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

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NonlinearBook10pt November 20, 2007

STABILITY THEORY FOR NONLINEAR SYSTEMS 185

(3.212) and (3.213) predicated on V (x

, x

). Plot D

in the x

-x

plane and

show the region wh ere

V (x

, x

) < 0, (x

, x

) ∈ R × R.

Problem 3.13. Consider the nonlinear system

¨x(t) + g(x(t)) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (3.214)

where xg(x) > 0, x 6= 0, and g(0) = 0. Using x

= x and x

= ˙x show

that th e trajectories of (3.214) satisfy V (x

, x

) = k

, where V (x

, x

) =

g(s)ds and k is a constant, and the origin is Lyapunov stable.

Plot sample trajectories for the cases i) g(x) = x, ii) g(x) =

x, and iii)

g(x) = x, x > 0, and g(x) =

x, x < 0.

Problem 3.14. Show that the rigid spacecraft dynamical system in

Example 3.1 is at an equilibrium if and only if at least two of the quantities

, x

) are equal to zero, and hence, the set of equilibria consists of the

union of the x

, x

, and x

axes. Furtherm ore, s how that equilibria of the

form (x

, 0, 0), x

6= 0, and (0, 0, x

), x

6= 0, are Lyapunov stable while

equilibria of the form (0, x

, 0), x

6= 0, are unstable. What does this

result imply in regards to the spacecraft spinning about its major, minor,

and intermediate axes?

Problem 3.15. Consider the controlled rigid spacecraft given by

˙x

(t) = I

(t)x

(t) + u

(t), x

(0) = x

, t ≥ 0, (3.215)

˙x

(t) = I

(t)x

(t) + u

(t), x

(0) = x

, (3.216)

˙x

(t) = I

(t)x

(t) + u

(t), x

(0) = x

, (3.217)

where I

= (I

− I

)/I

, I

= (I

− I

)/I

, I

= (I

− I

)/I

, and I

are the principal moments of inertia of the spacecraft such that

> I

> 0. Assume that the attitude control torques u

, u

, and

are given by u

(t) = α

(t), u

(t) = α

(t), and u

(t) = α

(t).

Using Lyapunov’s direct method determine suﬃcient conditions on α

, α

and α

such that the zero solution x(t) ≡ 0 to (3.215)–(3.217) is globally

asymptotically stable.

Problem 3.16. Consider the controlled rigid spacecraft given in Prob-

lem 3.15 with the bang-bang control law

(t) = f

)

△

= U

sgn(x

) =



, x

> 0

−U

, x

< 0,

(3.218)

where U

> 0, i = 1, 2, 3. Show that the feedback control law (3.218) globally

stabilizes the zero solution (x

(t), x

(t)) ≡ (0, 0, 0) to (3.215)–(3.217).

Furth ermore, show that the trajectories of the controlled spacecraft reach

the origin in ﬁnite time T ≤ kL(x

/U, where L(x

) is the initial angular

momentum vector of the rigid spacecraft and U = min{U

, U

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186 CHAPTER 3

Problem 3.17. Consider the controlled linearized approximation of

the attitude motion of a satellite given by

θ(t)+3ω

−I

)θ(t) = u(t), θ(0) = θ

θ(0) = θ

, t ≥ 0, (3.219)

where I

, I

, and I

are the moments of inertia of the satellite about the

pitch, roll, and yaw axis, respectively; ω is the satellite orbiting angu lar

rate; and θ is the pitch plane angular deviation from the local gravitational

vertical. Show that the nonlinear control torque u = −f (αθ + β

θ), where f

is continuous, f (0) = 0, and zf (z) > 0, z ∈ R, z 6= 0, can render the zero

solution (θ(t),

θ(t) ≡ (0, 0) to (3.219) asymptotically stable. To analyze the

stability of the closed-lo op system u se the Lyapunov fu nction candidate

V (θ,

θ) =

(θ

) + b

αθ+β

f(σ)dσ, (3.220)

where α, β, a, and b are parameters to be chosen. Can the f eedback control

law u = −f(β

θ) asymptotically stabilize (3.219)? Under what conditions

on f (·) will the origin be globally asymptotically stable?

Problem 3.18. Consider the nonlinear Lienard system given by

¨x(t) + f (x(t)) ˙x(t) + g(x(t)) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (3.221)

where g(0) = 0 and f and g are continuously diﬀerentiable, and deﬁne

F (x) =

f(s)ds. Show that the system can be written as

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (3.222)

˙x

(t) = −f (x

(t))x

(t) −g(x

(t)), x

(0) = x

, (3.223)

or, equivalently,

˙x

(t) = x

(t) −F (x

(t)), x

(0) = x

, t ≥ 0, (3.224)

˙x

(t) = −g(x

(t)), x

(0) = x

. (3.225)

Analyze the stability of both forms using the Lyapunov function candidates

V (x

, x

) =

g(s)ds (3.226)

and

V (x

, x

) =

+ F (x

)]

g(s)ds. (3.227)

Make sure you specify under what conditions (3.226) and (3.227) are valid

Lyapunov function candidates.

Problem 3.19. Consider the second-order nonlinear mechanical sys-

tem consisting of a unit mass with a nonlinear spring and a nonlinear damper

given by

¨x(t) + f ( ˙x(t)) + g(x(t)) = 0, x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (3.228)

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STABILITY THEORY FOR NONLINEAR SYSTEMS 187

where f(·) is the force due to friction and g(·) is the restoring force of the

nonlinear spring. Suppose that f (·) an d g(·) are continuous and f(0) = 0

and g(0) = 0. In addition, suppose there exists ε > 0 such that zf(z) > 0,

z ∈ B

(0), z 6= 0, and zg(z) > 0, z ∈ B

(0), z 6= 0.

i) Show that the origin is an equilibrium point.

ii) Show that th e zero solution (x(t), ˙x(t)) ≡ (0, 0) to (3.228) is asymp-

totically stable.

iii) Is the zero solution (x(t), ˙x(t)) ≡ (0, 0) to (3.228) globally asymp tot-

ically stable? If not, give conditions on f(·) and g(·) that guarantee

global asymptotic stability.

(Hint: Select as your Lyapunov function V (x, ˙x) the total mechanical energy

in the system and show that the α-sublevel set of V (x, ˙x) is bounded for

every α > 0. Clearly state any necessary assumptions needed to show this.)

Problem 3.20. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (3.229)

˙x

(t) = x

(t), x

(0) = x

, (3.230)

˙x

(t) = −f(x

(t)) − g(x

(t)) − ax

(t), x

(0) = x

, (3.231)

where f(0) = g(0) = 0 and f, g are continuously diﬀerentiable functions.

Show that the zero solution (x

(t), x

(t)) = (0, 0, 0) to (3.229)–(3.231)

is globally asymptotically stable if i) a > 0, ii) f (x

)/x

≥ ε

> 0, x

6= 0,

and iii) ag(x

)/x

− f

′

) ≥ ε

> 0, x

∈ R, x

6= 0. (Hint: Use the

Lyapunov function candidate V (x

, x

) = aF (x

) + f (x

+ G(x

) +

(ax

+ x

)

, wh ere F (x

) =

f(s)ds and G(x

) =

g(s)ds, to an alyze

(3.229)–(3.231).)

Problem 3.21. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (3.232)

˙x

(t) = x

(t), x

(0) = x

, (3.233)

˙x

(t) = −f (x

(t))x

(t) −ax

(t) −bx

(t), x

(0) = x

, (3.234)

where a > 0, b > 0, and f is continuous. Determine conditions on the

damping function f(·) so that the zero solution (x

(t), x

(t)) ≡ (0, 0, 0)

to (3.232)–(3.234) is globally asymptotically stable. (Hint: Consider the

Lyapunov function candidate

V (x

, x

) =

(bx

+ ax

)

+ bx

+ b

f(σ)σdσ, (3.235)

and show that (3.235) is a valid candidate.)

NonlinearBook10pt November 20, 2007

188 CHAPTER 3

Problem 3.22. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (3.236)

˙x

(t) = −2βx

(t) − 3x

(t) −αx

(t), x

(0) = x

, (3.237)

where α > 0 and β > 0. Show that (x

, x

) = (0, 0) and (x

, x

) =

(−

β, 0) are equ ilibr ium points of (3.236) and (3.237). Using the function

V (x

, x

) = βx

+ x

show that the region V (x

, x

) <

is the

union of two components Q

and Q

, where Q

is a bounded component

containing the origin to the right of (−

β, 0) and Q

is an unbounded

component to the left of (−

β, 0). Finally, show th at every solution starting

in Q

approaches the origin and hence Q

is the domain of attraction of the

equilibrium point (0, 0), and every solution starting in Q

approaches inﬁnity

and hence (−

β, 0) is unstable. Is the equilibrium point (0, 0) asymptotically

stable?

Problem 3.23. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + x

(t), x

(0) = x

, t ≥ 0, (3.238)

˙x

(t) = −2x

(t) + 3x

(t), x

(0) = x

. (3.239)

Using the Lyapunov f unction candidate V (x

, x

) =

show that the

zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.238) and (3.239) is asymptotically

stable and determine the largest ellipse contained in the domain of attraction

of (3.238) and (3.239).

Problem 3.24. Consider the nonlinear dynamical system

˙x

(t) = x

(t) − x

(t)f(x

(t), x

(t)), x

(0) = x

, t ≥ 0, (3.240)

˙x

(t) = −x

(t) −x

(t)f(x

(t), x

(t)), x

(0) = x

, (3.241)

where f : R

→ R is continuously diﬀerentiable and f(0, 0) = 0. Using

Lyapunov’s theorem obtain conditions on f(x

, x

) such that the zero

solution (x

(t), x

(t)) ≡ (0, 0) to (3.240) and (3.241) is Lyapunov stable,

asymptotically stable, and unstable.

Problem 3.25. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + 2x

(t) − 2x

(t), x

(0) = x

, t ≥ 0, (3.242)

˙x

(t) = −x

(t) −x

(t) + x

(t)x

(t), x

(0) = x

. (3.243)

Analyze the s tability of the zero s olution (x

(t), x

(t)) ≡ (0, 0) to (3.242)

and (3.243) using the Lyapunov function candidate V (x

, x

) = x

+ αx

where α, p, and q are parameters to be chosen.

Problem 3.26. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + 4x

(t), x

(0) = x

, t ≥ 0, (3.244)

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STABILITY THEORY FOR NONLINEAR SYSTEMS 189

˙x

(t) = −x

(t) −x

(t), x

(0) = x

. (3.245)

Analyze the stability of th e zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.244)

and (3.245) using the Lyapunov function candidate V (x

, x

) = x

+ αx

where α > 0 is a parameter to be chosen.

Problem 3.27. Consider the nonlinear dynamical system

˙x

(t) = x

(t) − x

(t), x

(0) = x

, t ≥ 0, (3.246)

˙x

(t) = −x

(t) −x

(t), x

(0) = x

. (3.247)

Analyze the stability of th e zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.246)

and (3.247) using the Lyapunov function candidate V (x

, x

) = αx

+ βx

where α > 0 and β > 0 are parameters to be chosen.

Problem 3.28. Consider the nonlinear dynamical system

¨x(t) + |x

(t) −1|x

(t) + x(t) − sin



πx(t)



= 0,

x(0) = x

, ˙x(0) = ˙x

, t ≥ 0. (3.248)

Show that (x

, ˙x

) = (1, 0), (x

, ˙x

) = (−1, 0), and (x

, ˙x

) = (0, 0) are

equilibrium points of (3.248). Use Lyapunov’s method to investigate the

stability of each of these equilibrium points.

Problem 3.29. Consider the scalar uncertain system

˙x(t) = αx(t) + u(t), x(0) = x

, t ≥ 0, (3.249)

where α is an unknown constant parameter. Show that the adaptive

controller u(t) = −k(t)x(t) with update law

k(t) = γx

(t), k(0) = k

, t ≥ 0, (3.250)

where γ > 0, guarantees that the equilibrium solution (x(t), k(t)) = (0, k

∗

where k

∗

> α, of the closed-loop system (3.249) and (3.250) is Lyapunov

stable and lim

t→∞

x(t) = 0 f or all x

∈ R. (Hint: Consider the Lyapunov

function candidate V (x, k) =

2γ

(k − k

∗

)

Problem 3.30. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + x

(t)[x

(t) + x

(t) −1], x

(0) = x

, t ≥ 0, (3.251)

˙x

(t) = x

(t) + x

(t)[x

(t) + x

(t) −1], x

(0) = x

. (3.252)

Is the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.251) and (3.252) asymptoti-

cally stable? Is the zero solution (x

(t), x

(t)) ≡ (0, 0) to (3.251) and (3.252)

globally asymptotically stable?

NonlinearBook10pt November 20, 2007

190 CHAPTER 3

Problem 3.31. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + g(x

(t)), x

(0) = x

, t ≥ 0, (3.253)

˙x

(t) = −g(x

(t)), x

(0) = x

, (3.254)

˙x

(t) = −αx

(t) + βx

(t) −γg(x

(t)), x

(0) = x

, (3.255)

where α, β, γ > 0 and g is continuously diﬀerentiable and satisﬁes g(0) = 0

and xg(x) > 0, 0 < |x| < σ, σ > 0. Show that the origin (0, 0, 0) is an

isolated equilibriu m point of (3.253)–(3.255). Using the Lyapun ov function

candidate

V (x

, x

) =

αx

βx

g(s)ds, (3.256)

show that the zero solution (x

(t), x

(t)) ≡ (0, 0, 0) to (3.253)–(3.255)

is asymptotically stable. If g is such that xg(x) > 0 for all x ∈ R, x 6= 0,

is the zero solution (x

(t), x

(t)) ≡ (0, 0, 0) globally asymp totically

stable?

Problem 3.32. Consider the nonlinear port-controlled Hamiltonian sy-

stem given by

˙x(t) = [J(x(t)) − R(x(t))]



∂H

∂x

(x(t))



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

, t ≥ 0,

(3.257)

y(t) = G

(x(t))



∂H

∂x

(x(t))



, (3.258)

where x(t) ∈ D ⊆ R

, D is an open set with 0 ∈ D, u(t) ∈ U ⊆ R

y(t) ∈ Y ⊆ R

, H : D → R is a continuously diﬀerentiable lower bounded

Hamiltonian (energy) function, J : D → R

n×n

is a skew-symmetric function,

R : D → R

n×n

is a nonnegative-deﬁnite dissipation function, and G : D →

n×m

i) Show th at the zero solution x(t) ≡ 0 of the uncontrolled (u(t) ≡ 0)

system (3.257) is Lyapunov stable.

ii) With u(t) = −Ky(t), where K is a positive-deﬁnite gain matrix, show

that the zero solution x(t) ≡ 0 to (3.257) is asymptotically stable.

Here, assume that the only solution of

˙x(t) = J(x(t))



∂H

∂x

(x(t))



(3.259)

that can stay in the set

(

x ∈ D : R(x)



∂H

∂x

(x)



= 0

)

(3.260)

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STABILITY THEORY FOR NONLINEAR SYSTEMS 191

is th e trivial solution x(t) ≡ 0.

iii) Under what conditions will the origin be globally asymptotically

stable?

Problem 3.33. Consider the controlled nonlinear dynamical system

given by

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

, t ≥ 0, (3.261)

u(t) = φ(x(t)), (3.262)

where x(t) ∈ D ⊆ R

, u(t) ∈ U ⊆ R

, t ≥ 0, f : D → R

, G : D → R

n×m

and φ : D → R

. Ass ume that f (0) = 0, φ(0) = 0, and f(·), G(·), and φ(·)

are continuously diﬀerentiable. Furthermore, let the controller u = φ(x) be

given by

u = φ(x) = −αR

−1

(x)G

(x)



∂V

∂x



, (3.263)

where α ∈ R and V : D → R is continuously diﬀerentiable, V (0) = 0,

V (x) > 0, x ∈ D, x 6= 0, and

∂V

∂x

satisﬁes the Hamilton-Jacobi-Bellman

equation

0 =

∂V

∂x

f(x) + L(x) −

∂V

∂x

G(x)R

−1

(x)G

(x)



∂V

∂x



, x ∈ D, (3.264)

where L : D → R and R

: D → R

m×m

is a positive-deﬁnite matrix-valued

function.

i) Show that if L(x) > 0, x ∈ D, x 6= 0, and α ≥

, then the zero solution

x(t) ≡ 0 of the closed-loop system (3.261) and (3.263) is asymptotically

stable.

ii) Show that if L(x) ≥ 0, x ∈ D, α >

, and the only solution ˙x = f(x)

that can stay identically in the set {x ∈ D : L(x) = 0} is the trivial

solution x(t) ≡ 0, then the zero solution x(t) ≡ 0 of the closed-loop

system (3.261) and (3.263) is asymptotically stable.

iii) Under what conditions will the origin be globally asymptotically

stable?

Problem 3.34. Consider the nonlinear dynamical system

˙x

(t) =

(t)[x

(t) −x

(t)] + x

(t)

(t) + x

(t)][1 + [x

(t) + x

(t)]

]

, x

(0) = x

, t ≥ 0, (3.265)

˙x

(t) =

(t)[x

(t) − 2x

(t)]

(t) + x

(t)][1 + [x

(t) + x

(t)]

]

, x

(0) = x

. (3.266)

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192 CHAPTER 3

Show that even though lim

t→∞

(t), x

(t)) = 0, the zero solution (x

(t),

(t)) ≡ (0, 0) to (3.265) and (3.266) is unstable.

Problem 3.35. Consider the nonlinear dynamical system

˙x

(t) = [x

(t) −x

(t)][1 −a

(t) −b

(t)], x

(0) = x

, t ≥ 0, (3.267)

˙x

(t) = [x

(t) + x

(t)][1 −a

(t) −b

(t)], x

(0) = x

. (3.268)

Analyze the stability of (3.267) and (3.268) using the Lyapunov function

candidate

V (x

, x

) = x

+ x

Is the ellipse E = {(x

, x

) ∈ R × R : a

+ b

= 1} a limit cycle or a

continuum of equilibrium points?

Problem 3.36. Consider the nonlinear dynamical system

˙x

(t) = x

(t) −x

(t)[x

(t) + 2x

(t) −10], x

(0) = x

, t ≥ 0, (3.269)

˙x

(t) = −x

(t) −3x

(t)[x

(t) + 2x

(t) −10], x

(0) = x

. (3.270)

i) Show that the set deﬁned by E

△

= {(x

, x

) ∈ R×R : x

+2x

−10 = 0}

is invariant.

ii) Wh at is the motion on E characterized by? Is this a limit cycle or a

continuum of equilibrium points?

iii) Examine the stability of E using the Barbashin-Krasovskii-LaSalle

theorem.

Problem 3.37. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (3.271)

˙x

(t) = −x

(t) + x

(t)[1 − x

(t) −x

(t)], x

(0) = x

. (3.272)

i) Show that the set deﬁned by E

△

= {(x

, x

) ∈ R ×R : x

+ x

−1 = 0}

is invariant.

ii) Wh at is the motion on E characterized by? Is this a limit cycle or a

continuum of equilibrium points?

iii) Examine the stability of E using the Barbashin-Krasovskii-LaSalle

theorem.

Problem 3.38. Consider the nonlinear d ynamical s ystem (3.1) and

assume that there exist a continuously diﬀerentiable function V : R

→ R

and positive scalars α, β such that

V (0) = 0, (3.273)

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STABILITY THEORY FOR NONLINEAR SYSTEMS 193

V (x) > 0, x ∈ R

, x 6= 0, (3.274)

V (x) → ∞ as kxk → ∞, (3.275)

′

(x(t))f(x(t)) ≤ αe

−βt

, t ≥ 0. (3.276)

Show that x(t), t ≥ 0, is bounded.

Problem 3.39. Consider the nonlinear dynamical sys tem (3.1) with

f(x

) = 0. Show that if there exists a continuously diﬀerentiable function

V : D → R such that

V (x

) = 0, (3.277)

V (x) > 0, x ∈ D, x 6= x

, (3.278)

′

(x)f(x) ≤ 0, x ∈ D, (3.279)

then the equilibrium solution x(t) ≡ x

to (3.1) is Lyapunov stable. If, in

addition,

′

(x)f(x) < 0, x ∈ D, x 6= x

, (3.280)

show that the equilibrium s olution x(t) ≡ x

to (3.1) is asymptotically stable.

Finally, show that if there exist scalars α, β, ε > 0, and p ≥ 1 such that

V : D → R satisﬁes

αkx −x

≤ V (x) ≤ βkx −x

, x ∈ D, (3.281)

′

(x)f(x) ≤ −εV (x), x ∈ D, (3.282)

then the equilibrium s olution x(t) ≡ x

to (3.1) is exponentially stable.

Problem 3.40. Consider the nonlinear dynamical system (3.1), as-

sume D

⊂ D is a compact invariant set with respect to (3.1), and assume

there exists a continuously diﬀerentiable function V : D

→ R such that

′

(x)f(x) ≤ 0, x ∈ D

. Furthermore, suppose that the largest invariant set

M contained in R

△

= {x ∈ D

: V

′

(x)f(x) = 0} consists of a ﬁnite collection

of isolated points. Show that lim

t→∞

x(t) exists and is equal to one of th e

isolated points contained in M.

Problem 3.41. Consider the nonlinear dynamical system

˙x

(t) = −x

(t)|x

(t)|

, x

(0) = x

, t ≥ 0, (3.283)

˙x

(t) = −x

(t)|x

(t)|

, x

(0) = x

, (3.284)

where α > 0 and β > 0. Show that lim

t→∞

(t), x

(t)) exists and is equ al

to one of the points (0, a), (0, −a), (a, 0), or (−a, 0), where a ∈ R.

Problem 3.42. A dynamical system on D ⊆ R

is the tr iple (D,

[0, ∞), s), w here s : [0, ∞) × D → D is such that th e following axioms

hold:

NonlinearBook10pt November 20, 2007

194 CHAPTER 3

i) (Continuity): s(·, ·) is jointly continuous on [0, ∞) × D.

ii) (Con s istency): For every x

∈ D, s(0, x

) = x

iii) (Semigroup property): s(τ, s(t, x

)) = s(t + τ, x

) for all x

∈ D and

t, τ ∈ [0, ∞).

Let G denote the dynamical system (D, [0, ∞), s) and let s(t, x

), t ≥ 0,

denote the trajectory of G corresponding to th e initial condition x

∈ D.

Furth ermore, let D

⊂ D be a compact invariant set with respect to G and

deﬁne V

−1

(γ)

△

= {x ∈ D

: V (x) = γ}, where γ ∈ R and V : D

→ R is a

continuous function. Show that if V (s(t, x

)) ≤ V (s(τ, x

)), for all 0 ≤ τ ≤ t

and x

∈ D

, then s(t, x

) → M

△

= ∪

γ∈R

as t → ∞, w here M

denotes

the largest invariant set (with respect to the dynamical system G) contained

in V

−1

(γ).

Problem 3.43. Consider the nonlinear dynamical system (3.1) where

f : D → R

is continuous and (3.1) possesses unique solutions in forward

time; that is, if the solutions agree at s ome time t

, then they agree at any

time t ≥ t

. The zero solution x(t) ≡ 0 to (3.1) is ﬁnite-time stable if the

origin is Lyapunov stable and there exists an open neighborhood N ⊆ D

of the origin and a function T : N \ {0} → (0, ∞), called a settling-time

function, s uch that for every x

∈ N \ {0}, s(·, x

) : [0, T (x

)) → N \ {0}

and s(t, x

) → 0 as t → T (x

). The zero solution x(t) ≡ 0 is globally

ﬁnite-time stable if it is ﬁnite-time stable with D = N = R

. Show that if

there exist a continuously diﬀerentiable function V : D → R, positive scalars

α ∈ (0, 1) and β > 0, and an open neighborhood N ⊆ D of the origin such

that

V (0) = 0, (3.285)

V (x) > 0, x ∈ D, x 6= 0, (3.286)

V (x) ≤ −βV (x)

, x ∈ N \ {0}, (3.287)

then the zero solution to (3.1) is ﬁnite-time stable. Furthermore, show that

T : N \ {0} → (0, ∞) is continuous on N and

T (x

) ≤

β(1 − α)

V (x)

1−α

, x ∈ N. (3.288)

Finally, show that if D = R

, V (x) → ∞ as kxk → ∞, and (3.287) is

satisﬁed on R

\ {0}, then the zero solution x(t) ≡ 0 to (3.1) is globally

ﬁnite-time stable. (Hint: Use the comparison principle (see Problem 2.107)

to construct a comparison system whose solution is ﬁnite-time stable and

relate this ﬁnite-time stability property to the stability property of (3.1).)

Problem 3.44. Consider the nonlinear dynamical system (3.1). An

equilibrium point x ∈ D is semistable if x ∈ D is Lyapunov s table and there