CDS 140a Winter 2015 Homework 4: Difference between revisions
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<!-- 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 --> | |||
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Determine the stability of the system | Determine the stability of the system | ||
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<!-- 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 --> | |||
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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 $\|x(t)\|\leq Me^{-\alpha t}\|x(0)\|$, $\forall\|x(0)\|\leq\epsilon,\,t\geq 0$. | ||
Latest revision as of 23:42, 7 February 2015
| R. Murray | Issued: 26 Jan 2015 |
| CDS 140, Winter 2015 | Due: 4 Feb 2015 at 12:30 pm In class or to box across 107 STL |
__MATHJAX__
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 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
<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>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.
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Determine the stability of the system
<amsmath> \aligned \dot{x}&=-y-x^3\\ \dot{y}&=x^5 \endaligned
</amsmath>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?
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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
<amsmath>k_1\|x\|^2\leq V(x)\leq k_2\|x\|^2,\quad \dot V(x)\leq -k_3\|x\|^2 </amsmath> for positive constants $k_1$, $k_2$ and $k_3$, then the equilibrium point at the origin is exponentially stable.
- 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
<amsmath> \aligned \dot{x}&=y\\ \dot{y}&=-y+\alpha x^2+xy \endaligned
</amsmath>for $\alpha\neq 0$ and for $\alpha=0$; i.e., follow the procedure in Example 1 after diagonalizing the system as in Example 3.
- Consider the following system in $\mathbb R^2$:
<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>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.
