
From Artstein-Sontag Theorem
to the Min-Projection Strategy
A. Bacciotti and L. Mazzi
Dipartimento di Matematica, Politecnico di Torino
Torino - Italy
Given a finite or countable family of continuous vector fields
exists a feedback
 : Rn  I
{ f i ( x )}iI , we show that there
such that the switched system x = f ( x ) ( x ) is globally
asymptotically stable whenever there exists a smooth control Liapunov function V such that
for all
x  0 , V ( x) fi ( x) < 0 , for some i  I .
Key Words: Families of vector fields, stability, discontinuous feedback, Filippov solutions
1. Introduction
One of the main problems in mathematical control theory is to design appropriate control
strategies, in such a way that all the trajectories of a given system are driven as near as
possible to an equilibrium (or, more generally, to a desired steady state). Control Liapunov
functions are one of the typical tools used to elaborate these strategies. To focus the problem
in a classical perspective, we first recall the case of a finite-dimensional, continuous-time,
time-invariant, affine system
m
x = f ( x, u ) = g 0 ( x )  ui gi ( x )
(1)
i =1
where x  R n represents the state variable, u = (u1 ,, um )  R m represents the input variable,
and the vector fields g0 , g1 ,, gm are locally Lipschitz continuous. Without loss of generality,
we assume that the desired equilibrium is the origin.
The so-called Artstein-Sontag Theorem (Artstein, 1983; Sontag, 1989 and 1990) states that
Address
for correspondence: Dipartimento di Matematica del Politecnico di Torino, C.so Duca degli Abruzzi,
24 - 10129 Torino – Italy. E-mail: andrea.bacciotti@polito.it, luisa.mazzi@polito.it
a smooth (except possibly at the origin) stabilizer  () : Rn  R m can be actually constructed
provided that a smooth (at least C 1 ) control Liapunov function V : R n  R can be found.
Roughly speaking, the point is that in the affine case, the inequality
min V ( x )  f ( x, u ) < 0
(2)
u
can be fulfilled by choosing a value of u which depends continuously on the value of x , for
each x = 0 . Note that the naive idea of determining  ( x ) by solving the minimum problem
V ( x )  f ( x,  ( x )) = min V ( x )  f ( x, u )
(3)
u
does not work, since it does not guarantee in general the continuity requirement.
Unfortunately, if we allow u to vary only on a preassigned set U , the Artstein-Sontag
Theorem is not always applicable, since its proof is based on the implicit assumption that u
varies freely on Rn . It often happens in applications that U is a finite set, for instance when
the control is exerted by means of a digital device. In this case, if we set
U = {(1,0, ,0), (0,1, ,0), , (0,0, ,1)},
we may interpret the system as a switched system, by defining fi = g0  gi .
More generally, consider a family F = { f i ( x)}iI of continuous and complete vector fields of
Rn , I being any finite or countable set of indices. Let U pc be the set of switching signals,
that is piecewise constant maps u : [0, )  I . According to Sun and Ge, 2005, one says
that F is pointwise stabilizable if it is asymptotically controllable and the input signal ux (t )
0
can be taken in U pc for each x0 .
Assuming now the existence of a smooth Liapunov function, it is natural to conjecture the
existence of a feedback law  : R n  I which stabilizes F at the origin. The delicate aspect
of this conjecture is that since  ( x ) is expected to be discontinuous, the general theory of
existence and continuability of solutions of ODE's cannot be applied to the closed loop system
x = f ( x ) = f ( x ) ( x ) .
(4)
One needs to resort to some generalized notion of solution: in the literature there are many
possible choices but a common agreement has not yet clearly emerged. As a matter of fact,
any result about discontinuous feedback stabilization inevitably depends on the adopted
notion of solution (see Ceragioli, 2006). Note that the sampling-and-holding approach
followed in Clarke et al., 1997 is useless here, since in general the constraint set U is
preassigned.
In Petterson and Lennarston, 2001, state depending discontinuous feedback laws have been
constructed using the so-called min-projection strategy (see also Geromel and Colaneri,
2006), which basically consists in exploiting (3). The authors assume that the control
Liapunov function is quadratic, and that its derivative with respect to F can be dominated by
a quadratic negative definite form. When I is finite, they prove that the origin is
exponentially stable with respect to Filippov solutions of the closed loop system.
The stabilization result we present in this paper can be viewed as an improvement of
Theorem 1 of Petterson and Lennarston, 2001, since our Theorem holds for countable
families of continuous vector fields (as opposed to finite families). Moreover, our Liapunov
function is positive definite but needs not to be quadratic; its derivative with respect to the
system is negative definite, but not necessarily dominated by a quadratic form as in Petterson
and Lennarston, 2001. As a consequence, we obtain asymptotic (not exponential) stability.
Finally, we deal with solutions in Krasowski sense. Therefore, we obtain asymptotic stability
with respect to a set of solutions larger than the one of Filippov solutions used in Petterson
and Lennarston, 2001. Although it is well known that Krasowski or Filippov solutions are not
always convenient for stabilization problems, they offer the advantage of a well developed
theory of existence and stability (see Filippov, 1988; Hájek, 1979; Clarke et al., 1998; Paden
and Sastry, 1987; see also Petterson, 2003; Pogromsky et al. 2003; Skafidas et al. 1999
for other applications of Filippov solutions in the switched systems literature).
The idea of looking for the existence of regions where at least one of the vector fields is such
that V ( x)  fi ( x) < 0 has been used in Hu et al., 2002 as well; however, the authors restrict
themselves to planar linear systems with centers or foci, and they use only quadratic
Liapunov functions.
In the next section we expose the problem with some details, and state the result. The proof
is given in Section 3 and Section 4 contains a few illustrative examples.
2. Statement of the result
Let F = { fi ( x )}iI be a family of continuous vector fields of Rn , where I is a finite or
countable set of indices, endowed with the discrete topology. As mentioned in the
introduction, we seek stabilizing feedback laws of the form  ( x) : R n  I . Let
Ai = {x  R n :  ( x ) = i} . Of course, Ai  Aj =  for i = j and i Ai = R n . It is clear that the right
hand side of the closed loop system (4) may be not continuous at the boundary points of the
Ai 's. Let us denote by B ( x,  ) the open ball of radius  and center x .
Recall that a Krasowski solution of (4) is any absolutely continuous curve satisfying the
differential inclusion
x  K [ f ( x )] = co{ f ( B( x,  ))} .
(5)
 >0
Recall also that the origin is globally asymptotically stable with respect to Krasowski solutions
of (4) if
(a)   such that |  (0) |<  implies |  (t ) |<  for all t  0 and all Krasowski
solutions  of (4);
(b) lim t  |  (t ) |= 0 for all Krasowski solutions  of (4).
Note that if (a) holds, then the origin is an equilibrium solution in the sense of Krasowski for
(4) (to this respect see also Xu et al., 2007); moreover, all the local Krasowski solutions
corresponding to a small initial state can be continued on [0,) .
Theorem 1 Together with the family of vector fields F , a positive definite, radially
unbounded and smooth (at least C 1 ) function V ( x) : Rn  R is given. Assume that
x = 0 i  I such that V ( x) fi ( x ) < 0 .
(6)
Then, there exists a discontinuous feedback  ( x) : Rn  I such that the origin is globally
asymptotically stable for (4) with respect to Krasowski solutions.
Remark 1 A function V ( x ) satisfying (6) is called a control Liapunov function for F . By
virtue of the converse theorem proven in Clarke et al., 1998 (see also Bacciotti and Rosier,
2001), the existence of a control Liapunov function is necessary for stabilization in Krasowski
sense.
3 The proof
In order to prove our main result, we need to recall two results concerning the Krasowski
differential inclusion K [ f ( x )] associated to a discontinuous system x = f ( x ) .
Lemma 1 Given a differential system x = f ( x ) , where f is measurable, we have the
following identity:
K [ f ( x )] =
co{ f ( B( x,  ))} =
 >0
= co{v  R n : x j  x s.t. f ( x j )  v}.
For a proof of this Lemma, see Shevitz and Paden, 1994. We remark that f (x ) defined in (4)
is measurable if and only if  ( x ) is measurable.
Lemma 2 Let K [ f ( x )] be the set valued map introduced in (5). If there exist a smooth (at
least C 1 ), radially unbounded, positive definite function V : R n  R and a real valued,
continuous, positive definite function W : R n  R such that x  Rn \ {0}, v  K [ f ( x)],
V ( x )  v  W ( x ) < 0,
then the origin is globally asymptotically stable with respect to Krasowski solutions of (4).
For a proof of this result, see for instance Filippov, 1988 or Bacciotti and Rosier, 2001.
Proof of Theorem 1. We consider the covering of R n \ {0} given by the family of closed
sphere hulls of radius 2i 1  2i , i  Z :
Ci = B(0,2i 1 ) \ B(0,2i ).
Claim. There exists i > 0 such that x  Ci , k  I such that V ( x) f k ( x)  i < 0 .
If this were not true, we should have that m  N \ {0} , xm  Ci such that k  I ,
1
V ( xm ) f k ( xm ) >  .
m
Since Ci is compact, there exists a subsequence xm , such that lim h xm = x  Ci . By
h
h
continuity,
lim V ( xmh ) f k ( xmh ) = V ( x ) f k ( x )  0
h  
for all k  I , a contradiction.
Therefore the claim is verified. We set
0 ,


 i = min {i ,2i ,  i 1},
 min { ,  },
i
i 1

i=0
i<0
i > 0.
We observe that the sequence { i , i  0} is decreasing, while { i , i  0} is increasing, and
lim i  i = 0 . We define now a function
0



  i   i 1 || x || 2 i   ,
i 1
 2 i 1  2 i

W ( x) = 
  1   1 || x || 1   1 ,


  i 1   i
i
 2 i 1  2 i || x || 2   i ,





x=0
x  Ci , i < 0
x  C0
x  Ci , i > 0 .
W ( x ) satisfies the following properties:
1. lim x0 W ( x ) = W (0) = 0 , since for all x  Ci , i < 0 , W ( x )   i  2i .
2. W ( x ) is positive definite and continuous for all x  R n \ {0} , by construction.
3. For all x  R n \ {0} there exists k  I such that
V ( x) f k ( x )  W ( x ) < 0.
(7)
This is true, since for all i  Z ,  W ( x)   i  i , and, by the claim,  i  V ( x) f k ( x ) , for
some k  I .
We define a covering of R n \ {0} given by the closed sets
i ,k ={x  Ci : V ( x ) f k ( x )  W ( x ) < 0},
where i  Z , k  I . We define now a feedback  : R n \ {0}  I as:
1

 ( x) =  k
0

x  i ,1 , i  Z
x  i ,k \ h < k i ,h , i  Z
x = 0.
We have Ak =  1 (k ) = iZ i ,k \ h<k i ,h . The set Ci  Ak is clearly measurable for each i
and for each k , so that  is a measurable function.
We observe that  ( x ) may be equivalently defined as  ( x) = min {k :V ( x) f k ( x )   i } , when
x  Ci .  ( x ) yields a discontinuous differential system
 f ( x),
x = f ( x) =   ( x )
0

x  R n \ {0}
x = 0.
By definition, for all x  R n \ {0} , there exists  i,k such that x  i ,k and
V ( x ) f ( x ) = V ( x ) f ( x ) ( x )  W ( x ) < 0.
In order to be able to apply Lemma 2, we need to show that for all x  R n \ {0} and for all
v  K [ f ( x )] , V ( x )  v  W ( x ) . First of all, we consider a vector v obtained as a limit
lim j f ( x j ) = v , for some x j  x . By construction, for all x j
V ( x j ) f ( x j ) = V ( x j ) f ( x ) ( x j )  W ( x j ).
j
Since V and W are continuous functions, we have that


lim V ( x j ) f ( x j ) ( x j ) = V ( x )  v 
x j x
 lim W ( x j ) = W ( x ).
x j x
By Lemma 1, w  K [ f ( x )] if it is a convex combination of vectors v obtained as above,
therefore the same inequality holds for any w  K [ f ( x )] . By Lemma 2 our main result is
proven.
4 Examples
The proof of Theorem 1 shows that, whenever it is possible to construct a function W ( x )
satisfying (7), we are able to construct a stabilizing feedback  ( x ) . (The claim basically states
that such a function exists whenever (6) is fulfilled.)
As illustrated by the following examples, in some cases it is not difficult to find a function
W ( x ) independently of the construction given in Theorem 1.
Given a family of vector fields F and a control Liapunov function V for F , we denote by  i
the set
 i ={x  R n : V ( x ) f i ( x ) < 0}.
Example 1 Let us consider the pair of planar linear systems
  x
f1 ( x, y ) =   ,
 0 
 0 
f 2 ( x, y ) =  
 y
and the candidate control Liapunov function V ( x, y ) = ( x 2  y 2 )/2 . It is easily seen that
1 = {( x, y) : x = 0} ,
2 = {( x, y) : y = 0}
so that condition (6) is fulfilled. If we choose W ( x, y ) = V ( x, y ) , we are led to the following
stabilizing feedback:
1 if | y || x |
2 otherwise.
 ( x) = 
Some trajectories of the closed loop system are plotted in Fig. 1. Stabilization of this system
can be also addressed with the method of Wicks et al., 1998: in particular, this system is
quadratically stable (see for instance Liberzon, 2003 for the definition of quadratic stability).
In Wicks et al., 1998, it is also shown how to avoid the sliding mode induced by the feedback
along the discontinuity surfaces, by resorting to memory-feedback and hysteresis.
Example 2 Let us consider now the pair of planar systems
 1 
f 2 ( x, y ) =  
 y
 1 
f1 ( x, y ) =   ,
 y
and take again V ( x, y ) = ( x 2  y 2 )/2 . We have
1 = {( x, y ) : x < y 2 } ,
 2 = {( x, y ) : x >  y 2 } .
Condition (6) is fulfilled. If we choose W ( x, y ) = 2V ( x, y ) , we obtain the stabilizing feedback
1 if x  0
2 otherwise.
 ( x) = 
Some trajectories of the closed loop system are plotted in Fig. 2. The existence of a
discontinuous stabilizing feedback can be also deduced from Ancona and Bressan, 1999;
indeed, the system is easily seen to be asymptotically controllable. However, we note that the
approach in Ancona and Bressan, 1999 does not lead to an easy and explicit construction.
Example 3 In this last example we consider the pair of planar linear systems
  2x  y 
 ,
f1 ( x, y ) = 
 4x  2 y 
 2x  4 y 
 .
f 2 ( x, y ) = 
 x  2y 
It is not difficult to show that in this case condition (6) is satisfied by no quadratic function
V ( x , y ) . On the contrary, taking V ( x, y ) = ( x 4  y 4 )/4 we have
1 = {( x, y ) : (2 x  y )(32 y  x ) > 0}
 2 = {( x, y ) : ( x  2 y )(32 x  y ) > 0} .
1
Regions

and

2
overlap,
 113 4 
 xy 
W ( x, y ) = 2  3 4 x 2  1 

20



3
and
Condition
(6)
is
fulfilled.
By
taking

4  3 2 y 2 , we obtain the stabilizing feedback
1 if (5 y  4 x)(4 y  5 x)  0
otherwise.
2
 ( x) = 
Some trajectories of the closed loop system are drawn in Fig. 3. Note that method of Wicks
et al., 1998 does not work in this case, and not even its generalization proposed in Bacciotti,
2004: in particular, we note that this system is not quadratically stable.
5 Final remarks
Besides the notion of pointwise stabilizability recalled above, in Sun and Ge, 2005 the
authors introduce also the notion of consistent stabilization, meaning that the system is
asymptotically controllable by a switching signal u : [0, )  U which is independent of the
initial state x0 . In this case, replacing u by u(t ) , (1) can be reviewed as an asymptotically
stable time-varying system. If, in addition, the stability is uniform, the existence of a timevarying, smooth Liapunov function is guaranteed (see Bacciotti and Rosier, 2001). Using the
same construction as in the proof of Theorem 1, it should be possible to define stabilizing
rules depending both on time and on the state variable and taking values in the constraint set
U.
References
Ancona F. and Bressan A., 1999: Patchy Vector
Fields and Asymptotic Stabilization, ESAIM:
Control, Optimisation and Calculus of Variations 4,
445-472
Artstein Z., 1983: Stabilization with Relaxed
Controls, Nonlinear Analysis, Theory, Methods and
Applications 7, 1163-1173
Bacciotti A., 2004: Stabilization by Means of State
Space Depending Switching Rules, Systems and
Control Letters 53, 195-201
Bacciotti A. and Rosier L., 2001: Liapunov
functions and Stability in Control Theory, Lecture
Notes in Control and Information Sciences 267,
Springer Verlag, London
Ceragioli F., 2006: Finite valued feedback laws
and piecewise classical solutions, Nonlinear
Analysis 65, 984-998
Clarke F.H., Ledyaev Yu.S., Sontag E.D. and
Subbotin A.I., 1997: Asymptotic Controllability
Implies Feedback Stabilization, IEEE Trans.
Automat. Control 42, 1394-1407
Clarke F.H., Ledyaev Yu.S. and Stern R.J.,
1998: Asymptotic Stability and Smooth Lyapunov
Functions, Journal of Differential Equations 149,
69-114.
Paden B.E. and Sastry S.S., 1987: A Calculus for
Computing Filippov's Differential Inclusion with
Application to the Variable Structure Control of
Robot Manipulators, IEEE Trans. on Circuits and
Systems 34, 73-81
Petterson S., 2003: Synthesis of Switched Linear
Systems, Proceedings of the 42nd IEEE-CDC,
5283-5288
Petterson S. and Lennarston B., 2001:
Stabilization of Hybrid Systems Using a MinProjection Strategy, Proceedings of the American
Control Conference, Arlington, 223-228
Pogromsky A.Yu., Heemels W.P.H. and
Nijmijer H., 2003: On Solution Concepts and WellPosedness of Linear Relay Systems, Automatica
39, 2139-2147
Shevitz D. and Paden B., 1994: Lyapunov
Stability Theory of Nonsmooth Systems, IEEE
Transactions on Automatic Control 39, 1910-1914
Skafidas E., Evans R.J., Savkin A.V. and
Petersen I.R., 1999: Stability Results for Switched
Controller Systems, Automatica 35, 553-564
Sontag E.D., 1990: Mathematical Control Theory,
Springer-Verlag, New York
Filippov A.F., 1988: Differential Equations with
Discontinuous Righthand Sides, Kluwer, Dordrecht
Geromel J.C. and Colaneri P., 2006:
Stability and Stabilization of Continuous-time
Switched Linear Systems, SIAM Journal Control
Optim. 45, 1915-1930
Hájek O., 1979: Discontinuous Differential
Equations, I, Journal of Differential Equations 32,
149-170
Hu B., Zhai G. and Michel A., 2002: Common
Quadratic Lyapunov-like Functions with Associated
Switching Regions for two Unstable Second-order
LTI Systems, International Journal of Control 75,
1127-1135
Liberzon D., 2003: Switching in Systems and
Control, Birkhäuser, Boston
Sontag E.D., 1989: A ``Universal'' Construction of
Artstein's Theorem on Nonlinear Stabilization,
Systems and Control Letters 13, 117-123.
Sun Z. and Ge S.S., 2005: Switched Linear
Systems, Springer-Verlag, London
Xu X., Zhai G. and He S., 2007: Stabilizability and
Practical Stabilizability of Continuous-Time Switched
Systems: A Unified View, Proceedings of American
Control Conference, New York
Wicks M., Peleties P. and DeCarlo R.A., 1998:
Switched Controller Synthesis for the Quadratic
Stabilization of a Pair of Unstable Linear Systems,
European Journal of Control 4, 140-147.
Figura 1: Some trajectories of the system (Example 1)
Figura 2. Some trajectories of the system (Example 2)
Figura 3. Some trajectories of the system (Example 3)