Internet Control Course
In the previous section a domain D was supposed to be a controllable foliation generated by an affine nonlinear system 
This assumption implies that on the domain 
is equivalent to the cart. In general, a domain
D is not a controllable foliation. For instance, in the following example the domain D is the whole plane 
and is covered by two controllable foliations.
Example 2.2 The system in question is of the form
The equilibrium set of the system is defined by
The vector field b(x) is equal to 
and generates the integral curve collection
The whole plane can be covered by two controllable foliations shown in Fig.1.
Figure 1: 
where 
are controllable foliations generated by the system.
The system is controllable both on 
and on 
Hence it is controllable on .
Thus, in order to investigate the controllability of an affine nonlinear system on a domain D such that
and
where 
is the collection of the connected component of 
we need to construct a cart garland defined as follows.
On each of the sets 
the system 
is equivalent to the cart. That is why, we call 
cart garland. Making use of the results of Chapter I,
one can relatively easy control the system on every from 
If cart garland covers D , i.e. 
then is controllable on D.
Proof. Let 
be any two points in the domain D. Since D is connected, there is a regular curve 
, such that 
is a compact subset in D and 
Hence there exists a natural number N such that
where 
is a controllable foliation generated by The covering 
of 
induces the covering of the interval 
where 
for 
and 
Thus 
and 
The system and the cart are equivalent on 
Therefore making use of Chapter I, one can construct controls 
such that 
steers the system from 
to 
Applying these controls one after another one can move the system from 
to 
Hence it is proved that the system is controllable both by C and by 
on D .
Q.E.D.
Theorem 2.2 turns out to be a criterion of controllability on a simply connected domain D.
Corollary 1. Let 
be a simply connected domain, and let be such that 
Then is controllable on D iff a cart garland representing on D covers D .
Proof. Sufficiency follows from Theorem 2.2.
Necessity. Let 
and let 
be the connected component of the intersection 
such that 
Then we have
By Jordan curve theorem [31], the curve splits the simply connected domain D into two disjointed parts. That means D has prisons and mysteries. Thus the system is not controllable on D .
Q.E.D.
Applying this corollary to the controllability analysis of an affine nonlinear system on 
we obtain the following proposition.
Corollary 2. Let be an affine nonlinear system such that
Then the system is controllable on if, and only if, for every 
there is 
such that 
To prove Corollary 2 is recommended to the reader as an easy exercise.
We now analyze controllability of a system on a domain D being not simply connected. We will follow the same scenario as that used in Chapter I to analyze the cart controllability on a domain with "holes". In the rest of this section, utilizing the apparatus of graph theory and the freetrap condition analogous to that of Chapter I, controllability conditions for on a domain with "holes" are formulated.
The digraph of on D will be denoted by 
(or by 
if 
is either C or 
Recall that a digraph is called strongly connected if from any vertex one can reach a preassigned one by moving along the edges of the digraph in accordance with their orientation. Clearly, if is controllable by a class of admissible controls on , then is strongly connected. The inverse proposition becomes true when the boundary 
satisfies certain trap free conditions.
For a domain D with the boundary satisfying strong trapfree condition the problem of controllability analysis is solved by
Proof. Let be a cart garland representing on D . Then is the union of segments of integral curves 
of the vector field b(x) emitted from 
and belonging to D . The sets are controllable foliations generated by Hence, on is equivalent to the cart and is controllable. Moreover, since is strongly
connected, there exists a set of curves 
where 
joins the sets and 
, such that is controllable on 
Then, it remains to prove that if the conditions of Theorem 2.3 are fulfilled, then one can reach 
from any point 
and vice versa.
Let us take any point 
Then the segment of the integral curve 
emitted from x and belonging to 
has a non empty intersection with some connected component 
By strong trapfree condition there is a control steering the system along . Hence going along one can reach
and vice versa any point 
is accessible from Therefore there exist controls 
such that
for some 
and 
Q.E.D.
It is an easy exercise to design the controls mentioned in the proof of Theorem 2.3 (see formula (2.3)). Strong trapfree condition does not allow to apply Theorem 2.3 to controllability analysis of a system 
with 
on a domain D with "holes" which do not break off the equilibrium set 
of
Trapfree condition can be weaken when the vector field b(x) is integrable on D . The latter implies that there exists a neighborhood 
of such that there is a two-times differentiable real function 
such that
for all 
,where
denotes the differential one-form corresponding to the first derivative of 
i.e., 
where 
denotes 
for 
It is easy to see that
for all 
Hence, 
is a constant on an integral curve of the vector field b(x).
is called an integral of the vector field b(x) on D .
The following theorem gives necessary conditions for a system to be controllable on 
The proof of Theorem 2.4 is analogous to the proof of Theorem 1.2, and is left for the reader as an exercise. It is clear from the geometrical point of view, that if the conditions of Theorem 2.4 are violated, then there are mysteries and prisons in D. Theorem 2.4 and Theorem 1.2 give only necessary conditions for controllability on D. In order to formulate more general sufficient conditions of controllability on D, we need to use weak trapfree condition and the notion of digraph of on D.
As an exercise it is recommended for the reader to prove Theorem 2.5 using the idea of the proof of Theorem 1.3.
Consider examples illustrating the application of the results obtained.
Example 2.3. The system
where 
is a differentiable function defined for all 
is equivalent to the cart and hence controllable on 
Indeed, under the coordinate transformation
the system becomes
From the geometrical point of view, is a controllable foliation generated by the system (Fig.2).
Figure 2: The integral curves of the vector field b(x) are vertical lines. It is easy to see, that is shadowed by the vertical lines emitted from 
Example 2.4. The system
is controllable on 
since all conditions of Theorem 2.4 are met. On the other hand, D is covered by a cart garland representing the system on D.
Example 2.5. Given the system
it is required to determine the values of the constant 
for which the system is controllable on In our case 
therefore is disconnected and since the domain 
is simply connected, it is necessary to check the condition of Corollary 1 of Theorem 2.2. We obtain
and if 
then the function 
will always be negative. Thus the conditions of Corollary 1 of Theorem 2.2 are valid. Since for 
any integral curve of the vector field b(x) intersects 
Example 2.6 We investigate the controllability of the system
on the domain 
For this system we have 
is connected; therefore, it is necessary to verify that the conditions of Theorem 2.1 are fulfilled. Since these conditions are met (Fig.3), we conclude that on D the system under consideration is equivalent to the cart and controllable.
Figure 3: 
is a controllable foliation generated by the system under consideration.