# Asymptotic Behavior of a Class of Evolution Variational Inequalities

Authors: Ailing Qi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We establish a new LaSalle's invariance principle and discuss the asymptotic behavior of a class of first-order evolution variational inequalities.


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## Body

## 1.  Introduction

Nonsmooth systems, roughly speaking, are those systems whose trajectories may not be differentiable everywhere. Usually nonsmooth dynamical systems are represented as differential inclusions, complementarity systems, evolution variational inequalities, and so on [1]. Since they play important roles in numerous fields, there appeared an increasing interest in the study of their dynamics in recent years.In this paper, we consider a class of typical nonsmooth dynamical systems given by the following first-order evolution variational inequalities:(1.1)〈dx(t)dt+f(x(t)),v-x(t)〉+φ(v)-φ(x(t))≥0,∀v∈Rn. It is known that many important mechanical systems arising from applications can be transformed into an variational inequality as above. In case φ is a proper convex and lower semicontinuous function from ℝn to ℝ1∪{+∞} and f is a continuous operator with f+ωI being monotone for some ω≥0, Adly and Goeleven [2] made a systematic study on the asymptotic behavior of the system (1.1). The existence and uniqueness of solutions were established, and the asymptotic behavior was discussed.In this present work, we are basically interested in the case wheref is only continuous. On the other hand, to avoid some technical difficulties, we will always assume φ is a proper convex and lower semicontinuous function from ℝn to ℝ1. Note that, in our case, (1.1) may fail to have uniqueness. The main purpose is to establish a LaSalle’s invariance principle and discuss asymptotic stability of the equilibria of the system.LaSalle’s invariance principle plays a key role in stability analysis and control. In the past decades, there appeared many important extensions. Results closely related to ours can be found in [2–6] and so forth.This paper is organized as follows. In Section2, we provide some basic definitions and auxiliary results. In Section 3, we develop a LaSalle’s invariance theorem and discuss the strong stability and strong asymptotic stability of the system.


## 2.  Preliminaries

This section is concerned with some preliminary works. For convenience, we will denote by〈·,·〉 the usual inner product in ℝn with the corresponding norm ∥·∥.


### 2.1.  Subdifferential

LetV be a function from ℝn to ℝ1. For x,v∈ℝn, define(2.1)DvV(x)=lim⁡h→0+V(x+hv)-V(x)h∈[-∞,+∞].DvV(x) is said to be the derivative ofVatxin the directionv.IfDvV(x) exists for all directions v, we say that V is differentiable at x.Definition 2.1.
The following closed convex subset (possibly empty)(2.2)∂V(x):={p∈Rn∣∀v∈Rn,〈p,v〉≤DvV(x)}
is called the subdifferential of V at x, and we say that the elements p of ∂V(x) are the subgradients of V at x.It is known that ifV is differentiable at x in the classical sense, then(2.3)∂V(x)={∇V(x)}.Proposition 2.2 (see [7]).
IfV is a convex function from ℝn to ℝ1, then, for each fixed x, the mapping v→DvV(x) is convex and positively homogeneous with the following inequalities hold:
(2.4)-∞≤V(x)-V(x-v)≤DvV(x)≤V(x+v)-V(x)≤+∞.
Furthermore,
(2.5)∂V(x)={p∈Rn∣V(x)-V(y)≤〈p,x-y〉,∀y∈Rn}.Proposition 2.3 (see [7]).
Let one assume thatV:ℝn→ℝ1 is convex and lower semicontinuous. Then,(a)
for eachx, ∂V(x) is a nonempty and bounded set,(b)
the mapping(x,v)∈ℝn×ℝn→DvV(x) is upper semicontinuous (and thus ∂V(·) is upper semicontinuous as well),(c)
the following regularity property holds:(2.6)DvV(x)=limsup⁡h→0+y→xV(y+hv)-V(y)h.


### 2.2.  Some Basic Facts on the Evolution Variational Inequality

Letφ:ℝn→ℝ be a convex and lower semicontinuous function, and let f:ℝn→ℝn be a continuous vector field. Consider the following evolution variational inequality.(VP)
For any givenx0∈ℝn, find a x(·)∈C([0,T);ℝn) with dx/dt∈Lloc∞(0,T;ℝn), such that
(2.7)〈dx(t)dt+f(x(t)),v-x(t)〉+φ(v)-φ(x(t))≥0,∀v∈Rn,a.e.t≥0,x(0)=x0.Thanks to Proposition2.2, one can easily rewrite (2.7) as the initial value problem of a differential inclusion:(2.8)dx(t)dt+f(x(t))∈-∂φ(x(t)),x(0)=x0.
By Definition 2.1 and Proposition 2.3, we see that, for each x∈ℝn, ∂φ(x) is a nonempty compact and convex subset of ℝn; moreover, the multifunction ∂φ(·) is upper semicontinuous in x. This guarantees by the basic theory on differential inclusions (see, e.g., [7–9], etc.) the local existence of solutions for (2.7).Letx(·) be a solution to differential inclusion (2.7) defined on [0,+∞). Then, the ω-limit setω(x(·)) is defined as(2.9)ω(x(⋅)):={y∈Rn∣∃tn⟶+∞suchthatx(tn)⟶y}.We infer from [10] that the following basic facts on ω-limit sets hold.Proposition 2.4.
If a solutionx(·) of (2.7) is bounded on [0,+∞), then ω(x(·)) is a nonempty compact weakly invariant set, namely, for each y∈Ω, there is a complete solution x(·) on ℝ1 which is contained in ω(x(·)) with x(0)=y. Moreover,
(2.10)lim⁡t→+∞d(x(t),ω(x(⋅)))=0.


## 2.1.  Subdifferential

LetV be a function from ℝn to ℝ1. For x,v∈ℝn, define(2.1)DvV(x)=lim⁡h→0+V(x+hv)-V(x)h∈[-∞,+∞].DvV(x) is said to be the derivative ofVatxin the directionv.IfDvV(x) exists for all directions v, we say that V is differentiable at x.Definition 2.1.
The following closed convex subset (possibly empty)(2.2)∂V(x):={p∈Rn∣∀v∈Rn,〈p,v〉≤DvV(x)}
is called the subdifferential of V at x, and we say that the elements p of ∂V(x) are the subgradients of V at x.It is known that ifV is differentiable at x in the classical sense, then(2.3)∂V(x)={∇V(x)}.Proposition 2.2 (see [7]).
IfV is a convex function from ℝn to ℝ1, then, for each fixed x, the mapping v→DvV(x) is convex and positively homogeneous with the following inequalities hold:
(2.4)-∞≤V(x)-V(x-v)≤DvV(x)≤V(x+v)-V(x)≤+∞.
Furthermore,
(2.5)∂V(x)={p∈Rn∣V(x)-V(y)≤〈p,x-y〉,∀y∈Rn}.Proposition 2.3 (see [7]).
Let one assume thatV:ℝn→ℝ1 is convex and lower semicontinuous. Then,(a)
for eachx, ∂V(x) is a nonempty and bounded set,(b)
the mapping(x,v)∈ℝn×ℝn→DvV(x) is upper semicontinuous (and thus ∂V(·) is upper semicontinuous as well),(c)
the following regularity property holds:(2.6)DvV(x)=limsup⁡h→0+y→xV(y+hv)-V(y)h.


## 2.2.  Some Basic Facts on the Evolution Variational Inequality

Letφ:ℝn→ℝ be a convex and lower semicontinuous function, and let f:ℝn→ℝn be a continuous vector field. Consider the following evolution variational inequality.(VP)
For any givenx0∈ℝn, find a x(·)∈C([0,T);ℝn) with dx/dt∈Lloc∞(0,T;ℝn), such that
(2.7)〈dx(t)dt+f(x(t)),v-x(t)〉+φ(v)-φ(x(t))≥0,∀v∈Rn,a.e.t≥0,x(0)=x0.Thanks to Proposition2.2, one can easily rewrite (2.7) as the initial value problem of a differential inclusion:(2.8)dx(t)dt+f(x(t))∈-∂φ(x(t)),x(0)=x0.
By Definition 2.1 and Proposition 2.3, we see that, for each x∈ℝn, ∂φ(x) is a nonempty compact and convex subset of ℝn; moreover, the multifunction ∂φ(·) is upper semicontinuous in x. This guarantees by the basic theory on differential inclusions (see, e.g., [7–9], etc.) the local existence of solutions for (2.7).Letx(·) be a solution to differential inclusion (2.7) defined on [0,+∞). Then, the ω-limit setω(x(·)) is defined as(2.9)ω(x(⋅)):={y∈Rn∣∃tn⟶+∞suchthatx(tn)⟶y}.We infer from [10] that the following basic facts on ω-limit sets hold.Proposition 2.4.
If a solutionx(·) of (2.7) is bounded on [0,+∞), then ω(x(·)) is a nonempty compact weakly invariant set, namely, for each y∈Ω, there is a complete solution x(·) on ℝ1 which is contained in ω(x(·)) with x(0)=y. Moreover,
(2.10)lim⁡t→+∞d(x(t),ω(x(⋅)))=0.


## 3.  LaSalle’s Invariance Principle

We are now ready to establish a LaSalle’s invariance principle for (2.7). For convenience, we will denote by EΩ(f,φ,V) the set(3.1)EΩ(f,φ,V):={x∈Ω∣〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))=0}.
In case Ω=ℝn, we simply write EΩ(f,φ,V) as E(f,φ,V).


### 3.1.  Invariance Principle

In this subsection, we provide a LaSalle’s invariance principle for the system (2.7) involving a mapping f that is only assumed continuous. The approach followed by Adly and Goeleven [2] has been proved with f being continuous and f+ωI monotone.The main result is contained in the following theorem. The weak invariance ofω-limit set plays an important role in the proof of the theorem.Theorem 3.1.
LetΩ⊂ℝn be closed. Assume that there exists V∈C1(Ω;ℝ) such that
(3.2)〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0,x∈Ω.
Let ℳΩ be the largest weakly invariant set of EΩ(f,φ,V).
Then, for eachx0∈Ω and each bounded solution x(·) of (2.7) in Ω, we have(3.3)lim⁡t→+∞d(x(t),MΩ)=0.Proof.
For eachp∈-∂φ(x), by Proposition 2.2, we find that
(3.4)〈p,x-y〉+φ(x)-φ(y)≤0,∀y∈Rn.
Taking y=x-∇V(x) in (3.4), one gets
(3.5)〈∇V(x),p〉≤φ(x-∇V(x))-φ(x).
Equation (3.2) then implies
(3.6)φ(x-∇V(x))-φ(x)≤〈f(x),∇V(x)〉.
Now, by (3.5) and (3.6), we deduce that
(3.7)max⁡p∈-∂φ(x)〈∇V(x),p〉≤〈∇V(x),f(x)〉,x∈Ω.
Set(3.8)E:={x∈Ω∣〈∇V(x),p-f(x)〉=0forsomep∈-∂φ(x)}.
Denote by ℳ the largest weakly invariant set of E. In the following, we will check that, for each x0∈Ω and each bounded solution x(·) of (2.7) in Ω, we have
(3.9)lim⁡t→+∞d(x(t),M)=0.
By Proposition2.4, we know that ω(x(·)) is a nonempty compact weakly invariant set, and(3.10)lim⁡t→+∞d(x(t),ω(x(⋅)))=0.
To prove (3.9), it suffices to check that
(3.11)ω(x(⋅))⊂M.
Note thatΩ is closed, we have ω(x(·))⊂Ω. Since for every y∈ω(x(·)), by the definition of ω-limit set, there is tn→+∞ such that x(tn)→y. Here, x(·) is a bounded solution of (2.7) in Ω, and Ω is closed, hence y∈Ω.
In what follows we first show that(3.12)V(y)≡const.,y∈ω(x(⋅)).
Indeed, by (2.8) and (3.7), we see that
(3.13)dV(x(t))dt=〈∇V(x(t)),ẋ(t)〉≤max⁡p∈-∂φ(x(t))〈∇V(x(t)),p-f(x(t))〉≤0,a.e.t≥0.
It then follows from the proof of Lemma  2 in [2] that V is nonincreasing on [0,+∞). Moreover, V is bounded from below on [0,+∞) since x([0,+∞))⊂Ω and V is continuous on the closed set Ω. This provides with an existence of the limit of V. Hence,
(3.14)lim⁡t→+∞V(x(t)):=λ
exists.
For eachy∈ω(x(·)), by definition of ω-limit set, there exists tn→+∞ such that(3.15)x(tn)⟶y,n⟶+∞.
Further, by continuity, we deduce that
(3.16)V(y)=V(lim⁡n→+∞x(tn))=lim⁡n→+∞V(x(tn))=λ.
This verifies the validity of (3.12).
Now, we check thatω(x(·))⊂E. Let y0∈ω(x(·)). As ω(x(·)) is weakly invariant, there exists a complete solution y(·) starting from y0 with y(t)∈ω(x(·)) for all t≥0. By what we have just proved, it holds that(3.17)V(y(t))≡λ,∀t≥0.
Take a sequence tn→0 such that y(t) is differentiable at each tn with
(3.18)ẏ(tn)+f(y(tn))∈-∂φ(y(tn)).
Then,
(3.19)dV(y(tn))dt=〈∇V(y(tn)),ẏ(tn)〉=0.
This implies that
(3.20)〈∇V(y(tn)),p-f(y(tn))〉=0,forsomep∈-∂φ(y(tn)).
Thus, one deduces that y(tn)∈E. By continuity of y(·), we know that y0∈E. This proves what we desired, and (3.11) follows directly from the weak invariance of ω(x(·)).
Finally, we verify thatE⊂EΩ(f,φ,V), which implies ℳ⊂ℳΩ and completes the proof of the theorem. Let x∈E. Then, by (3.8), we have(3.21)〈∇V(x),p〉=〈∇V(x),f(x)〉,forsomep∈-∂φ(x).
Invoking (3.5) and (3.6), we find that
(3.22)〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))=0.
Thus, x∈EΩ(f,φ,V).As a particular case of Theorem3.1, we have the following.Theorem 3.2.
Assume that there existsV∈C1(ℝn;ℝ1) such that(1)
〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0, x∈ℝn,(2)
V(x)→+∞ as ∥x∥→+∞, x∈ℝn. Denote by ℳ the largest weakly invariant set of E(f,φ,V).
Then, for eachx0∈ℝn, every solution x(·) of (2.7) is bounded and(3.23)lim⁡t→+∞d(x(t),M)=0.Proof.
Letx0∈ℝn be given. We set
(3.24)Ω:={x∈Rn∣V(x)≤V(x0)}.
Then, by assumption (2), we see that Ω is a bounded closed subset of ℝn. We infer from the proof of Theorem 3.1 that V is decreasing along any solution of (2.7). Thus, Ω is actually positively invariant. Therefore, by Theorem 3.1, one concludes that
(3.25)lim⁡t→∞d(x(t),MΩ)=0
for each solution x(·), where ℳΩ is the largest weakly invariant subset of E(f,φ,V)∩Ω. Clearly, ℳΩ⊂ℳ, and the conclusion follows.


### 3.2.  Asymptotic Stability of Equilibria

As simple applications of the LaSalle’s invariance principle established above, we make some further discussions on the asymptotic behavior of the system (1.1). For this purpose, we denote by ℰ(f,φ) the set of stationary solutions to (1.1), that is,(3.26)E(f,φ):={z∈Rn∣〈f(z),v-z〉+φ(v)-φ(z)≥0,∀v∈Rn}.
In what follows, we will always assume that(3.27)f(0)∈-∂φ(0),
so that 0∈ℰ(f,φ) is the trivial stationary solution of (1.1).Let us first prove the strong stability of the trivial stationary solution 0. Forr>0, we denote by B¯r the closed ball of radius r,(3.28)B¯r:={x∈Rn:‖x‖≤r}.Theorem 3.3.
Suppose that there existsσ>0 and V∈C1(B¯σ;ℝ) such that(1)
V(x)≥a(∥x∥) for all x∈B¯σ, where a∈C([0,σ]) satisfies a(t)>0  (∀t∈(0,σ)),(2)
V(0)=0,(3)
〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0, x∈B¯σ. Then, the stationary solution 0 is strongly stable, that is, for any ɛ>0, there exists a δ>0 such that for any solution x(t) of (1.1) with ∥x(0)∥≤δ, one has
(3.29)‖x(t)‖≤ɛ,∀t≥0.Proof.
For anyδ<σ, let m be the minimum value of V on the boundary of B¯δ. Then, by assumptions (1) and (2), we find that m>0. Let
(3.30)U={x∈B¯δ∣V(x)≤m}.
It is clear that U is a neighborhood of 0. Since V is decreasing along each solution in B¯δ, one trivially checks that U is strongly positive invariant. This implies the desired result.Propositions3.4–3.6 below can be proved by the same arguments as the ones in [2]. We omit the details.Proposition 3.4.
Suppose thatΩ is a subset of ℝn, and there exists V∈C1(Ω;ℝ) such that
(3.31)〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0,x∈Ω.
Then,
(3.32)(E(f,φ)∩Ω)⊂EΩ(f,φ,V).Proposition 3.5.
Suppose that there exist aσ>0 and V∈C1(B¯σ;ℝ) such that(1)
〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0, x∈B¯σ,(2)
E(f,φ,V)∩B¯σ={0}. Then, the stationary solution 0 is isolated in ℰ(f,φ).Proposition 3.6.
Suppose that there existsV∈C1(ℝn;ℝ) such that
(3.33)〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0,x∈Rn.
Then, ℰ(f,φ)⊂E(f,φ,V).Now, we can easily prove the following result.Theorem 3.7.
Suppose that there existsσ>0 and V∈C1(B¯σ;ℝ) such that (1)–(3) in Theorem 3.3 hold; moreover,
(3.34)E(f,φ,V)∩B¯σ={0}.
Then, the trivial stationary solution 0 is strongly asymptotically stable.Proof.
The strong stability is readily implied in Theorem3.3. Define
(3.35)Ω={x∈B¯σ∣V(x)≤m}.
Then, as in the proof of Theorem 3.3, we know that Ω is a strongly positively invariant neighborhood of 0. Applying Theorem 3.1, we deduce that, for x0∈Ω,
(3.36)lim⁡t→+∞d(x(t),M)=0.
On the other hand, by (3.34), we see that ℳ={0}. Therefore, the trivial stationary solution 0 is strongly asymptotically stable.Theorem 3.8.
Suppose that there existsV∈C1(ℝn;ℝ) such that(1)
V(x)≥a(∥x∥) for all x∈ℝn, where a:ℝ+→ℝ is a continuous strictly increasing function with a(0)=0,(2)
V(0)=0,(3)
〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0, x∈ℝn,(4)
E(f,φ,V)={0}. Then, the trivial stationary solution to (2.7) is globally strongly asymptotically stable.Proof.
The strong asymptotic stability can be directly deduced from Theorem3.8. Repeating the same argument as in Theorem 3.2, we can show that each solution of the system approaches ℳ={0}.


## 3.1.  Invariance Principle

In this subsection, we provide a LaSalle’s invariance principle for the system (2.7) involving a mapping f that is only assumed continuous. The approach followed by Adly and Goeleven [2] has been proved with f being continuous and f+ωI monotone.The main result is contained in the following theorem. The weak invariance ofω-limit set plays an important role in the proof of the theorem.Theorem 3.1.
LetΩ⊂ℝn be closed. Assume that there exists V∈C1(Ω;ℝ) such that
(3.2)〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0,x∈Ω.
Let ℳΩ be the largest weakly invariant set of EΩ(f,φ,V).
Then, for eachx0∈Ω and each bounded solution x(·) of (2.7) in Ω, we have(3.3)lim⁡t→+∞d(x(t),MΩ)=0.Proof.
For eachp∈-∂φ(x), by Proposition 2.2, we find that
(3.4)〈p,x-y〉+φ(x)-φ(y)≤0,∀y∈Rn.
Taking y=x-∇V(x) in (3.4), one gets
(3.5)〈∇V(x),p〉≤φ(x-∇V(x))-φ(x).
Equation (3.2) then implies
(3.6)φ(x-∇V(x))-φ(x)≤〈f(x),∇V(x)〉.
Now, by (3.5) and (3.6), we deduce that
(3.7)max⁡p∈-∂φ(x)〈∇V(x),p〉≤〈∇V(x),f(x)〉,x∈Ω.
Set(3.8)E:={x∈Ω∣〈∇V(x),p-f(x)〉=0forsomep∈-∂φ(x)}.
Denote by ℳ the largest weakly invariant set of E. In the following, we will check that, for each x0∈Ω and each bounded solution x(·) of (2.7) in Ω, we have
(3.9)lim⁡t→+∞d(x(t),M)=0.
By Proposition2.4, we know that ω(x(·)) is a nonempty compact weakly invariant set, and(3.10)lim⁡t→+∞d(x(t),ω(x(⋅)))=0.
To prove (3.9), it suffices to check that
(3.11)ω(x(⋅))⊂M.
Note thatΩ is closed, we have ω(x(·))⊂Ω. Since for every y∈ω(x(·)), by the definition of ω-limit set, there is tn→+∞ such that x(tn)→y. Here, x(·) is a bounded solution of (2.7) in Ω, and Ω is closed, hence y∈Ω.
In what follows we first show that(3.12)V(y)≡const.,y∈ω(x(⋅)).
Indeed, by (2.8) and (3.7), we see that
(3.13)dV(x(t))dt=〈∇V(x(t)),ẋ(t)〉≤max⁡p∈-∂φ(x(t))〈∇V(x(t)),p-f(x(t))〉≤0,a.e.t≥0.
It then follows from the proof of Lemma  2 in [2] that V is nonincreasing on [0,+∞). Moreover, V is bounded from below on [0,+∞) since x([0,+∞))⊂Ω and V is continuous on the closed set Ω. This provides with an existence of the limit of V. Hence,
(3.14)lim⁡t→+∞V(x(t)):=λ
exists.
For eachy∈ω(x(·)), by definition of ω-limit set, there exists tn→+∞ such that(3.15)x(tn)⟶y,n⟶+∞.
Further, by continuity, we deduce that
(3.16)V(y)=V(lim⁡n→+∞x(tn))=lim⁡n→+∞V(x(tn))=λ.
This verifies the validity of (3.12).
Now, we check thatω(x(·))⊂E. Let y0∈ω(x(·)). As ω(x(·)) is weakly invariant, there exists a complete solution y(·) starting from y0 with y(t)∈ω(x(·)) for all t≥0. By what we have just proved, it holds that(3.17)V(y(t))≡λ,∀t≥0.
Take a sequence tn→0 such that y(t) is differentiable at each tn with
(3.18)ẏ(tn)+f(y(tn))∈-∂φ(y(tn)).
Then,
(3.19)dV(y(tn))dt=〈∇V(y(tn)),ẏ(tn)〉=0.
This implies that
(3.20)〈∇V(y(tn)),p-f(y(tn))〉=0,forsomep∈-∂φ(y(tn)).
Thus, one deduces that y(tn)∈E. By continuity of y(·), we know that y0∈E. This proves what we desired, and (3.11) follows directly from the weak invariance of ω(x(·)).
Finally, we verify thatE⊂EΩ(f,φ,V), which implies ℳ⊂ℳΩ and completes the proof of the theorem. Let x∈E. Then, by (3.8), we have(3.21)〈∇V(x),p〉=〈∇V(x),f(x)〉,forsomep∈-∂φ(x).
Invoking (3.5) and (3.6), we find that
(3.22)〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))=0.
Thus, x∈EΩ(f,φ,V).As a particular case of Theorem3.1, we have the following.Theorem 3.2.
Assume that there existsV∈C1(ℝn;ℝ1) such that(1)
〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0, x∈ℝn,(2)
V(x)→+∞ as ∥x∥→+∞, x∈ℝn. Denote by ℳ the largest weakly invariant set of E(f,φ,V).
Then, for eachx0∈ℝn, every solution x(·) of (2.7) is bounded and(3.23)lim⁡t→+∞d(x(t),M)=0.Proof.
Letx0∈ℝn be given. We set
(3.24)Ω:={x∈Rn∣V(x)≤V(x0)}.
Then, by assumption (2), we see that Ω is a bounded closed subset of ℝn. We infer from the proof of Theorem 3.1 that V is decreasing along any solution of (2.7). Thus, Ω is actually positively invariant. Therefore, by Theorem 3.1, one concludes that
(3.25)lim⁡t→∞d(x(t),MΩ)=0
for each solution x(·), where ℳΩ is the largest weakly invariant subset of E(f,φ,V)∩Ω. Clearly, ℳΩ⊂ℳ, and the conclusion follows.


## 3.2.  Asymptotic Stability of Equilibria

As simple applications of the LaSalle’s invariance principle established above, we make some further discussions on the asymptotic behavior of the system (1.1). For this purpose, we denote by ℰ(f,φ) the set of stationary solutions to (1.1), that is,(3.26)E(f,φ):={z∈Rn∣〈f(z),v-z〉+φ(v)-φ(z)≥0,∀v∈Rn}.
In what follows, we will always assume that(3.27)f(0)∈-∂φ(0),
so that 0∈ℰ(f,φ) is the trivial stationary solution of (1.1).Let us first prove the strong stability of the trivial stationary solution 0. Forr>0, we denote by B¯r the closed ball of radius r,(3.28)B¯r:={x∈Rn:‖x‖≤r}.Theorem 3.3.
Suppose that there existsσ>0 and V∈C1(B¯σ;ℝ) such that(1)
V(x)≥a(∥x∥) for all x∈B¯σ, where a∈C([0,σ]) satisfies a(t)>0  (∀t∈(0,σ)),(2)
V(0)=0,(3)
〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0, x∈B¯σ. Then, the stationary solution 0 is strongly stable, that is, for any ɛ>0, there exists a δ>0 such that for any solution x(t) of (1.1) with ∥x(0)∥≤δ, one has
(3.29)‖x(t)‖≤ɛ,∀t≥0.Proof.
For anyδ<σ, let m be the minimum value of V on the boundary of B¯δ. Then, by assumptions (1) and (2), we find that m>0. Let
(3.30)U={x∈B¯δ∣V(x)≤m}.
It is clear that U is a neighborhood of 0. Since V is decreasing along each solution in B¯δ, one trivially checks that U is strongly positive invariant. This implies the desired result.Propositions3.4–3.6 below can be proved by the same arguments as the ones in [2]. We omit the details.Proposition 3.4.
Suppose thatΩ is a subset of ℝn, and there exists V∈C1(Ω;ℝ) such that
(3.31)〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0,x∈Ω.
Then,
(3.32)(E(f,φ)∩Ω)⊂EΩ(f,φ,V).Proposition 3.5.
Suppose that there exist aσ>0 and V∈C1(B¯σ;ℝ) such that(1)
〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0, x∈B¯σ,(2)
E(f,φ,V)∩B¯σ={0}. Then, the stationary solution 0 is isolated in ℰ(f,φ).Proposition 3.6.
Suppose that there existsV∈C1(ℝn;ℝ) such that
(3.33)〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0,x∈Rn.
Then, ℰ(f,φ)⊂E(f,φ,V).Now, we can easily prove the following result.Theorem 3.7.
Suppose that there existsσ>0 and V∈C1(B¯σ;ℝ) such that (1)–(3) in Theorem 3.3 hold; moreover,
(3.34)E(f,φ,V)∩B¯σ={0}.
Then, the trivial stationary solution 0 is strongly asymptotically stable.Proof.
The strong stability is readily implied in Theorem3.3. Define
(3.35)Ω={x∈B¯σ∣V(x)≤m}.
Then, as in the proof of Theorem 3.3, we know that Ω is a strongly positively invariant neighborhood of 0. Applying Theorem 3.1, we deduce that, for x0∈Ω,
(3.36)lim⁡t→+∞d(x(t),M)=0.
On the other hand, by (3.34), we see that ℳ={0}. Therefore, the trivial stationary solution 0 is strongly asymptotically stable.Theorem 3.8.
Suppose that there existsV∈C1(ℝn;ℝ) such that(1)
V(x)≥a(∥x∥) for all x∈ℝn, where a:ℝ+→ℝ is a continuous strictly increasing function with a(0)=0,(2)
V(0)=0,(3)
〈f(x),∇V(x)〉+φ(x)-φ(x-∇V(x))≥0, x∈ℝn,(4)
E(f,φ,V)={0}. Then, the trivial stationary solution to (2.7) is globally strongly asymptotically stable.Proof.
The strong asymptotic stability can be directly deduced from Theorem3.8. Repeating the same argument as in Theorem 3.2, we can show that each solution of the system approaches ℳ={0}.



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*Source: 905871-2012-05-09.xml*