How do I prove that the system is asimptotically stable under a control law?

Question

I am consiering the model of a unicycle in polar coordinates:

$\dot{r}=-vcos\gamma$

$\dot{\gamma}=v\frac{sin\gamma}{r}-\omega$

$\dot{\delta}=v\frac{sin\gamma}{r}$

where $r$ is the distance from the origin, $\gamma$ is a sort of pointing error, so it is the angle between $\theta$, which is the orientation of the unicycle with respect to the world frame and $r$, and $\delta$ is the orientation at the origin.

with the following velocity inputs inputs:

$v=k_1rcos\gamma$

$\omega=k_2\gamma+k_1\frac{sin\gamma cos\gamma}{\gamma}(\gamma+k_3\delta)$

which is obtained by a Lyapunov argument, choosing the Lyapunov function:

$V=\frac{1}{2}(r^2+\gamma^2+\delta^2)$

and computing its derivative and using the control inpute above, I obtain the following:

$\dot{V}=-k_1cos^2\gamma r^2-k_2\gamma^2<0$

so, these control inputs $v$ and $\omega$ drive the robot to the origin:

now, I would like to prove mathematically that this control law drives the unicycle to the origin or the cartesian plane, but I have not clear how to do this.

I have seen that it is possible to use the Barbalat lemma, which I have not studied in my coruse of studies, and also the La Salle's Theorem.

I really don't know how to do this, and it would be really interesting to prove that the control law works.

Can somebody help me?

I will add some papers iin which this problem is used:

• http://www.centropiaggio.unipi.it/sites/default/files/vehicles-RA-MGZ95.pdf
• http://www.diva-portal.org/smash/get/diva2:662268/FULLTEXT01
• https://people.kth.se/~dimos/pdfs/IET15_Zambelli.pdf

In particular, what I am really interested to prove is what is said in the first paper in the list above, in which says that the only possible point of convergence is $(x,y,\theta)=(0,0,0)$

[EDIT] I am not sure of what I am doing, but I have seen that for the barbalt Lemma I need to have the following conditions:

1. $V$ is lower bounded
2. $\dot{V}$ negative definite
3. $\dot{V}$ continuos

so, am I going in the correct direction? And it seems that the first two conditions are satisfied, but how do I know if the third condition is satisfied?

I have seen that La Salle's Theorem is only for autonomous systems, and I think that this is not my case.

[EDIT]As suggested in the comments, I will use LaSalle Invariance principle to prove asymptotic stability. This is what I have done so far:

I have plugged the inputs $v$ and $\omega$ into the system, and I obtain:

$\dot{r}=-k_1rcos^2\gamma$

$\dot{\gamma}=k_1rcos\gamma\frac{sin\gamma}{\gamma}-k_2\gamma+k_1\frac{sin\gamma cos\gamma}{\gamma}(\gamma+k_3\delta)$

$\dot{\delta}=k_1cos\gamma sin\gamma$

$(1)$

now, I define the set of points in which :

$\dot{V}=-k_1cos^2\gamma r^2-k_2\gamma^2=0$

which is:

$E=(r,\gamma : \dot{V}=0)$

which is the one given by $r=0$ and $\gamma=0$

so, if I plug these values into the system $(1)$ I obtain:

$\dot{r}=0$

$\dot{\gamma}=0$

$\dot{\delta}=0$

so $M=(0,0,0)$ ,which is the largest invariant set in $E$, and so $(r,\gamma,\delta)->(0,0,0)$ meaning that $r,\gamma$ and $\delta$ tends asimptotically to zero.

can somebody tell me if this is correct?