Asymptotic Stability of Uniformly Bounded Nonlinear Switched Systems

We study the asymptotic stability properties of nonlinear switched systems under the assumption of the existence of a common weak Lyapunov function. We consider the class of nonchaotic inputs, which generalize the different notions of inputs with dwell-time, and the class of general ones. For each of them we provide some sufficient conditions for asymptotic stability in terms of the geometry of certain sets. The results, which extend those of Balde, Jouan about linear systems, are illustrated by many examples.


Introduction.
This paper is concerned with the asymptotic stability properties of nonlinear switched systems defined by a finite collection {f 1 , . . . , f p } of smooth vector fields in R d . They are assumed to share a smooth weak Lyapunov function V , hence to be stable at the origin, and we are looking for sufficient conditions of asymptotic stability for certain classes of switching laws. Our stability hypothesis is similar to the ones of [3-5, 9, 11]. In particular in [4,5,11] the vector fields are linear, the Lyapunov function is quadratic, and the asymptotic stability properties are closely related to the geometry of the union of some linear subspaces of R d .
In the present paper, we introduce in the same way two geometric subsets of R d which turn out to be fundamental in the sense that they contain all the limit sets for two classes of inputs. More accurately let K i = {L f i V = 0} be the set where the vector field f i is tangent to a level set of the Lyapunov function V . We show in Section 8 that the limit sets are always contained in the union of the K i 's. On the other hand, for the class of nonchaotic inputs (which are a generalization of inputs with dwell-time), the limit sets are included in the smaller set p i=1 M i where M i is roughly speaking the largest f i -invariant subset of K i .
The geometry of the union of the M i 's and the K i 's is consequently the key point of our results. Of course these sets are no longer linear whenever the vector fields are not, and we are lead to approximate them by linear subspaces or cones.
In the paper [3] the authors deal with another set included in the union of the K i 's but generally much larger than the union of the M i 's (see Example 1 in Section 4). However they consider only dwell-time inputs and their results are improved by the distinction we introduce between the two categories of sets and inputs.
The paper is organized as follows: basic definitions and notations are stated in Section 2, and some preliminary and technical results in Section 3. In Section 4, we state a general result of asymptotic stability for nonchaotic inputs. We provide sufficient conditions of stability by means of some tangent objects in Section 5. The results are then refined in the analytic case (Section 6) and in the linear one (Section 7). Section 8 is devoted to asymptotic stability for general inputs and the particular case of two globally asymptotically stable vector fields is dealt with in Section 9. Finally, in Section 10 a necessary and sufficient condition for a planar switched system to be GUAS is stated. The results are illustrated by some examples in Section 11.

Statement of the Problem
As explained in the introduction, we deal with a finite collection {f 1 , . . . , f p } of smooth vector fields on R d vanishing at the origin and assumed to share a smooth weak Lyapunov function V : R d −→ R + . This means that V is positive definite and for each i ∈ {1, . . . , p} the Lie derivative L f i V is nonpositive, so that V is nonincreasing along the solutions ofẋ = f i (x), i = 1, . . . , p.
An input, or switching signal, is a piecewise constant and right-continuous function u from [0, +∞[ into {1, ..., p}. We denote by (a n ) n≥0 the sequence of switching times (with a 0 = 0). In the particular case where the number of switches is finite we adopt the following convention: if u is discontinuous at a 0 < · · · < a N and constant from a N , then we set a n = a N + (n − N) for n > N. The sequence (a n ) n≥0 is thus strictly increasing to +∞ in any case, and u(t) is constant on each interval [a n , a n+1 [. This value will be denoted by u n ∈ {1, ..., p}, and the duration a n+1 − a n by δ n .
As the input is entirely defined by the switching times and the values taken at these instants we can write u = (a n , u n ) n≥0 .
We consider the switched systeṁ Fixing a switching signal u gives rise to the dynamical system defined in Its solution for the initial condition x will be denoted Φ u (t, x) and Φ i (t, x) will stand for the flow of f i . As we are concerned with the stability analysis at the origin of the switched system given by (S), let us recall some stability notions : Definition 1 The switched system (S) is said to be 1. uniformly stable at the origin if for every δ > 0, there exists ε > 0 such that for every x ∈ R d , x ≤ ε, all inputs u and t ≥ 0, Φ u (t, x) ≤ δ; 2. locally attractive at the origin if there exists δ > 0 such that for all x ∈ R d , x ≤ δ and all inputs u, Φ u (t, x) converges to the origin as t goes to infinity; 3. globally attractive at the origin if for every input u and all x ∈ R d , Φ u (t, x) converges to the origin as t goes to infinity; 4. asymptotically stable at the origin if it is uniformly stable and locally attractive at the origin; 5. globally asymptotically stable at the origin if it is uniformly stable and globally attractive at the origin.
6. globally uniformly asymptotically stable (GUAS) at the origin if it is uniformly stable and globally uniformly attractive at the origin i.e.
It is a well-known fact that the set {f 1 , . . . , f p } being finite, hence compact, global asymptotic stability is equivalent to GUAS (see [2]). The vector fields are not necessarily forward complete but V being a weak common Lyapunov function, the switched system is uniformly stable, and all solutions of (S) starting in some neighborhood of the origin are forward bounded, hence defined for all positive times (however the f i 's are forward complete whenever V is radially unbounded, see Proposition 1). Before concluding this section, let us define some classes of inputs that we will consider in the sequel.
Definition 2 (see [5]) An input u is said to be chaotic if there exists a sequence ([t k , t k + τ ]) k∈N of intervals which satisfies the following conditions: 2. For all ε > 0 there exists k 0 ∈ N such that for all k ≥ k 0 , the input u is constant on no subinterval of [t k , t k + τ ] of length greater than or equal to ε.
An input that does not satisfy these conditions is called a nonchaotic input.
In the litterature, one often encounters inputs with dwell-time (an input u is said to have a dwell-time δ > 0 if for each n ∈ N, δ n ≥ δ). Such an input is clearly nonchaotic.
Definition 3 An input u is said to satisfy the assumption H(i) if there exist a subsequence (a n k ) k∈N and δ > 0 such that ∀k ≥ 0, u n k = i and δ n k ≥ δ.
It is said to be regular if it is nonchaotic and satisfies the assumption H(i) for i = 1, . . . , p.

Limit Sets and Convergence Results
For an input u and an initial condition x ∈ R d , we denote by Ω u (x) the set of ω-limit points of {Φ u (t, x); t ≥ 0}, that is the set of limits of sequences x)) is nonincreasing, and we have the following result: Proposition 1 There exists a neighborhood W of the origin such that for all x ∈ W and all input u, the solution starting at x is forward bounded and the ω-limit set Ω u (x) is a compact and connected subset of a level set {V = R} for some R ≥ 0.
In particular if the Lyapunov function is radially unbounded (i.e. V (x) −→ +∞ as x goes to +∞), we can choose W to be equal to R d .
Proof : V has a positive minimum m on the unit sphere. Let us fix r ∈]0, m[ and let W be the connected component of {V ≤ r} which contains 0 (note that W is compact). In the case where V is radially unbounded r can be chosen arbitrarily in ]0, +∞[. Let x ∈ W and u be an input. Since V is nonincreasing along the solution Φ u (·, x), the trajectory Φ u (t, x), t ≥ 0, is included in W, and is forward bounded. It follows that Ω u (x) is a compact and connected subset of W (see [10]).
It remains to show that Ω u (x) is contained in a level set of V . The map t ∈ R + −→ V (Φ u (t, x)) being nonincreasing and nonnegative, has a limit R ≥ 0 as t goes to +∞. It is clear that Ω u (x) ⊆ {V = R}.
From now on W will be a connected neighborhood of the origin satisfying the properties of Proposition 1.
Next, we prove a uniform convergence result that will be used many times in the sequel and which is based on the notion of weak convergence of the inputs.
Proposition 2 Let x ∈ W, u an input, (t k ) k∈N ⊂ R + a sequence increasing to +∞ and τ > 0. Consider the sequence (ϕ k ) k∈N defined by Then (ϕ k ) k∈N admits a subsequence that converges uniformly to an absolutely continuous function ϕ : Remark 1 Proposition 2 shows that a "limit trajectory" is a solution of the convexified system.

Proof :
The switched systemẋ = f u(t) (x) can be rewritten as the affine control oneẋ where v : [0, +∞[−→ B, and B = {e 1 , . . . , e p } is the canonical basis of R p . Consider the following functions: Since the solution Φ u (·, x) is forward bounded, up to a subsequence (ϕ k (0)) k∈N converges to a limit ℓ ∈ Ω u (x) and by Proposition 10.1.5 of [12], (w k ) k∈N converges weakly to a measurable function w : [0, τ ] −→ U where U is the convex hull of B (this is equivalent to saying that w i ≥ 0 and p i=1 w i = 1 almost everywhere).
The following lemma will also be useful in the study of the stability for general inputs (Section 8). The notations are the same as above.
) be a weakly convergent sequence and w ∈ L ∞ ([0, τ ], U) its limit. The following statements are equivalent: where m stands for the Lebesgue's measure on the real line.
Proof : It suffices to write and to conclude using the weak convergence.

Stability Results for Regular Inputs
In this section we state and prove one of the main results of the paper; but first we introduce very important subsets of R d and establish relations between them and the ω−limit sets. For each i ∈ {1, . . . , p} we define the set M i by The following lemma is the centerpiece of the part of the paper concerned with nonchaotic inputs because it enables us to establish two fundamental facts. On the one hand the ω-limit sets intersect M i for an input satisfying Assumption H(i) (see Corollary 1) and on the other hand, for a nonchaotic input every ω-limit set is contained in the union of the M i 's (see Proposition 3). A fortiori the ω-limit sets for regular inputs satisfy both statements.
It follows that ℓ and Φ i (τ, ℓ) are two ω-limit points of Ω u (x), and by Propo- Since V is nonincreasing along the trajectory Φ i (·, ℓ), we get This implies that for each k ≥ 1, Finally ℓ ∈ M i .

Corollary 1 If an input u satisfies Assumption
Proposition 3 Let x ∈ W and let u be a nonchaotic input. Then Let T > 0 and set By Proposition 2 and up to a subsequence, we can assume that (ϕ k ) k∈N converges uniformly on [0, T ] to a continuous function ϕ : The input u being nonchaotic, there exist a number τ > 0 and a sequence (s k ) k∈N such that for all k ∈ N, Up to a subsequence, we can assume that there exists j ∈ {1, . . . , p} such that u is equal to j on each interval [s k , s k + τ ], and the sequence (ε k ) k∈N = (t k − s k ) k∈N converges to a limit ε ∈ [0, T ]. By continuity of ϕ and by uniform convergence of (ϕ k ) k∈N , the sequence (ϕ k (ε k )) k∈N converges to ϕ(ε). According to Lemma 2 the point ϕ(ε) belongs to M j , which contradicts In [3] the authors showed that for all initial conditions x in W and for every input u with dwell-time, the ω-limit set Ω u (x) is contained in the union of all the compact and weakly invariant sets which are included in Proposition 3 is therefore a generalization of Bacciotti and Mazzi's result because the set p i=1 M i is in general strictly contained in the set defined in [3]. Moreover it applies to a larger class of inputs.
Example 1 (see [3,9]) Consider the linear switched system consisting in Bacciotti and Mazzi's result says that for inputs with dwell-time, the ω-limit sets are contained in {x 1 = 0}. On the other hand, computing L 3 According to Proposition 3, the switched system is asymptotically stable for every nonchaotic input (recall that the class of nonchaotic inputs is larger than the class of inputs with dwell-time). Notice that the set {x 1 = 0} is what we define as K 1 ∪ K 2 (see Section 8).
Now we give a sufficient condition of asymptotic stability for regular inputs.
Theorem 1 If the sets M i satisfy the condition there exists a neighborhood V of the origin such that, for all R > 0, no connected component of the set then for every regular input u, the switched system is asymptotically stable. In particular if V = W = R d the switched system is globally asymptotically stable for every regular switching signal.
Proof : Since (S) is uniformly stable at the origin, it suffices to check its attractivity for regular inputs. Let V be a neighborhood of the origin satisfying Condition (C). We can assume that V is included in W and is the connected component of {V ≤ r} containing 0 for some r > 0.
Let x ∈ V and let u be a regular input. Since V is nonincreasing along the solution Φ u (·, x), we have Ω u (x) ⊆ V. We know from Proposition 1 that Ω u (x) is a connected subset of a level set {V = R} for some R ≥ 0. Let us show that R = 0. Since u is a regular input, the two following facts hold: Therefore Remark 2 Note that Condition (C) cannot be fulfilled if the origin is not isolated in p i=1 M i . This provides a necessary condition for (C) to hold.

Approximation of the M i 's
In general the sets M i are difficult to compute and Condition (C) may be hard to check directly. It is of interest to approximate the M i 's by some "tangent" sets easier to compute like linear subspaces or cones. However the sets M i are not necessarily submanifolds. Moreover dL k f i V (0) may be equal to zero, it is for instance the case for k = 1 because L f i V has a maximum at 0. For these reasons we define l k as the smallest integer (which depends on i) such that d l k L k f i V (0) = 0. If such an integer does not exist we set l k = 1. Note that l 1 is necessarily even since L f i V has a maximum at 0.
For each i ∈ {1, . . . , p} consider the set The sets ker d l k L k f i V (0) are cones but not linear subspaces in general.

Lemma 4
We have the following results: In other words, every tangent vector to Proof : 1. For all k, n ≥ 1, Taking the limit as n goes to +∞, we get Up to a subsequence, ( x n x n ) n≥1 converges to a limit x = 0 which belongs to M i ∩ M j by the previous item. A straightforward computation gives M = {α = 0} (this set can be any closed subset of R 2 by Borel's Theorem). One also has d k L f V (0) = 0 for k = 1, 2, 3 so that M ⊆ ker d 4 L f V (0). But However, the previous computation gives M ⊆ {x 2 = ±x 1 }.

The Analytic Case
In this section we assume the vector fields and the Lyapunov function to be analytic. In that case the set M i can also be defined as where I x is the maximal interval of definition of the solution Φ i (·, x). It is shown in the next proposition that whenever V i is a centre manifold for f i (see [7] for the definition and basic properties of centre manifolds), then locally around the origin the inclusion M i ⊆ V i holds and enables us to give sufficient conditions for asymptotic stability. Thanks to the principle of approximation at any order of centre manifolds (see [7]), these conditions are checkable in practice.
Proposition 4 Let V i be a centre manifold for f i . Then there exists a neighborhood O of the origin such that Proof : Since f i is stable at the origin, its differential df i (0) at 0 has no positive eigenvalue. Therefore we can choose a basis such that f i writes where A ≡ df i (0) is a block diagonal matrix consisting in two blocks, a n × n block A 1 , the eigenvalues of which lies on the imaginary axis, and a m × m Hurwitz block A 2 . According to this decomposition we will write each x in R d as x = (x 1 , x 2 ) ∈ R n × R m . By definition V i is locally the graph of a smooth map h : R n −→ R m defined in a neighborhood of the origin such that h(0) = 0 and dh(0) = 0 (see [7]).
Let O be the connected component of {V ≤ r} containing the origin for some r > 0. According to Lemma 1, page 20 of [7], and for r small enough, there exist two numbers C, µ > 0 such that for all t ≥ 0 and all We can also assume that O is bounded and that h is defined on O∩(R n ×{0}). The map h being continuous on the compact set O ∩ (R n × {0}), there exists a number M > 0 such that In order to prove that x ∈ V i , we show that x 2 = h(x 1 ). By analycity V (Φ i (t, x)) = V (x) ≤ r for all t ∈ I x . It follows that the trajectory {Φ i (t, x); t ∈ I x } cannot leave the compact set O in backward time (it is therefore defined for all t ≤ 0). For each k ∈ N, set x k = Φ i (−k, x) and C 1 = CM. Then for all t ≥ 0, and replacing t by k we get Finally x 2 = h(x 1 ) and x ∈ V i . From Proposition 4, we deduce that all tangent vectors to M i at the origin are contained in the tangent space to V i at the origin which will be denoted by V i . Consequently in the analytic case we get the following results. As well as in [5] we deduce from Theorem 3 some geometric sufficient conditions for asymptotic stability. They deal with linear objects and are quite easily checkable.
Then, the switched system is asymptotically stable for all regular switching signals as soon as one of the following conditions holds: Proof : Each condition implies Condition (C) (see [5]).

Example 3 Let us consider the vector field
where ϕ and ψ are analytic functions with ψ(0) = 0, the variable z belongs to R n+m , and B is a Hurwitz matrix of the following form: We assume the n × n matrix A to satisfy A T + A = 0, the m × m matrix D to satisfy D T + D < 0, and the pair (C, A) to be observable. These conditions ensure that B is Hurwitz (see [4]). The positive definite function V defined by V (x 1 , x 2 , y, z) = x 2 1 + x 2 2 + y 2 + z T z is a weak Lyapunov function for f . Indeed a straightforward computation gives: As B T + B ≤ 0, we obtain L f V ≤ 0 and: On the other hand the plane P = {y = 0, z = 0} is a centre manifold for f . According to Proposition 4 we obtain Thanks to LaSalle's invariance Principle and because z(t) tends to zero as t goes to +∞ the inclusions are global. This example can be refined in the following way. Let us first denote by f 0 the vector field obtained by applying to f the diffeomorphism Now let g be the vector field obtained by applying the same diffeomorphism to the vector field F defined by: The conclusion is that the switched system defined by f 0 , f 1 is asymptotically stable for any regular input. Indeed the positive function

The Linear Case
The paper [5] deals with the linear case but consider quadratic Lyapunov functions only. Actually Theorems 3 and 4 of [5] are direct consequences of the results of the previous sections. However we will show that our results also apply to linear systems with nonquadratic, but analytic and unbounded, Lyapunov functions. So let us assume the fields f i to be linear and the Lyapunov function to be analytic and radially unbounded. Fix a vector field f i . For every real or complex eigenvalue a of f i , let us denote by L(a) = ker(f i − aI) d the characteristic subspace of C d for this eigenvalue. We define where a is the complex conjugate of a, and we set where ℜ(a) stands for the real part of a. It is a well known fact that as soon as the real part of one of the eigenvalues of f i is positive, or in the case where the index of one of the eigenvalues whose real part vanishes is greater than one (that is the characteristic subspace is strictly larger than the eigenspace for this eigenvalue), there exists an x such that Φ i (t, x) tends to infinity as t goes to +∞.
As by hypothesis f i is stable at the origin, the space The subspaces V i and V S i are f i -invariant, so that V i is also invariant under the flow of f i . It is consequently a centre manifold for f i , since the restriction of f i to V S i is Hurwitz. Moreover, according to the previous decomposition of R d , we can write where the real parts of A 1 's eigenvalues vanish and A 2 is Hurwitz. The flow of f i is then Φ i (t, x) = e tA 1 x 1 e tA 2 x 2 .
In order to prove that x ∈ V i it suffices to show that x 2 = 0. Since A 2 is Hurwitz, there exist C, µ > 0 such that for all t ∈ R and all y ∈ V S i , e tA 2 y ≤ Ce −µt y . For each n ≥ 1, set Since x ∈ M i and by analycity, V (x n ) = V (x) for each n ≥ 1. As V is radially unbounded (x n ) n≥1 is bounded, and so is (x n 2 ) n≥1 . But It follows that x 2 = 0 and x ∈ V i . Conversely, since A 1 is semi-simple the matrices e tA 1 are rotation ones in some Euclidean structure on V i . It is a well known fact that every point of the trajectory t −→ e tA 1 is recurrent. In particular there exists a sequence (t k ) k≥0 increasing to +∞ such that and V (e tA 1 x) being not increasing and by analycity we obtain x ∈ M i .
Then, the linear switched system is globally asymptotically stable for all regular switching signals as soon as one of the conditions (1-4) of Corollary 2 holds.

Stability for general inputs
In this section we deal with general inputs, more accurately the stability results of [5] for possibly chaotic inputs are extended to nonlinear switched systems using the convergence results of Proposition 2 and Lemma 1 (see Section 3).
For each i ∈ {1, . . . , p}, we introduce the closed set containing the origin and for a given input u, the subset J u of {1, . . . , p} defined by (recall that m stands for the Lebesgue's measure).

Proposition 6
For all x ∈ W and all switching signals u Proof : Assume by contradiction that there exists ℓ ∈ Ω u (x) such that ℓ ∈ i∈Ju K i . Let (t k ) k≥0 ⊂ R + be a sequence increasing to +∞ and such that Fix T > 0 and consider the sequence (ϕ k ) k≥0 defined by By Proposition 2 and Lemma 1, and up to a subsequence, (ϕ k ) k≥0 converges uniformly on [0, T ] to an absolutely continuous function ϕ :

] is a measurable function and
But for all t ∈ [0, T ], ϕ(t) ∈ i∈Ju K i and L f i V (ϕ(t)) < 0 for i ∈ J u . It follows that for each i ∈ J u , w i = 0 almost everywhere which contradicts i∈Ju w i = 1 almost everywhere.

Proposition 7
For all x ∈ W and all inputs u: Proof : Assume by contradiction that there exists i ∈ J u such that Since Ω u (x) is compact and K i is closed, the distance between those two sets Since K := {y ∈ R d / d(y, Ω u (x)) ≤ ε} is a compact set which does not intersect K i there exists α > 0 such that For each t ≥ 0, As Ω u (x) is contained in a level set {V = R} for some R > 0, this integral converges as t goes to infinity and we get From Propositions 6 and 7, we deduce for general inputs a theorem similar to Theorem 1 for regular ones.
then for every input u such that J u = J, the switched system is asymptotically stable. In particular if V = W = R d then the switched system is globally asymptotically stable for every input u such that J u = J.
Using the same technique of linearization as in the regular input case we can give sufficient conditions for (K) to hold. Set K i = ker d 2 L f i V (0), i = 1, . . . , p, and recall that these sets are linear subspaces (see Lemma 4).
. . , p} be a nonempty set and assume that Then, the switched system is asymptotically stable for all switching signals u such that J u = J as soon as one of the following conditions holds: 1. there exists i ∈ J such that dim K i = 0; 2. there exists i ∈ J such that dim K i = 1 and K i ⊆ K j =⇒ K i = K j for each j ∈ J; 3. #J = 2; 4. #J > 2 and dim( i∈J K i ) > i∈J dim K i − #J + 1.

The case of two GAS vector fields
In this section, we are interested in asymptotic stability for all inputs of a system consisting in a pair of vector fields. They are of course assumed to be asymptotically stable at the origin (consider constant inputs) and denoted by f 0 and f 1 so that the switched system (S) writeṡ Since the vector fields are asymptotically stable, we can choose a neighborhood W of the origin such that ∀x ∈ W and for i = 0, 1, Φ i (t, x) goes to 0 as t goes to +∞. We consider the set K = K 0 ∩ K 1 and we get: For all x ∈ W and all inputs u, Moreover, if the vector fields and the Lyapunov function are analytic then If the vector fields are globally asymptotically stable and V is unbounded, one can take W = R d .
Since Ω u (x) ⊆ K 0 ∪ K 1 we can assume without loss of generality that ℓ ∈ K 0 \ K 1 . Fix τ > 0 and consider the sequence (ϕ k ) k∈N defined by By Proposition 2, up to a subsequence, (ϕ k ) k∈N uniformly converges on [0, τ ] to an absolutely continuous function ϕ : We can choose τ so small that As Ω u (x) is contained in a level set {V = r} for some r > 0, we get for Consequently λ(t) = 0 for almost all t ∈ [0, τ ] anḋ In the analytic case this implies ∀t ≥ 0 V (Φ 0 (t, ℓ)) = r, and Φ 0 (t, ℓ) cannot converge to 0 as t goes to +∞.
In the smooth case let us assume that Ω u (x) ∩ K = ∅. Let us first notice that the limit trajectory can be extended to any interval [0, T [ hence to [0, +∞[. The assumption Ω u (x) ∩ K = ∅ implies then that ϕ(t) ∈ K 0 \ K 1 for t ≥ 0, hence that ϕ(t) = Φ 0 (t, ℓ) for t ≥ 0. The conclusion comes from the same contradiction that Φ 0 (t, ℓ) cannot converge to 0 as t goes to +∞.
Corollary 4 Assume that the vector fields f 0 and f 1 are globally asymptotically stable and that V is radially unbounded. If K = {0} then the switched system is GUAS.

Planar analytic switched systems
In this section we give a necessary and sufficient condition for a planar analytic switched system consisting in two globally asymptotically stable subsystems to be GUAS under our usual assumption of existence of a weak common Lyapunov function.
Consider the analytic planar switched systeṁ where f 0 , f 1 are two globally asymptotically stable analytic vector fields and V is a weak analytic Lyapunov function.
We assume V to be radially unbounded and dV (x) to vanish at x = 0 only, which implies that for each R > 0, the set {V = R} is a one-dimensional submanifold. It is a well-known fact that (P) is GUAS if and only if the convexified planar switched system, obtained by replacing u ∈ {0, 1} by u ∈ [0, 1], is GUAS (see [2,6]). In [6] the authors consider the set where the vector fields are colinear. Under generic conditions they give a necessary and sufficient condition for a planar nonlinear switched system to be GUAS, a necessary condition being that the set {x ∈ Z, f 0 (x), f 1 (x) < 0} is empty. This condition turns out to be also sufficient under our hypothesis.
There exists u 0 ∈ [0, 1] such that (1 − u 0 )f 0 (y) + u 0 f 1 (y) = 0. The point y is therefore an equilibrium of the convexified switched system for the input u ≡ u 0 . The convexified system is not GUAS, and neither is System (P).
The subsystems and the Lyapunov function being analytic, we know from Proposition 8 that Ω u (x) is contained in K so that Ω u (x) ⊆ Z. Indeed, for each y ∈ K \ {0}, L f 0 V (y) = L f 1 V (y) = 0 and this is equivalent to f 0 (y), f 1 (y) ∈ ker dV (y). But dV (y) = 0; it follows that f 0 (y) and f 1 (y) are colinear and that K ⊆ Z. In particular, ϕ([0, +∞[) ⊆ Z.
As the next example shows, the previous theorem is no longer true without the analycity hypothesis.

Example 4 Consider the smooth vector field
whose trajectories are circles around the origin, and the closed sets By Borel's Theorem, there exists two smooth functions ϕ i : . Then we consider the vector fields Thus all hypothesis of Theorem 6 are satisfied except analycity of the vector fields. However the switched system is not GUAS since it admits periodic trajectories.

Examples
Example 5 Consider the following analytic vector fields on R 2 , Then V (x) = x 2 1 + x 2 2 is a weak common Lyapunov function which is radially unbounded, But all these points are equilibrium so that Similarly L f 2 V (x) = 0 if and only if x 2 2 − x 3 1 = 0, and For 0 < R < 2, no connected component of {V = R} ∩ (M 1 ∪ M 2 ) intersects M 1 and M 2 . By Theorem 1 the switched system is asymptotically stable for all regular input. Notice that we cannot conclude using centre manifolds because the linear parts of f 1 and f 2 at the origin vanish.
Example 6 Consider the following vector fields on R 2 , The weak Lyapunov function V (x) = x 2 1 + x 2 2 is radially unbounded, and , follows that M 1 = {x 2 = 0} and M 2 = {0}, and that the switched system is globally asymptotically stable for all regular switchings. We could conclude using centre manifolds but the conclusion would be only local. Indeed V 1 = {x 2 = 0} and V 2 is the graph of a map x 2 = −x 2 1 + O(x 4 1 ) (see [7]). It follows that Condition 1 of Theorem 3 is satisfied and that the switched system is asymptotically stable for all regular inputs. However we cannot say anything about the region of attraction.
Example 8 (see [3]) Consider the switched system consisting in the two following vector fields where k is an odd integer. The map V (x) = x 2 1 + x 2 2 is a weak common Lyapunov function. Indeed, Then M 1 and M 2 are contained in the set K 1 = K 2 = {x 2 = 0}.
But each limit set is connected and contained in a level set of V , hence reduced to a point. If u satisfies Assumption H(i) for at least one i ∈ {1, 2} then for each x ∈ R 2 , Ω u (x) consists in an equilibrium point for the vector field f i which implies that Ω u (x) = {0}. So the switched system is globally asymptotically stable for all inputs satisfying H(i) for at least one i ∈ {1, 2}.
Example 9 Consider the switched system consisting in the two following analytic vector fields The function V (x) = x 4 1 +2x 2 2 is a weak analytic common Lyapunov function. Indeed, L f 1 V (x) = −4x 6 1 and L f 2 V (x) = −8x 6 1 .
It follows that K = {x 1 = 0}, so that Condition (K) is not fulfilled. However the LaSalle invariance principle tells us that the vector fields are globally asymptotically stable, and on the other hand, for each x ∈ K, f 1 (x), f 2 (x) = x 2 2 ≥ 0. According to Theorem 6 the switched system is GUAS.
Example 10 Consider the three following analytic vector fields: and the analytic Lyapunov function V (x) = x 2 1 + 2x 2 2 + x 2 3 . It is easy to see that V 1 = {x 3 = 0}, V 2 = {x 1 = x 2 = 0} and V 3 = {x 1 = x 3 = 0} so that the switched system is asymptotically stable for regular inputs by Theorem 3 (or Condition 2 of Corollary 2). We have thus K 1 = V 1 , K 2 = V 2 and K 3 = {x 1 = x 3 }. It follows by Theorem 5 that the switched system is globally asymptotically stable for all inputs u such that J u = {1, 2, 3}.