gronwell's inequality because of a negative constant instead of positive. They were told they could do the problem without Gronwell's inequality, but if they could use Gronwell's inequality that would be good too. -->		<!-- 2014 TA comments: Some of the students mentioned they couldn't use Gronwell's inequality because of a negative constant instead of positive. They were told they could do the problem without Gronwell's inequality, but if they could use Gronwell's inequality that would be good too. -->
			<!-- 2015 comments: If this problem is used again, ask students to compute M and alpha -->
	<li>		<li>
	Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.		Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\\|x(t)\\|\leq Me^{-\alpha t}\\|x(0)\\|$, $\forall\\|x(0)\\|\leq\epsilon,\,t\geq 0$.

Murray at 23:27, 7 February 2015

2015-02-07T23:27:35Z

← Older revision		Revision as of 23:27, 7 February 2015
Line 29:		Line 29:

	<!-- 2014 TA comments: there was some unclearness as to whether this Lyapunov function could show whether or not this system was asymptotically stable. For sure it could be shown to be stable. Picking a=... and b=... (or the opposite a=..., b=...), one could obtain Vdot = ... The problem was that if y=..., and x=..., Vdot=... still, even though the system was not necessarily at an equilibrium point. We did not have a good course solution for this problem from last year (we had half of an unlabeled solution that looking back does not correct). -->		<!-- 2014 TA comments: there was some unclearness as to whether this Lyapunov function could show whether or not this system was asymptotically stable. For sure it could be shown to be stable. Picking a=... and b=... (or the opposite a=..., b=...), one could obtain Vdot = ... The problem was that if y=..., and x=..., Vdot=... still, even though the system was not necessarily at an equilibrium point. We did not have a good course solution for this problem from last year (we had half of an unlabeled solution that looking back does not correct). -->
			<!-- 2015 note: this problem is most easily solved using Lasalle's theorem, which we usually haven't covered at this point. Either ask student to find a full Lyapunov function or pick a different problem -->
	<li>		<li>
	Determine the stability of the system		Determine the stability of the system

Murray at 06:48, 27 January 2015

2015-01-27T06:48:22Z

← Older revision		Revision as of 06:48, 27 January 2015
Line 8:		Line 8:
	\| pdf = cds140-wi15_hw4.pdf		\| pdf = cds140-wi15_hw4.pdf
	}} __MATHJAX__		}} __MATHJAX__
	{{warning\|This problem set is under construction. This banner will be removed when the problems are finalized.}}		<!-- {{warning\|This problem set is under construction. This banner will be removed when the problems are finalized.}} -->

	'''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		'''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

Murray: Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #4 | issued = 26 Jan 2015 | due = 4 Feb 2015 at 12:30 pm
In ..."

2015-01-25T02:29:07Z

Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #4 | issued = 26 Jan 2015 | due = 4 Feb 2015 at 12:30 pm<br>In ..."

New page

{{CDS homework
| instructor = R. Murray
| course = CDS 140
| semester = Winter 2015
| title = Problem Set #4
| issued = 26 Jan 2015
| due = 4 Feb 2015 at 12:30 pm<br>In class or to box across 107 STL
| pdf = cds140-wi15_hw4.pdf
}} __MATHJAX__
{{warning|This problem set is under construction. This banner will be removed when the problems are finalized.}}

'''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).



<ol>

<li>'''Perko, Section 2.9, problem 3'''
Use the Lyapunov function $V(x)=x_1^2+x_2^2+x_3^2$ to show that the origin is an asymptotically stable equilibrium point of the system
<center><amsmath>
\dot{x}=\begin{bmatrix}-x_2-x_1x_2^2+x_3^2-x_1^3\\
x_1+x_3^3-x_2^3\\
-x_1x_3-x_3x_1^2-x_2x_3^2-x_3^5\end{bmatrix}
</amsmath></center>
Show that the trajectories of the linearized system $\dot{x}=Df(0)x$ for this problem lie on circles in planes parallel to the $x_1,\,x_2$ plane; hence, the origin is stable, but not asymptotically stable for the linearized system.
</li>


<li>
Determine the stability of the system
<center><amsmath>
\aligned
\dot{x}&=-y-x^3\\
\dot{y}&=x^5
\endaligned
</amsmath></center>
Hint: motivated by the first equation, try a Lyapunov function of the form $V(x,y)=\alpha x^6+\beta y^2$. Is the origin asymptotically stable? Is the origin globally asymptotically stable?
</li>


<li>
Definition: An equilibrium point is ''exponentially stable'' if $\exists\,M,\,\alpha>0$ and $\epsilon>0$ such that $\|x(t)\|\leq Me^{-\alpha t}\|x(0)\|$, $\forall\|x(0)\|\leq\epsilon,\,t\geq 0$.

Let $\dot x = f(x)$ be a dynamical system with an equilibrium point at $x_e = 0$. Show that if there is a function $V(x)$ satisfying
<center><amsmath>k_1\|x\|^2\leq V(x)\leq k_2\|x\|^2,\quad \dot V(x)\leq -k_3\|x\|^2
</amsmath></center>
for positive constants $k_1$, $k_2$ and $k_3$, then the equilibrium point at the origin is exponentially stable.

</li>


<li>'''Perko, Section 2.12, problem 2'''
Use Theorem 1 [Center Manifold Theorem] to determine the qualitative behaviour near the non-hyperbolic critical point at the origin for the system
<center><amsmath>
\aligned
\dot{x}&=y\\
\dot{y}&=-y+\alpha x^2+xy
\endaligned
</amsmath></center>
for $\alpha\neq 0$ and for $\alpha=0$; i.e., follow the procedure in Example 1 after diagonalizing the system as in Example 3.
</li>

<li> Consider the following system in $\mathbb R^2$:
<center><amsmath>\aligned
\dot{x}&=-\frac{\alpha}{2}(x^2+y^2)+\alpha (x+y)-\alpha\\
\dot y&=-\alpha xy+\alpha (x+y)-\alpha
\endaligned</amsmath></center>
Determine the stable, unstable, and centre manifold of the equilibrium point at $(x,y)=(1,1)$, and determine the stability of this equilibrium point for $\alpha\neq 0$. For determining stability, note that near the equilibrium point there are two 1-dimensional invariant linear manifolds of the form $M=\{(a_1,a_2)\in\mathbb R^2\|a_2=ka_1\}$; determine the flow on these invariant manifolds.
</li>


</ol>

CDS 140a Winter 2015 Homework 4 - Revision history

Murray at 23:42, 7 February 2015

Murray at 23:27, 7 February 2015

Murray at 06:48, 27 January 2015

Murray: Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #4 | issued = 26 Jan 2015 | due = 4 Feb 2015 at 12:30 pmIn ..."

Murray: Created page with "{{CDS homework | instructor = R. Murray | course = CDS 140 | semester = Winter 2015 | title = Problem Set #4 | issued = 26 Jan 2015 | due = 4 Feb 2015 at 12:30 pm
In ..."