Tohoku University Policy Design Lab Discussion Paper

TUPD-2021-003

Lyapunov’s direct method for stability of a set andits transitivity under a differential inclusion

Dai ZUSAI
Policy Design Lab and Graduate School of Economics and Management
Tohoku University

May 2021

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Lyapunov’s direct method for stability of a set and its
transitivity under a differential inclusion*
Dai Z USAI†
May 14, 2021

Abstract
We present a version of Lyapunov’s direct method for stability of a set under adifferential inclusion. We pay careful attention to the assumption of forward invariance of a basin of attraction, which is often overlooked when applying the method
to local stability. Even if the value of a local Lyapunoov function monotonicallychanges in some neighborhood of the limit set, this alone does not prevent a trajectory from escaping from this particular neighborhood. In this note, we verify that
we can construct a smaller but forward invariant neighborhood. As a corollary, we
obtain a transitivity theorem on basins of attractions without requiring forward invariance.
Keywords: Lyapunov function, stability of a set, forward invariance, evolutionary
dynamics.

1

Introduction

We present a version of Lyapunov’s direct method for stability of a set under a differential inclusion. Specifically, our version in this paper differs from known versions fordifferential inclusions as i) the limit can be a set, not necessarily a single point as in
Smirnov (2001); Bacciotti and Ceragioli (1999) and ii) a sufficient condition is based on
only local information (derivative with respect to state variable) of a Lyapunov function
at each point around the limit, not requiring information about intertemporal changes inthe value of the Lyapunov function or its time derivative as in Benaı̈m et al. (2005). Our
first motivation for this note is to serve as a handy reference for applications by reducing
the burden of proof and making these points explicit. Our version is particularly usefulfor a practitioner (say, a game theorist) to prove local stability of a set when the basin of* A chat with Bill Sandholm at Stony Book in 2016 led me to paying attention to the assumption ofinvariance and to realizing the merits of theorems in this paper. (I still remember the chat at a parking lotof Hilton Garden Inn after our lunch.) As well as uncountably many inputs and encouragements, I owe somuch to Bill.
† Graduate School of Economics and Management and Policy Design Lab, Tohoku University. E-mail:ZusaiDPublic@gmail.com.

1attraction, which must be forward invariant, is not known for the practitioner. Furthermore, this allows us to verify transitivity of stability of sets from Lyapunov functions.
The idea of Lyapunov stability theorem or Lyapunov’s direct method appeals to intuition: if we find a mapping (Lyapunov function) from the current state of a dynamic toa real number such that i) the function attains a local minimum only at an equilibrium(possibly a set) and ii) its value decreases as long as the current state has not reached
the equilibrium, then the equilibrium is stable under the dynamic. With this idea on
hand, we hope to tell local stability without identifying a solution path; we just find a
Lyapunov function and see how its value changes in a neighborhood of an equilibrium.
So, we typically find a neighborhood where the decrease in the Lyapunov function is
guaranteed, which we call here a monotone decreasing neighborhood; one might expect thisneighborhood to be a basin of attraction.
However, a basin of attraction must be forward invariant. (This does not matterfor global stability, of course.) Even if we find a monotone decreasing neighborhood, asolution path may escape from this neighborhood and eventually the Lyapunov function
may not decrease after the escape. This imposes an additional burden of proof, losing anappeal of the theorem to intuition since we eventually need to identify a solution path.
Similarly, we would expect transitivity of stable sets based on Lyapunov functions.
That is, if we find a Lyapunov function that decreases in X1 and attains the minimumin X and another that decreases in X and attains the minimum in X ∗ , then we expect2

2

X ∗ to be stable in X . Again, the known version of the transitivity theorems as in Con1

ley (1978) requires X1 to be forward invariant and X2 to be forward and also strongly
negative (i.e., backward) invariant.1
In applications to economics or game theory, we hope to find a Lyapunov function
from economic intuition. Under an evolutionary dynamic in a game, an aggregate ofagents’ possible gains from adjustment of their choices can be used as a candidate for a
Lyapunov function once we find a neighborhood where an agent’s revision of the choice
incurs negative payoff externality to others’ gains from further changes; generality of thisproof strategy is verified by Zusai (2020). This neighborhood is a monotone decreasingneighborhood. However, forward invariance needs more mathematical examination of
the dynamic system itself, which may not be appealing to economic intuition.
In this paper, we reduce the burden of proof by showing that we can construct a
forward invariant (smaller) neighborhood from a monotone decreasing neighborhood.
This is a unique issue for local stability and has been overlooked when one applies atheorem on global stability to local stability by analogy; as far as the author finds, thisissue is also not explicitly solved in known proofs as in Smirnov (2001). Our proof fixes
this hole. Further, this helps us to establish a transitivity theorem without requiringforward or negative invariance.
We consider a differential inclusion (a set-valued differential equation) and conver1 Strong negative invariance of X means that, if a solution path (starting at time 0) visits X at any timet ≥ 0, then it must have started from X at time 0.

2

gence to an equilibrium set, not necessarily a point. This generalization is needed tocover evolutionary dynamics in games, since Nash equilibrium may constitute a (connected) set and also a transition vector may not be uniquely specified when there are
multiple best responses.
In the next section, we present our theorems with quick summary of necessary notation and concepts. We discuss applications in evolutionary game theory in Section 3. In
Section 4, this leads to comparison of our theorems with those in preceding literature tohighlight significance of our theorem, exemplified in those applications. The proofs are
given in Section 5.

2

Definitions and theorems

We consider an autonomous differential inclusion V : X ⇒ T X on a compact metric
A-dimensional real space X ⊂ R A with A < ∞, such asẋ ∈ V (x)at each x ∈ X .

T X stands for the tangent space of X .2 As a solution concept for the differential inclusion, we adopt a Carathéodory solution; that is, a solution trajectory {xt }t≥0 must be
Lipschitz continuous at every t ≥ 0 and also differentiable with derivative ẋt ∈ V (xt )at almost every t. To guarantee the existence of a solution trajectory from an arbitrary
point x0 ∈ X , we assume that correspondence (set-valued mapping) V is bounded andupper semicontinuous, with compact and convex values (Smirnov, 2001, Ch. 4).
Let X ∗ be a nonempty closed set. We say X ∗ is Lyapunov stable under V if for any
open neighborhood O of X ∗ there exists a neighborhood O0 of A such that every solution
path {xt }that starts from O0 remains in O.3 X ∗ is attracting if there is a neighborhoodt ≥0

O of X ∗ such that every solution that starts in O converges to X ∗ ; O is called a basin of
attraction to X ∗ . If it is the entire space X , then we say X ∗ is globally attracting. X ∗ isasymptotically stable if it is Lyapunov stable and attracting; it is globally asymptoticallystable if it is Lyapunov stable and globally attracting. In this paper, we prove Theorem 1
and corollary 1.

Lyapunov’s direct method
Theorem 1 (Lyapunov stability theorem). Let X ∗ be a non-empty closed set in a compact
metric space X ⊂ R A with tangent space T X , and X 0 be an open neighborhood of X ∗ . Supposethat continuous function W : X → R and lower semicontinuous function W̃ : X → R satisfy(a) W (x) ≥ 0 and W̃ (x) ≤ 0 for all x ∈ X 0 and (b) cl X 0 ∩ W −1 (0) = cl X 0 ∩ W̃ −1 (0) = X ∗ .
2 Henceforth the definitions follow Sandholm (2010b).

3 This allows a solution path to transit between points in X ∗ . It is indeed a case for the best response

dynamic in a connected set of Nash equilibria.) Because of this, our set-wise stability notion is weaker thanstability of each point in X ∗ .

3

In addition, assume that W is Lipschitz continuous in x ∈ X 0 . If a differential inclusion V :X → T X satisfies4
DW (x)ẋ ≤ W̃ (x)for any ẋ ∈ V (x)(1)

whenever W is differentiable at x ∈ X 0 , then X ∗ is asymptotically stable under V .
W is a Lyapunov function; we call W̃ a decaying rate function and X 0 a monotone
decreasing neighborhood. Note that we allow for multiplicity of transition vectors,
while requiring functions W and W̃ to be well defined (the uniqueness of the values)
as functions of state variable x, independently of the choice of transition vector ẋ fromV ( x ).
Generally speaking, in Lyapunov’s direct method, monotone decrease in the value ofthe Lyapunov function W along with a solution trajectory to zero implies convergenceto zeros of W. In our version, condition (1) guarantees monotone decrease along withany solution path starting from some neighborhood of X ∗ , possibly smaller than X 0 ,only based on local information on the derivative of W at each state x without requiring
explicit identification of the solution trajectory. Notice that we do not assume forward
invariance of X 0 . Thus, it is possible that a solution trajectory may escape from X 0 andthen W may eventually increase; see Fig. 1. Therefore, X 0 itself may not be a basin of
attraction to X ∗ . Yet, our theorem guarantees asymptotic stability of X ∗ since we canconstruct a forward invariant set from a subset of X 0 .
In a standard Lyapunov stability theorem (e.g. Robinson (1998, §5.5.3)) for a differential equation, a decaying rate function W̃ is not explicitly required while Ẇ is assumed tobe (strictly) negative until x reaches the limit set X ∗ . The most significant difference is therequirement of lower semicontinuity of W̃. This assures the existence of a lower boundon the decaying rate Ẇ (x) ≤ w̄ < 0 over any solution trajectories in a hypothetical casein which x remained out of an arbitrarily small neighborhood of X ∗ for an arbitrarily
long period of time. This excludes the possibility that x would stay out there forever andguarantees convergence to X ∗ (not only Lyapunov stability, i.e., no asymptotic escapefrom X ∗ ).

Lyapunov-based transitivity theorem.
Corollary 1 (Transitivity theorem). Let X1 , X2 , and X ∗ be three non-empty subsets of a compact metric space X such that X ⊃ X ⊃ X ∗ ; assume that X ∗ is closed and X is open. Suppose12

1that two Lipschitz continuous functions W1 , W2 : X → R and two lower semicontinuous functions W̃1 , W̃2 : X → R satisfy the following assumptions: for any x ∈ X1 ,a) i) W1 (x) ≥ 0, ii) W̃1 (x) ≤ 0, and iii) cl X1 ∩ W1 −1 (0) = cl X1 ∩ W̃1−1 (0) = cl X2 ;b) i) W2 (x) ≥ 0, ii) x ∈ X2 ⇒ W̃2 (x) ≤ 0 , and iii) cl X2 ∩ W2 −1 (0) = cl X2 ∩ W̃2−1 (0) =X∗;
4 D denotes differentiation, so DW ( x ) = dW/dx ( x ) = [ ∂W/∂x ( x ), . . . , ∂W/∂x ( x )].
1
A

4

W= w1
W= w2

W= w0

~
X '={x : Ẇ ( x)≤ W ( x)}x0X*Ẇ ( x)≤0

Figure 1: A worrisome situation. Here w0 > w1 > w2 and Ẇ is negative in the area
within the red dashed curve (inside toward X ∗ ); X 0 is contained in this area. Alonga green solution trajectory, the value of W decreases until it crosses the red curve but
increases after it and escapes from X ∗ . Notice that, without forward invariance of X 0 ,the solution trajectory leaves it while W is still decreasing.
c) W̃1 (x) + W̃2 (x) ≤ 0.
Furthermore, assume thata-iv) DW1 (x)ẋ ≤ W̃1 (x),b-iv) DW2 (x)ẋ ≤ W̃2 (x)for any ẋ ∈ V (x),whenever W1 and W2 are differentiable at x ∈ X1 . Then, X ∗ is asymptotically stable under V .
One might expect the set of conditions in b) to imply asymptotic stability of X ∗thanks to Theorem 1. But here X2 may not be an open set. In particular, as we will
see in Section 3, we want to cover a case where X2 is just a hyperplane in X as in Figure 2; for example on evolution in a population game, X ∗ can be a set of Nash equilibria
and X2 is a set of states where strictly dominated strategies are not used. Then, X2 is notopen. Of course, X2 may not be invariant; indeed, the known transitivity theorem as inConley (1978) would require an interim set X2 to be both forward and backward (negative) invariant. However, to appeal to intuition, a practitioner may hope to establish
asymptotic stability of X ∗ by first constructing a Lyapunov function on X for stability
1of X2 and then another that would work on the hyperplane X2 .
We fix this gap by imposing c). Since we assume openness of X1 in a), the set of
conditions in a) indeed implies asymptotic stability of cl X2 . W1 stops decreasing oncea solution trajectory reaches X2 , while W2 may have increased before that. So, either
one of them alone is not enough to serve as a Lyapunov function for convergence to X ∗ .
Condition c) essentially implies that a proper linear combination of these two functions
resolves these two issues and serves as a Lyapunov function for the entire region.

5

X
X1

X*

X2

Figure 2: Pseudo-transitivity theorem. Here X ∗ is a singleton of a point on the line
segment X2 . Notice that X2 is not open in X .

3

Applications in game theory

The merit of our theorems lies in allowing a practitioner to verify a Lyapunov function
just by finding a neighborhood of the limit set where the value of the Lyapunov function decreases, without worry about possibilities that a solution trajectory might escapefrom this neighborhood. This reduces a burden of proof significantly and leads to an
intuitively natural proof, as we discuss here.
Stability of an equilibrium has been a fundamental issue in economic theory andgame theory. An economic model typically focuses on the payoff structure, i.e., how payoffs (returns of choices) are determined from choices of agents in the model. Economists
argue that if payoffs negatively respond to perturbation of agents’ choices from equilibrium, it should drive agents to returning to equilibrium. Economists regard such anegative feedback of payoffs as a (sufficient) condition for stability, calling static stability.
By explicitly formulating agents’ processes of revising their choices (that is, an evolutionary dynamic in economic sense, see Sandholm (2010b)), evolutionary game theorists
have attempted to test if such a negative feedback of payoffs indeed assure asymptotic
stability of an equilibrium.
Concrete formulation of static stability differs by specification of a game and the context of analysis. Here we specifically consider a population game in a strategic form, apredominant playground in evolutionary game theory. A population game consists ofa large population of agents, each of which makes an independent decision. In a strategic form, an agent’s decision is a choice of a “strategy” from a given strategy set, say

S = {1, . . . , S}. Each strategy s ∈ S yields payoff Fs (x) ∈ R, depending on distribution of agents over the strategies x = ( xs )s∈S ∈ ∆S := {x = ( xs )s∈S ∈ RS | xs ≥0∀s ∈ S and ∑s xs = 1}. State x∗ ∈ ∆S is a Nash equilibrium if (x − x∗ ) · F(x∗ ) ≤0 for all x ∈ ∆S . Static stability in this setting is conceptualized as a property of contraction of payoff function F = ( Fs )s∈S , namely negative semidefiniteness of its Jacobian
DF: z · DF(x)z ≤ 0 for all z ∈ RS .. A game is called a contractive game if contraction
property holds globally, i.e., at any x ∈ ∆S .
While it is proven for each of canonical evolutionary dynamics such as the best response dynamic and excess payoff dynamics that contraction guarantees asymptotic sta6

bility of the set of Nash equilibria (Hofbauer and Sandholm, 2009), Zusai (2020) presents
a unifying general proof that covers any evolutionary dynamic as long as agents’ revision of strategies is “rationalizable”, i.e., a new strategy after each revision can be explained as a constrained optimal choice, albeit the new strategy may not maximize thepayoff, by introducing a (possibly stochastic) switching cost to rationalize reluctance toswitch and also a (possibly stochastic) restriction to available strategies to rationalize aswitch to a suboptimal strategy. Then, Zusai (2020) defines a net gain from revision by
subtracting the switching cost from payoff improvement by the switch and finds that theaggregate net gain G attains zeros (only) at Nash equilibria and also it is coupled with anegative definite function H : ∆S → R such that5
DG (x)ẋ = H (x) + ẋ · DF (x)ẋ

for any x ∈ ∆S , ẋ ∈ V(x).

Contraction implies DG (x)ẋ ≤ H (x), so we can apply Theorem 1 to G as a Lyapunov
function and H as a decay rate function. Our theorem covers both a global case (i.e., acontraction game) and a local case in which the contraction property holds only in someneighborhood of an equilibrium set. So, this argument provides a single proof for global
stability of the set of Nash equilibria in the former case and local stability of a Nash equilibrium in the latter case. The neighborhood for local contraction is identified only frominformation about the game, especially, F, though a forward invariant neighborhoodneeds identification of a solution trajectory.
Speaking about local stability in a game, an evolutionary stable state (ESS) is theclassical static stability concept in particular to evolutionary game theory. While thecontraction property is locally satisfied at an ESS (or its weaker version, a neutrally stablestate) if it lies in the interior of ∆S , a weaker version of contraction has been considered
to cover the boundary case Taylor and Jonker (1978): x is called a regular ESS if 1)
Fu (x∗ ) < max{ Fs (x∗ ) | s ∈ S}

for any u ∈ U ∗ := {s ∈ S | xs∗ = 0}.

and 2) there exists a neighborhood of x∗ in which each point x satisfies
z · DF(x)z ≤ 0SS∗for any z ∈ RU
∗ : = { z ∈ R | x u = 0 ⇒ z u = 0} .

Here U ∗ is the set of strategies that are unused at x∗ . 1) means that none of those strategies is indeed optimal at x∗ . Condition 2) essentially means contraction but it is weakened to require the contraction only when those strategies in U ∗ are kept unused.
Sandholm (2010a) suggests for pairwise comparison dynamics and the standard best
response dynamic that, once we can somehow obtain a Lyapunov function for a contractive game, we can use it to construct that for a regular ESS. To fill the gap by weakening
the requirement of contraction, he suggests to add (a sufficiently large scalar multiple of)the mass of players of those strategies in U ∗ , i.e., M ∗ (x) := ∑∗ x , to the LyapunovU

5 See Zusai (2020) for an exact definition of G.

7

u ∈U

u

function. By applying Corollary 1 to the combination of G in Zusai (2020) and MU ∗ in
Sandholm (2010a), we can generalize dynamic stability of a regular ESS to a wide classof rationalizable evolutionary dynamics.

4

Literature review

Lyapunov’s direct method
Here we clarify the merit of our own version of the Lyapunov’s direct method theorem
from comparison with preceding versions of the theorem.
As we have repeatedly stated, the technical difference lies in not requiring the priorknowledge on a forward invariance neighborhood. Actually this issue has been overlooked. Lyapunov’s direct method is typically proven for global stability (i.e., condition(1) holds globally); see Bacciotti and Ceragioli (1999) and Clarke et al. (1998, Theorem
5.5) for global stability of a single point. Then, we easily expect its analogous result tohold for a local case. It is not trivial as we discussed and highlighted.—It is indeed amatter when trying to use a transitivity theorem.
When local stability of a set is explicitly argued, a similar condition such thatW ( x t ) ≤ W ( x t0 )whenever t ≥ t0 ≥ 0 and x0 ∈ X(2)Ẇ (xt ) ≤ W̃ (xt )at any t ≥ 0 whenever x0 ∈ X

(3)

or

is imposed; see Benaı̈m et al. (2005, Prop. 3.25) and Zusai (2018, Theorem 7). If X 0 is
forward invariant, then (1) implies these conditions. The current version weakens theassumption of the forward invariance and thus applies to cases in which the neighborhood X is not known a priori. Yet, we retrieve asymptotic stability of X ∗ by obtaining aforward invariant subset of X 0 .
It may be possible to restrict the domain of W and W̃ to X 0 . Then, one might easily
conclude that, with some constant w̄, a lower contour set W −1 ([0, w̄]) should serve as aforward invariant set since condition (1) would imply that W cannot increase as in (2)and (3); see Hale (1980, pp.313–4) for a proof for local stability of an isolated equilibrium
point under a differential equation and Smirnov (2001, Theorem 8.2) for that under adifferential inclusion, both presuming that this argument works. However, this argument presumes that condition (1)always holds along with a solution trajectory to derive(3) from (1) as argued above. This argument still misses the possibility that a trajectory
starting from X 0 may escape from X 0 . We are questioning this presumption and fix thehole, while also extending pointwise stability as in Smirnov (2001) to set-wise stability.

8Transitivity theorem
One might (wrongly) argue that conditions a) and b) in Corollary 1 imply asymptotic
stability of X with X as a monotone decreasing neighborhood and that of X ∗ with X ,2

2

1

by applying only Theorem 1. From this chain relationship, one might want to use thestandard transitivity theorem as in Conley (1978) as we argued in Section 1. However,Conley’s theorem requires both forward and strongly negatively invariance of interim
set X2 . However, a monotone decreasing neighborhood may not be invariant and thus
we cannot apply Conley’s theorem. Furthermore, we do not assume openness of theinterim set X2 ; in the application to a regular ESS, it is indeed a hyperplane defined by∑u∈U ∗ xu = 0. The openness of monotone decreasing set X 0 is crucial for our Theorem
1, since we eventually construct a smaller forward invariant set within the monotone
decreasing set.6 So, we cannot apply Theorem 1 to obtain asymptotic stability of X ∗from condition b). We resolve these issues by reconstructing a Lyapunov function.

5

Proofs

Proof of Theorem 1
Proof. Here we prove the difference from Zusai (2018, Theorem 7), which assumes (3).
For this, we focus on the case of X 0 ( X and find a forward invariant subset of X 0 . Once
we find it, any Carathéodory solution starting from the forward invariant subset remains
there and thus satisfies (3) as (1) holds for x = xt at each time t ∈ R+ . Then, Zusai (2018,
Theorem 7) is applied and assures asymptotic stability of X ∗ while having the forward
invariant subset as a basin of attraction.
First, construct a distance from point x ∈ X to X ∗ based on the metric on X , say
d : X × X → R+ , byd∗ (x) := min d(x, x∗ ).
x∗ ∈ X ∗Since X ∗ is a non-empty compact set and d(x, x∗ ) is continuous in x∗ when x is fixed,
Weierstrass theorem assures the existence of the minimum in the above definition ofd∗ (x). This d∗ satisfiesd∗ (x) ≥ 0;d∗ (x) = 0 ⇔ x ∈ X ∗ .

Let d¯ be the shortest distance from the complement of X 0 to X ∗ :d¯ := min d∗ (x).
x∈X \ X 0

(4)

6 Specifically, in our proof, we use openness of monotone decreasing neighborhood X 0 to confirm theexistence and (nonzero) positiveness of d,¯ i.e., the shortest distance from X ∗ to get out of X 0 . In case X 0 is
not open, we could define d¯ by replacing minimum with infimum. But this may be zero especially if X 0 is
only a hyperplane.

9

X'
d

cl

W=w
X'0

X''
X*
d/2

xT’

x0

xT

x T̄

Figure 3: Sets in the proof of Theorem 1. X ∗ is the black area in the center and X 00 is thelight gray area. cl X 0 is the entire oval, with the outermost outline. X00 is the dark grayarea, including the both boundaries.
Maximum theorem guarantees continuity of d∗ : X → R+ by continuity of d(x, x∗ )in both x and x∗ . Besides, X \ X 0 is a non-empty compact subset by X 0 ( X and theopenness of X 0 . Hence, the minimum in (4) exists. It follows thatd∗ (x) < d¯ ⇒ x ∈ X 0 .

d¯ > 0;(5)

Define set X00 ⊂ cl X 0 by¯ ∞)).
X00 := cl X 0 ∩ d∗ −1 ([d/2,¯ ∞)) are closed, X 0 is closed and thus compact in X . It
Since both cl X 0 and d∗ −1 ([d/2,0¯is not empty, as proven here. Suppose X 0 = ∅; then, any x ∈ X with d∗ (x) ≥ d/2
0must be out of cl X 0 . On the other hand, since cl X 0 is not empty, X 0 has at least one
boundary point x0 ; then, d∗ (x0 ) ≥ d.¯ 7 By the former statement, this implies x0 ∈/ cl X 0
but it contradicts with x0 being on the boundary of X 0 ; hence, X00 cannot be empty.
Let w̄ be the minimum of W in X00 ;w̄ := min0 W (x).
x ∈ X0Since X00 is compact and nonempty and W is (Lipschitz) continuous, the minimum exists.
Furthermore, it is positive; we have X00 ⊂ cl X 0 by construction and W (x) ≥ 0 for allx ∈ cl X 0 by condition (a) and continuity of W, while no element x ∈ X 0 belongs to X ∗0since d∗ (x) ≥ d¯ > 0 for any x ∈ X00 . Because X ∗ = cl X 0 ∩ W −1 (0) by condition (b) and
X00 ⊂ cl X 0 , this implies x ∈ X00 ⇒ W (x) > 0. Hence we have w̄ > 0. By the definition of
7 We can make a sequence converging to x0 from elements out of X 0 , whose distance from X ∗ cannot be
smaller than d¯ by (4).

10w̄, we have¯x ∈ cl X 0 and d∗ (x) ≥ d/2⇒ W (x) ≥ w̄.
|
{z
}



(6)

i.e., x∈ X00

Define set X 00 ⊂ X 0 by
X 00 = W −1 ([0, w̄) ∩ X 0 .

(7)This set is an open neighborhood of X ∗ by X ∗ ⊂ X 00 , since W = 0 at anywhere in X ∗1¯and X ∗ ⊂ X 0 . Now we prove that X 00 is wholly contained in set d−∗ ([0, d/2)). Assume00¯
that there exists x ∈ X such that d∗ (x) ≥ d/2.
These jointly imply W (x) ≥ w̄ by (6)0000since x ∈ X ⊂ X ⊂ cl X . However, this contradicts with W (x) ∈ [0, w̄) for x to belong
to X 00 . Hence, we have¯x ∈ X 00 ⇒ d∗ (x) < d/2.

(8)

Now we prove X 00 is forward invariant; remind that X 0 may not be forward invariant
and thus it is not trivial to confirm that a trajectory from X 00 stays in X 0 . To verify it bycontradiction, assume that there is a Carathéodory solution trajectory {xt } starting from
X 00 but escaping X 00 at some moment of time:x0 ∈ X 00 ,and x T ∈/ X 00 at some T > 0.

(9)

The statement x T ∈/ X 00 means x T ∈/ X 0 or W (x T ) ≥ w̄ by (7). In the former case, we have¯ by (8). By continuity of d∗ (x) in x and of xt in td∗ (x T ) ≥ d¯ by (4) while d∗ (x0 ) < d/2
on a Carathéodory solution trajectory {xt }, d∗ (xt ) is continuous in t; hence, there exists
a moment of time T 0 ∈ (0, T ) such that d∗ (x T 0 ) = 0.9d¯ ∈ (0.5d,¯ d¯) ⊂ (d∗ (x0 ), d∗ (x T )). At
this point, x T 0 ∈/ X 00 by (8) while x T 0 ∈ X 0 by (5); thus, W (x T 0 ) ≥ w̄ by (7). Hence, the firstcase of escaping X 00 implies the existence of T 0 > 0 such that
W (x T 0 ) ≥ w̄and

xT0 ∈ X 0 .

In the second (but not the first) case, we have W (x T ) ≥ w̄ but x T ∈ X 0 ; that is, the above
statement holds with T 0 = T.
The above two conditions on T 0 implies the existence of T̄ ∈ (0, T 0 ] such that
W (x T̄ ) ≥ w̄,



and


xt ∈ X 0 for all t < T̄ .

(10)

To prove it, assume xt0 ∈/ X 0 at some t0 < T 0 , i.e., the negation of the latter condition
with T̄ = T 0 ; if there is no such t0 ≤ T 0 , then it suggests that the claim (10) holds at
T̄ = T 0 by the fact W (x T 0 ) ≥ w̄. By (5), the hypothesis xt0 ∈/ X 0 implies d∗ (xt0 ) ≥ d.¯
Again, by continuity of d∗ (xt ) in t, the set {t ≤ t0 | d∗ (xt ) ≥ d¯} is closed and thus
compact. This implies the existence of the minimum T̄ in this set, and further T̄ > 0¯
by the fact d∗ (x0 ) < d/2.
That is, we have d∗ (xt ) < d¯ for all t < T̄ while d∗ (x ) = d.¯T̄The former implies xt ∈ X 0 for all t < T̄ by (5) and the latter implies W (x T̄ ) ≥ w̄ by

11

x T̄ = limt→T̄ xt ∈ cl X 0 and (6). Thus, the above claim (10) holds at this T̄ ∈ (0, T 0 ].
Since condition (a) and (1) hold almost everywhere in X 0 , we have Ẇ (xτ ) ≤ W̃ (xτ ) ≤0 at almost all τ < T̄;8 thus, we haveW (x T̄ ) ≤ W (x0 ) +Z T̄
0W̃ (xτ )dτ ≤ W (x0 ).

Since W (x0 ) < w̄ by x0 ∈ X 00 , we have W (x T̄ ) < w̄ in (10). This contradicts with W (x T̄ ) ≥w̄.
Therefore, the hypothesis (9) cannot hold: any Carathéodory solution trajectory {xt }starting from X 00 cannot escape X 00 at any moment of time. That is, X 00 is forward invariant.

Proof of Corollary 1
Proof. Define a Lyapunov function W : X → R and a decaying rate function W̃ : X → R
byW (x) := 2W1 (x) + W2 (x),W̃ (x) := 2W̃1 (x) + W̃2 (x)for each x ∈ X1 .

Lipschitz continuity of W1 and W2 and lower semicontinuity of W̃1 and W̃2 are succeeded
to those of W and W̃, respectively. It is immediate from assumptions a-i,iv), b-i,iv) andc) to see that
W (x) = 2W1 (x) + W2 (x) ≥ 0,W̃ (x) = W̃1 (x) + {W̃1 (x) + W̃2 (x)} ≤ 0,
DW (x)ẋ = 2DW1 (x)ẋ + DW2 (x)ẋ ≤ 2W̃1 (x) + W̃2 (x) = W̃ (x)(11)

for any x ∈ X1 , ẋ ∈ V (x) (for the last equation assuming that W1 and W2 are differentiable at x).
Further, since X ∗ ⊂ X2 , it follows assumptions a-iii) and b-iii) that W (x) = W̃ (x) = 0
if x ∈ X ∗ ; thus X ∗ is contained in cl X ∩ W −1 (0) and cl X ∩ W̃ −1 (0) by X ∗ ⊂ X ⊂1

1

2

X1 ⊂ cl X1 . In contrary, assume W (x) = 0 at x ∈ cl X1 first. By assumptions a-i) andb-i), it must be the case that W1 (x) = 0 and W2 (x) = 0. The former implies x ∈ cl X2
by assumption a-iii). Together with this, the latter implies x ∈ X ∗ by assumption b-iii).
Separately from this, now assume W̃ (x) = 0 at x ∈ cl X1 . By assumptions a-ii) andc), it must be the case that W̃1 (x) = 0 and W̃1 (x) + W̃2 (x) = 0.9 The former impliesx ∈ cl X2 by assumption a-iii); besides, by plugging the former into the latter, we haveW̃ (x) = 0. These two statements jointly imply x ∈ X ∗ by assumption b-iii). In sum, we
2

8 A Carathéodory solution trajectory is differentiable at almost all moments of time, though it may not

be so at all moments.
9 Note that the latter condition alone cannot assure W̃ ( x ) = 0, since W̃ ( x ) could take a positive value
22
unless x is in X2 .

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have verified
cl X1 ∩ W −1 (0) = cl X1 ∩ W̃ −1 (0) = X ∗ .

(12)

Note that the first equality is due to the fact that X ∗ ⊂ X1 and thus X ∗ ∩ bd X1 = ∅since X1 is open.

We have verified all the assumptions in Theorem 1; therefore, X ∗ is asymptoticallystable. Notice that X1 may not be forward invariant, but part i) of Theorem 1 assures that
we can make some subset of X as a basin of attraction to X ∗ .
1

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