Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

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ADVANCED STABILITY THEORY 245

with u = 0, the zero solution (x

(t), x

(t)) ≡ 0 to (4.129) and (4.130) is

globally attractive, that is, lim

t→∞

(t), x

(t)) = (0, 0) for all (x

, x

) ∈

. △

4.5 Input-to-State Stability

In the previous sections we examined the stability and boundedness of

undisturbed dynamical systems. In this section, we introduce the notion

of input-to-state stability involving the stability of disturbed nonlinear

dynamical s ystems. In p articular, we consider n on linear dynamical systems

of the form

˙x(t) = F (x(t), u(t)), x(0) = x

, t ≥ 0, (4.133)

where x(t) ∈ R

, t ≥ 0, u(t) ∈ R

, t ≥ 0, and F : R

× R

→ R

Lipschitz continuous on R

×R

. The input u is assumed to be a piecewise

continuous fun ction of time with values in R

, that is, u : [0, ∞) → R

Now, suppose that (4.133) with u(t) ≡ 0 is globally asymptotically stable,

then we are interested in whether a bounded input u(t), t ≥ 0, implies that

the state x(t), t ≥ 0, is bounded. This is precisely the notion of input-to-

state stability.

For the linear dynamical sy s tem

˙x(t) = Ax(t) + Bu(t), x(0) = x

, t ≥ 0, (4.134)

where A is Hurwitz, it follows that

x(t) = e

x(0) +

A(t−τ)

Bu(τ)dτ. (4.135)

Using the fact that for an asymptotically stable matrix A ∈ R

n×n

, ke

k ≤

γe

−βt

, t ≥ 0, where k · k is a su bmultiplicative matrix norm, γ > 0, and

0 < β < −α(A), where α(A)

△

= max{Re λ : λ ∈ spec(A)}, it follows that

kx(t)k ≤ γe

−βt

k +

γe

−β(t−τ)

kBkku(τ)kdτ

≤ γe

−βt

k +

kBk sup

0≤τ≤t

ku(τ)k, (4.136)

which shows th at if u(t), t ≥ 0, is bounded, then x(t), t ≥ 0, is also

bounded. Note that in the case where x

= 0, the state response is

proportional to the input bound. However, for nonlinear asymptotically

stable dynamical systems a bounded input does not necessarily imply that

the state is boun ded. To see this, consider

˙x(t) = −x(t) + x

(t)u(t), x(0) = x

, t ≥ 0, (4.137)

which is globally asymptotically stable for u(t) ≡ 0. However, in the case

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246 CHAPTER 4

where x(0) = 2 and u(t) ≡ 1, the solution to (4.137) is given by x(t) =

1/(1−0.5e

), which is unbounded and, in fact, possesses a ﬁnite escape time.

In light of the above, we introduce the notion of input-to-state stability for

nonlinear dynamical systems.

Deﬁnition 4.6. A nonlinear dynamical system given by (4.133) is said

to be input-to-state stable if for every x

∈ R

and every continuous and

bounded input u(t) ∈ R

, t ≥ 0, the solution x(t), t ≥ 0, of (4.133) exists

and satisﬁes

kx(t)k ≤ η(kx

k, t) + γ



sup

0≤τ≤t

ku(τ)k



, t ≥ 0, (4.138)

where η(s, t), s > 0, is a class KL fun ction and γ(s), s > 0, is a class K

function.

The input-to-state inequality (4.138) guarantees that for a bounded

input u(t) ∈ R

, t ≥ 0, the s tate x(t), t ≥ 0, is bounded. In particular,

as t increases, the state x(t), t ≥ 0, is bounded by a class K function of

sup

t≥0

ku(t)k. In the case where u(t) ≡ 0, (4.138) reduces to

kx(t)k ≤ η(kx(t

)k, t), (4.139)

and hence, input-to-state stability implies that the zero solution x(t) ≡ 0

of (4.133) (with u(t) ≡ 0) is globally asymptotically stable. Additionally,

(4.138) implies that if lim

t→∞

u(t) = 0, then lim

t→∞

x(t) = 0. To see this,

let µ > 0 be such that γ(µ) ≤ ε/2 for a given ε > 0. Since lim

t→∞

u(t) = 0,

it follows th at there exists t

> 0 s uch th at ku(t)k ≤ µ, t ≥ t

. Now, since

x(t), t ≥ 0, is bounded, it follows that

kx(t)k ≤ η(kx(t

)k, t −t

) + γ(µ) ≤ η(δ, t − t

) + ε/2, t ≥ t

, (4.140)

for some δ > 0. Since η(δ, t−t

) → 0 as t → ∞, there exists t

> 0 such th at

η(δ, t) ≤ ε/2, t ≥ t

. Thus, it follows from (4.140) that kx(t)k ≤ ε, t ≥ τ,

where τ = max(t

, t

), which implies that lim

t→∞

x(t) = 0. The following

theorem gives necessary and suﬃcient conditions for input-to-state stability

of a nonlinear d y namical system.

Theorem 4.15. The nonlinear dynamical system (4.133) is input-to-

state stable if and only if there exist a continuously diﬀerentiable radially

unbounded, positive-deﬁnite function V : R

→ R and continuous functions

, γ

∈ K such that for every u ∈ R

′

(x)F (x, u) ≤ −γ

(kxk), kxk ≥ γ

(kuk). (4.141)

Proof. Assum e (4.141) holds and let u(·) be such that u(t) ∈ R

t ≥ 0. With f(t, x) = F (x, u), V (t, x) = V (x), W (x) = γ

(kxk), and

µ = γ

(|||u|||), where |||u|||

△

= sup

t∈[0,∞)

ku(t)k, it follows from Corollary 4.2

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ADVANCED STABILITY THEORY 247

that there exists t

> 0 such that

kx(t)k ≤ γ(|||u|||), t ≥ t

, (4.142)

where γ = α

−1

◦β ◦γ

and where α(·), β(·) are class K

∞

functions su ch that

α(kxk) ≤ V (x) ≤ β(kxk), x ∈ R

(see Problem 3.72). Next, without loss of

generality, let t

> 0 be s uch that kx(t)k ≥ γ

(|||u|||), t ≤ t

. Thus ,

dV (x(t))

= V

′

(x)F (x, u(t))|

x=x(t)

≤ −γ

(kx(t)k)

≤ −γ

◦ β

−1

(V (x(t)), a.e. t ≤ t

. (4.143)

Note that (4.143) guarantees that x(t) is deﬁned for all t ≥ 0. Furthermore,

it follows from the comparison principle (see Problem 2.107) that (4.143)

implies that there exists a class KL fun ction ˆη such that V (x(t)) ≤

ˆη(V (x

), t), t ≤ t

. Hence,

kx(t)k ≤ η(kx

k, t), t ≤ t

, (4.144)

where η(s, t) = α

−1

ˆη(β(s), t). Now, it follows from (4.142) and (4.144) that

kx(t)k ≤ η(kx

k, t)+γ(|||u|||), t ≥ 0. Since x

and u(·) are arbitrary, it follows

that (4.133) is input-to-state stable.

Necessity is considerably more involved and requires concepts not

introduced in this book. For details, see [409].

The next result provides suﬃcient conditions for input-to-state stabil-

ity as an immediate consequence of the converse global exponential stability

theorem.

Proposition 4.1. Consider the nonlinear d ynamical system (4.133)

where F : R

×R

→ R

is continuously diﬀerentiable and globally Lipschitz

continuous on R

× R

. If the zero solution x(t) ≡ 0 of the undistur bed

(u(t) ≡ 0) system (4.133) is globally exponentially stable, then (4.133) is

input-to-state stable.

Proof. Since the zero solution x(t) ≡ 0 of the undisturbed (u(t) ≡ 0)

system (4.133) is globally exponentially stable, it follows from Theorem

3.11 that there exist a continuously diﬀerentiable positive-deﬁnite function

V : R

→ R and scalars α, β, and ε > 0 such that

αkxk

≤ V (x) ≤ βkxk

, x ∈ R

, (4.145)

′

(x)F (x, 0) ≤ −εV (x), x ∈ R

. (4.146)

Next, using the fact that kV

′

(x)k ≤ νkxk, x ∈ R

, ν > 0, (see Problem 3.60)

and the global Lipschitz continuity of F (·, ·) on R

×R

, for every u ∈ R

, it

follows that the derivative of V (x) along the trajectories of (4.133) through

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248 CHAPTER 4

x ∈ R

at t = 0 is given by

V (x) = V

′

(x)F (x, u)

= V

′

(x)F (x, 0) + V

′

(x)[F (x, u) − F (x, 0)]

≤ −εαkxk

+ νLkxkkuk, (4.147)

where L > 0 denotes the Lipschitz constant of F (x, ·). Rewriting (4.147) as

V (x) ≤ −εα(1 −µ)kxk

− εαµkxk

+ νLkxkkuk, (4.148)

where µ ∈ (0, 1), it follows that

V (x) ≤ −εα(1 −µ)kxk

, kxk ≥

νL

αεµ

kuk. (4.149)

Now, the result is a direct consequence of Theorem 4.15 with γ

(σ) = εα(1−

µ)σ

and γ

(σ) =

νL

αεµ

σ.

Example 4.10. Consider the nonlinear d y namical system

˙x

(t) = −x

(t) + x

(t), x

(0) = x

, t ≥ 0, (4.150)

˙x

(t) = −x

(t) + u(t), x

(0) = x

. (4.151)

To show that (4.150) and (4.151) is input-to-state stable consider the radially

unbounded, positive deﬁnite function V (x

, x

) =

, α > 0, and

note that

V (x

, x

) = x

(−x

+ x

) + αx

(−x

+ u)

= −x

+ x

− α(1 − ε)x

− αεx

+ αx

u, ε ∈ (0, 1),

≤ −x

+ x

− α(1 − ε)x

, |x

| ≥

|u|

≤ −









, |x

| ≥

|u|

, (4.152)

where



1 −1/2

−1/2 α(1 − ε)



Now, choosing α > 1/4(1−ε) ensures that R

> 0, and hence,

V (x

, x

) < 0,

| ≥

|u|

. Next, for |x

| ≤

|u|

, it follows that

V (x

, x

) ≤ −x

+ x

−αx

+ α|u|

/ε

= −(1 −ε)x

+ x

− αx

− εx

+ α|u|

/ε

≤ −









, |x

| ≥ α

1/2

|u|

, (4.153)

where



1 −ε −1/2

−1/2 α



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ADVANCED STABILITY THEORY 249

Note that with α > 1/4(1 −ε), R

> 0, and hence,

V (x

, x

) < 0, |x

| ≤

|u|

Next, let δ = min{λ

min

), λ

min

)} and let γ

(s) =

1/2

, s ≥ 0. Note

that γ

∈ K

∞

. Now, (4.152) and (4.153) imply

V (x

, x

) ≤ −δ(x

+ x

kxk

∞

≥ γ

(kuk), and hence, it follows from Theorem 4.15 that (4.150) and

(4.151) is input-to-state stable. △

The following propositions address stability of cascade and intercon-

nected dynamical systems. Here, we provide the global versions of these

propositions; the local cases are identical except for restricting the domain

of analysis.

Proposition 4.2. Consider the nonlinear cascade d ynamical system

˙x

(t) = f

(t), x

(t)), x

(0) = x

, t ≥ 0, (4.154)

˙x

(t) = f

(t)), x

(0) = x

, (4.155)

where f

: R

×R

→ R

and f

: R

→ R

are Lipschitz continuous and

satisfy f

(0, 0) = 0 and f

(0) = 0. If (4.154), with x

viewed as the input,

is input-to-state s table and the zero solution x

(t) ≡ 0 to (4.155) is globally

asymptotically stable, then the zero solution (x

(t), x

(t)) ≡ (0, 0) of the

cascade d ynamical system (4.154) and (4.155) is globally asymptotically

stable.

Proof. Since (4.154) is input-to-state stable and the zero solution

(t) ≡ 0 of (4.155) is globally asymptotically stable it follows that there

exist KL functions η

(·, ·) and η

(·, ·) and a class K function γ(·) such that

(t)k ≤ η

(kx

(s)k, t −s) + γ



sup

s≤τ≤t

(τ)k



, (4.156)

(t)k ≤ η

(kx

(s)k, t −s), (4.157)

where t ≥ s ≥ 0. Setting s = t/2 in (4.157) yields

(t)k ≤ η

(kx

(t/2)k, t/2) + γ

sup

t/2≤τ≤t

(τ)k

. (4.158)

Next, setting s = 0 and replacing t by t/2 in (4.156) yields

(t/2)k ≤ η

(kx

(0)k, t/2) + γ

sup

0≤τ≤t/2

(τ)k

. (4.159)

Now, using (4.157), it follows that

sup

0≤τ≤t/2

(τ)k ≤ η

(kx

(0)k, 0), (4.160)

sup

t/2≤τ≤t

(τ)k ≤ η

(kx

(0)k, t/2). (4.161)

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250 CHAPTER 4

Next, substituting (4.159)–(4.161) into (4.158) and using the fact that

(0)k ≤ kx(0)k, kx

(0)k ≤ kx(0)k, and kx(t)k ≤ kx

(t)k + kx

(t)k, t ≥ 0,

where x(t)

△

= [x

(t), x

(t)]

, it follows that

kx(t)k ≤ η(kx(0)k, t), (4.162)

where

η(r, s) = η

(η

(r, s/2) + γ(η

(r, 0)), s/2) + γ(η

(r, s/2)) + η

(r, s). (4.163)

Finally, since η(·, t) is a strictly increasing function with η(0, t) = 0 and

η(r, ·) is a decreasing function of time such that lim

t→∞

η(r, t) = 0, r > 0,

it follows that the zero s olution x(t) ≡ 0 of the interconnected dynamical

system (4.154) and (4.155) is globally asymptotically stable.

Proposition 4.3. Consider the nonlinear interconnected dynamical sy-

stem

˙x

(t) = f

(t), x

(t)), x

(0) = x

, t ≥ 0, (4.164)

˙x

(t) = f

(t), x

(t)), x

(0) = x

, (4.165)

where f

: R

× R

→ R

is such that, for every x

∈ R

, f

(·, x

)

is Lipschitz continuous in x

, and f

: R

× R

→ R

is such that

for every x

∈ R

, f

, ·) is Lip s chitz continuous in x

. If (4.165) is

input-to-state stable with x

viewed as the input and (4.164) and (4.165) is

globally asymptotically s table with respect to x

uniformly in x

, then the

zero solution (x

(t), x

(t)) ≡ (0, 0) of the interconnected dynamical system

(4.164) and (4.165) is globally asymptotically stable.

Proof. Since (4.165) is input-to-state stable with x

viewed as the

input and (4.164) and (4.165) is globally asymptotically stable with respect

to x

uniformly in x

, it follows that there exist class KL fu nctions η

(·, ·)

and η

(·, ·) and a class K function γ(·) (see Problem 4.6) such that

(t)k ≤ η

(kx

(s)k, t −s), (4.166)

(t)k ≤ η

(kx

(s)k, t −s) + γ



sup

s≤τ≤t

(τ)k



, (4.167)

where t ≥ s ≥ 0. Setting s = t/2 in (4.167) yields

(t)k ≤ η

(kx

(t/2)k, t/2) + γ

sup

t/2≤τ≤t

(τ)k

. (4.168)

Next, setting s = 0 and replacing t by t/2 in (4.167) yields

(t/2)k ≤ η

(kx

(0)k, t/2) + γ

sup

0≤τ≤t/2

(τ)k

. (4.169)

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ADVANCED STABILITY THEORY 251

Now, using (4.166), it follows that

sup

0≤τ≤t/2

(τ)k ≤ η

(kx

(0)k, 0), (4.170)

sup

t/2≤τ≤t

(τ)k ≤ η

(kx

(0)k, t/2). (4.171)

Next, substituting (4.169)–(4.171) into (4.168) and using the fact that

(0)k ≤ kx(0)k, kx

(0)k ≤ kx(0)k, and kx(t)k ≤ kx

(t)k + kx

(t)k, t ≥ 0,

where x(t)

△

= [x

(t), x

(t)]

, it follows that

kx(t)k ≤ η(kx(0)k, t), (4.172)

where

η(r, s) = η

(r, s) + η

(η

(r, s/2) + γ(η

(r, 0)), s/2) + γ(η

(r, s/2)). (4.173)

Finally, since η(·, t) is a strictly increasing function with η(0, t) = 0 and

η(r, ·) is a decreasing function such that lim

t→∞

η(r, t) = 0, r > 0, it follows

that the zero solution x(t) ≡ 0 of the interconnected dynamical system

(4.164) and (4.165) is globally asymptotically stable.

Example 4.11. To illustrate the utility of Pr oposition 4.2 for feedback

stabilization, consider the non linear system with a linear input subsystem

˙x(t) = f(x(t)) + G(x(t))ˆx(t), x(0) = x

, t ≥ 0, (4.174)

ˆx(t) = Aˆx(t) + Bu(t), ˆx(0) = ˆx

, (4.175)

where x ∈ R

, ˆx ∈ R

ˆn

, f : R

→ R

and satisﬁes f (0) = 0, G : R

→ R

n×ˆn

u ∈ R

, A ∈ R

ˆn×ˆn

, B ∈ R

ˆn×m

, and (A, B) is controllable. Fur th ermore,

assume (4.174) is inpu t-to-state stable with ˆx viewed as the input. Now,

since (A, B) is controllable, th ere exists K ∈ R

m×ˆn

such that with u = K ˆx,

(4.175) is asymp totically stable, that is, the zero solution of

ˆx(t) = (A +

BK)ˆx(t), ˆx(0) = ˆx

, t ≥ 0, is globally asymptotically stable. Hence, it

follows from Proposition 4.2 that the zero solution (x(t), ˆx(t)) ≡ (0, 0) of the

cascade connection (4.174) and (4.175) is globally asymptotically stable. △

Finally, we present a proposition on ultimate boundedness of intercon-

nected systems.

Proposition 4.4. Consider the nonlinear interconnected dynamical sy-

stem (4.164) and (4.165). If (4.165) is input-to-state stable with x

viewed as

the inpu t and (4.164) and (4.165) is ultimately bounded with respect to x

uniformly in x

, th en the solution (x

(t), x

(t)), t ≥ 0, of the interconnected

dynamical system (4.164) and (4.165) is ultimately bounded.

Proof. Let δ > 0. Since (4.164) and (4.165) is ultimately bounded with

respect to x

(uniformly in x

), for every kx

k < δ, there exist positive

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252 CHAPTER 4

constants δ, ε and T = T (δ, ε) such that if kx

k < δ, then kx

(t)k < ε,

t ≥ T . Furthermore, since (4.165) is input-to-state stable with x

viewed as

the input, it follows that x

(T ) is ﬁnite, and hence, there exist a class KL

function η(·, ·) and a class K function γ(·) such that

(t)k ≤ η(kx

(T )k, t − T ) + γ

sup

T ≤τ ≤t

(τ)k

≤ η(kx

(T )k, t − T ) + γ(ε)

≤ η(kx

(T )k, 0) + γ(ε), t ≥ T, (4.176)

which proves that the solution (x

(t), x

(t)), t ≥ 0, to (4.164) and (4.165) is

ultimately bounded.

4.6 Finite-Time Stability of Nonlinear Dynamical Systems

The notions of asymptotic and exponential stability in dynamical system

theory imply convergence of the system trajectories to an equilibrium state

over the inﬁnite h orizon. In many applications, however, it is desirable that

a dynamical system possesses the pr operty that trajectories that converge

to a Lyapunov stable equilibriu m state must do so in ﬁnite time rather

than merely asymptotically. The stability theorems presented in Ch apter

3 involve system dynamics with L ipschitz continuous vector ﬁelds, which

implies uniqueness of system solutions in forward and b ackward times.

Hence, convergence to an equilibrium state is achieved over an inﬁnite time

interval. In order to achieve convergence in ﬁnite time, the system dynamics

need to be non-Lipschitzian, giving rise to nonuniqueness of solutions in

backward time. Uniqueness of solutions in forward time, however, can be

preserved in the case of ﬁnite-time convergence. Suﬃcient conditions that

ensure uniqueness of solutions in forward time in the absence of Lipschitz

continuity are given in [4, 118, 232, 474]. In addition, it is shown in [96,

Theorem 4.3, p. 59] that uniqueness of solutions in forward time along

with continuity of the sys tem dynamics ensure that the system solutions are

continuous functions of the system initial conditions even when the dynamics

are n ot Lipschitz continuous.

In this section, we develop Lyapunov and converse Lyapunov theorems

for ﬁnite-time s tability of autonomous systems. Speciﬁcally, consider the

nonlinear dynamical system given by

˙x(t) = f (x(t)), x(t

) = x

, t ∈ I

, (4.177)

where x(t) ∈ D ⊆ R

, t ∈ I

, is the system state vector, I

is th e maximal

interval of existence of a solution x(t) of (4.177), D is an open set, 0 ∈ D,

f(0) = 0, and f(·) is continuous on D. We assume that (4.177) possesses

unique solutions in forward time f or all initial conditions except possibly the

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ADVANCED STABILITY THEORY 253

origin in the following sense. For every x ∈ D\{0} there exists τ

> 0 such

that, if y

: [0, τ

) → D and y

: [0, τ

) → D are two solutions of (4.177) with

(0) = y

(0) = x, then τ

≤ min{τ

, τ

} and y

(t) = y

(t) for all t ∈ [0, τ

Without loss of generality, we assume that for each x, τ

is chosen to be the

largest such number in R

. Suﬃcient conditions for forward uniqueness in

the absen ce of Lipschitz continuity can be found in [4, 118,232,474].

The next result presents the classical comparison principle for nonlin-

ear d y namical systems.

Theorem 4.16. Consider the nonlinear dynamical system (4.177).

Assume there exists a continuously diﬀerentiable function V : D → R such

that

′

(x)f(x) ≤ w(V (x)), x ∈ D, (4.178)

where w : R → R is a continuous function and

˙z(t) = w(z(t)), z(t

) = z

, t ∈ I

, (4.179)

has a unique solution z(t), t ∈ I

. If [t

, t

+ τ ] ⊆ I

∩ I

is a compact

interval and V (x

) ≤ z

, z

∈ R, then V (x(t)) ≤ z(t), t ∈ [t

, t

+ τ ].

Proof. Consider the family of dynamical systems given by

˙z(t) = w(z(t)) +

, z(t

) = z

, (4.180)

where n ∈ Z

and t ∈ I

, and denote the solution to (4.180) by s

(n)

(t, z

t ∈ I

. Now, it follows from [98, p. 17, Theorem 3] that there exists a

compact interval [t

, t

+ τ ] ⊆ I

∩ I

such that s

(n)

(t, z

), t ∈ [t

, t

+ τ ],

is deﬁned for all suﬃciently large n. Furthermore, it follows from Lemma

3.1 of [179] that s

(n)

(t, z

) → z(t) as n → ∞ uniformly on [t

, t

+ τ],

where z(t), t ∈ I

, is a solution to (4.179). Next, we show that V (x(t)) ≤

(n)

(t, z

), n > m, t ∈ [t

, t

+ τ ], where m is suﬃciently large so that

(n)

(t, z

) is well deﬁned on [t

, t

+ τ] f or all n > m. Note that at t = t

V (x

) = V (x(t

)) ≤ z

= s

(n)

, z

). Now, suppose, ad absurdum, that

there exist t

, t

∈ [t

, t

+ τ] such that V (x(t)) > s

(n)

(t, z

), t ∈ (t

, t

and V (x(t

)) = s

(n)

, z

) f or some n > m. Since V (·) is continuously

diﬀerentiable, it follows that

V (x(t

)) ≥ ˙s

(n)

, z

)

= w(s

(n)

, z

)) +

= w(V (x(t

))) +

> w(V (x(t

))), (4.181)

which is a contradiction. Thus , V (x(t)) ≤ s

(n)

(t, z

), t ∈ [t

, t

+ τ], n > m.

NonlinearBook10pt November 20, 2007

254 CHAPTER 4

Since s

(n)

(t, z

) → z(t) uniformly on [t

, t

+ τ ], this p roves the result.

The next deﬁnition introduces the notion of ﬁnite-time stability. For

this deﬁn ition and the remainder of the section we assume, without loss of

generality, that t

= 0.

Deﬁnition 4.7. Consider the nonlinear dynamical system (4.177). The

zero solution x(t) ≡ 0 to (4.177) is ﬁnite-time stable if there exist an open

neighborhood N ⊆ D of the origin and a fu nction T : N\{0} → (0, ∞),

called the settling-time function, such that the following statements hold:

i) Finite-time convergence. For every x ∈ N\{0}, s

(t) is deﬁn ed on

[0, T (x)), s

(t) ∈ N\{0} for all t ∈ [0, T (x)), and lim

t→T (x)

s(x, t) = 0.

ii) Lyapunov stability. For every ε > 0 there exists δ > 0 such that

(0) ⊂ N and for every x ∈ B

(0)\{0}, s(t, x) ∈ B

(0) for all t ∈

[0, T (x)).

The zero solution x(t) ≡ 0 of (4.177) is globally ﬁnite-time stable if it is

ﬁnite-time stable with N = D = R

Note that if the zero solution x(t) ≡ 0 to (4.177) is ﬁnite-time stable,

then it is asymptotically stable, and hence, ﬁnite-time stability is a stronger

notion than asymptotic stability. Next, we show that if the zero solution

x(t) ≡ 0 to (4.177) is ﬁnite-time stable, then (4.177) has a unique solution

s(·, ·) deﬁned on R

×N f or every initial condition in an open n eighborhood

of the origin, including the origin, and s(t, x) = 0 for all t ≥ T (x), x ∈ N,

where T (0)

△

= 0.

Proposition 4.5. Consider the nonlinear dynamical s ystem (4.177).

Assume that the zero solution x(t) ≡ 0 to (4.177) is ﬁnite-time stable and

let N ⊆ D and T : N\{0} → (0, ∞) be as in Deﬁnition 4.7. Then, s(·, ·) is

a unique s olution of (4.177) and is deﬁned on R

× N, an d s(t, x) = 0 for

all t ≥ T (x), x ∈ N, where T (0) , 0.

Proof. It follows from Lyapunov stability of the origin that x(t) ≡ 0

is the unique solution x(·) of (4.177) satisfying x(0) = 0. This proves that

×{0} is contained in the domain of the deﬁnition of s(·, ·) and s

(t) ≡ 0.

Next, let N ⊆ D and T (·) be as in Deﬁnition 4.7, and let x

∈ N\{0}.

Deﬁne

x(t)

△



s(t, x

), 0 ≤ t ≤ T (x

0, T(x

) ≤ t.

(4.182)