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

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NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 435

r(u

, y

) = u

−

ˆα+

−

ˆα

ˆα+

, where ˆα = α + δ,

β = β − δ, and

δ ∈ R such that 0 < 2δ < β − α.

Deﬁnition 6.4. Let α, β ∈ R be such that 0 < α ≤ 1 ≤ β < ∞. Then

the nonlinear dynamical system G given by (6.62) and (6.63) is said to have

a structured disk margin (α, β) if the negative feedback interconnection of G

and ∆(·) is globally asymptotically stable for all dynamic operators ∆(·) such

that ∆(·) is zero-state observable, ∆(u

) = diag[δ

), . . . , δ

)], and

(·), i = 1, . . . , m, is dissipative with respect to the supply rate r(u

, y

) =

−

ˆα+

−

ˆα

ˆα+

, where ˆα = α + δ,

β = β − δ, and δ ∈ R such that

0 < 2δ < β −α.

Note that if G has a disk margin (or structured disk margin) (α, β),

then G has gain and sector margins (α, β). To see this, let ∆(u

) = σ(u

where σ : R

→ R

is such that σ(0) = 0 and

σ(u

) = [σ

), . . . , σ

)]

Now, if ∆(u

) is dissipative with respect to the supply rate r(u

, y

) =

−

ˆα+

−

ˆα

ˆα+

, then σ(·) satisﬁes

0 ≤ σ

−

ˆα+

)σ(u

) −

ˆα

ˆα+

, (6.64)

or, equivalently,

0 ≥

i=1

(σ

) − ˆαu

)(σ

) −

βu

). (6.65)

Hence, if G has a disk margin (α, β), then G has a sector margin (α, β).

Similarly, if ∆(·) is a linear operator such that ∆(u

) = ∆u

, where ∆ =

diag[k

, . . . , k

], and ∆(u

) is dissipative with r espect to the supply rate

r(u

, u

) = u

−

ˆα+

−

ˆα

ˆα+

, it follows that

0 ≥

i=1

− ˆα)(k

−

β). (6.66)

Thus , if G has a disk m argin (α, β), then G has a gain margin (α, β). Finally,

letting β → ∞ in Deﬁnition 6.3 we recover the deﬁnition of disk margins

given in [395].

In the case where ∆(·) is a dynamic operator we assum e that ∆(·) can

be characterized as (6.3) and (6.4) where x

denotes the internal state of

the operator ∆(·), u

denotes the input, and y

denotes the output of the

operator. If ∆(·) is a single-input/single-output, linear dy namic operator it

follows from Parseval’s theorem that if ∆(·) is dissipative with respect to

NonlinearBook10pt November 20, 2007

436 CHAPTER 6

the supply rate r(u

, y

) = u

−

ˆα+

−

ˆα

ˆα+

, then

|∆(ω)|

− (ˆα +

β)Re ∆(jω) + ˆα

β < 0. (6.67)

In this case, it can be easily shown that if G has a disk margin of (α, β),

then G has a gain margin (α, β) and a phase margin ϕ, where cos(ϕ) =

αβ

α+β

Hence, the concept of the d isk margin f or nonlinear systems p rovides a

nonlinear hybrid analog to the concepts of gain and phase margins of linear

systems.

The following results provide algebraic su ﬃcient conditions that

guarantee disk margins for the nonlinear dynamical system G given by (6.62)

and (6.63).

Theorem 6.5. Consider the n onlinear dynamical system G given by

(6.62) and (6.63). Let α, β ∈ R be such that 0 < α ≤ 1 ≤ β < ∞. Suppose

there exist a positive-deﬁnite diagonal matrix Z ∈ R

m×m

and functions

: R

→ R, ℓ : R

→ R

, and W : R

→ R

p×m

such that V

(·) is

continuously diﬀerentiable, V

(0) = 0, V

(x) > 0, x ∈ R

, x 6= 0, and for all

x ∈ R

0 = V

′

(x)f(x) −

αβ

α+β

(x)Zφ(x) + ℓ

(x)ℓ(x), (6.68)

0 = V

′

(x)G(x) + φ

(x)Z + 2ℓ

(x)W(x), (6.69)

0 =

α+β

Z − W

(x)W(x). (6.70)

Then the nonlinear system G has a s tructured disk margin (α, β). Alterna-

tively, if (6.68)–(6.70) are satisﬁed with Z = I

, then the nonlinear system

G has a d isk margin (α, β).

Proof. With h(x) = −φ(x), J(x) ≡ 0, Q =

αβ

α+β

Z, R =

α+β

and S =

Z, it follows from Theorem 5.6 that the nonlinear system G is

dissipative with respect to the supply rate r(u, y) = u

Zy +

αβ

α+β

Zy +

α+β

Zu. Next, let ∆ (·) be zero-state observable and let ∆(u

) =

diag[δ

), . . . , δ

)] be such that δ

(·), i = 1, . . . , m, is dissipative

with respect to the supply rate r(u

, y

) = u

−

ˆα+

−

ˆα

ˆα+

, where

ˆα = α + δ,

β = β − δ, and δ ∈ R is such that 0 < 2δ < β − α. Now,

noting that in th is case ∆(·) is dissipative with respect to the supply rate

r(u

, y

) = u

−

ˆα+

−

ˆα

ˆα+

, it follows from Deﬁnition 6.4

and Corollary 6.2 that the nonlinear sys tem G has a structured disk margin

(α, β). Finally, if (6.68)–(6.70) are satisﬁed with Z = I

, then it follows

from Theorem 5.6 that the nonlinear system G is dissipative with respect

to the supply r ate r(u, y) = u

y +

αβ

α+β

y +

α+β

u. Hence, it follows

from Deﬁnition 6.3 and Corollary 6.2 that the nonlinear system G has a disk

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STABILITY OF FEEDBACK SYSTEMS 437

margin (α, β).

Corollary 6.3. Consid er the nonlinear dynamical system G given by

(6.62) and (6.63). Let α, β ∈ R be such that 0 < α ≤ 1 ≤ β < ∞. Suppose

there exist a positive-deﬁnite diagonal matrix Z ∈ R

m×m

and a continuously

diﬀerentiable function V

: R

→ R such that V

(0) = 0, V

(x) > 0, x ∈ R

x 6= 0, and for all x ∈ R

0 ≥ V

′

(x)f(x) −

αβ

α+β

(x)Zφ(x)

α+β

[φ

(x)Z + V

′

(x)G(x)]Z

−1

[φ

(x)Z + V

′

(x)G(x)]

. (6.71)

Then the nonlinear system G has a structured d isk margin (α, β). Alterna-

tively, if (6.71) is satisﬁed with Z = I

, then the n onlinear system G has a

disk margin (α, β).

Proof. It follows from Theorem 5.6 th at (6.68)–(6.70) are equivalent

to (6.71). Now, the result is a direct consequence of Theorem 6.5.

The following theorem gives the nonlinear version of the results of

[316].

Theorem 6.6. Consider the n onlinear dynamical system G given by

(6.62) and (6.63). Let α, β ∈ R be such that 0 < α ≤ 1 ≤ β < ∞. Suppose

there exist a positive-deﬁnite diagonal matrix Z ∈ R

m×m

, a continuously

diﬀerentiable function V : R

→ R, and a scalar q > 0 such that V (0) = 0,

V (x) > 0, x ∈ R

, x 6= 0, and for all x ∈ R

0 ≥ V

′

(x)f(x) −

αβ

q(α+β)

′

(x)G(x)Z

−1

(x)V

′T

(x). (6.72)

Then, with φ(x) = −

q(α+β)

−1

(x)V

′T

(x), the nonlinear system G h as

a structured disk margin (α, β). Alternatively, if (6.72) is satisﬁed with

Z = I

, then the nonlinear s ystem G has a disk margin (α, β).

Proof. The result is a direct consequence of C orollary 6.3 with V

(x) =

q(α+β)

V (x). Speciﬁcally, sin ce φ(x) = −

q(α+β)

−1

(x)V

′T

(x) = −Z

−1

·G

(x)V

′T

(x), it follows fr om (6.72) that for all x ∈ R

0 ≥ V

′

(x)f(x) −

αβ

q(α+β)

′

(x)G(x)Z

−1

(x)V

′T

(x)

= q(α + β)



′

(x)f(x) −

αβ

α+β

(x)Zφ(x)



= q(α + β)



′

(x)f(x) −

αβ

α+β

(x)Zφ(x)

α+β

(φ(x)

Z + V

′

(x)G(x))Z

−1

(φ

(x)Z + V

′

(x)G(x))



which implies (6.71), so that the conditions of Corollary 6.3 are satisﬁed.

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438 CHAPTER 6

6.5 Control Lyapunov Functions

In this section, we consider a feedback control problem and introduce the

notion of control Lyapunov functions. Furthermore, using the concept of

control Lyapun ov functions we provide necessary and suﬃcient conditions

for nonlinear system stabilization.

Consider the nonlinear controlled dynamical system given by

˙x(t) = F (x(t), u(t)), x(t

) = x

, t ≥ 0, (6.73)

where x(t) ∈ D ⊆ R

, t ≥ 0, is the state vector, D is an open set with

0 ∈ D, u(t) ∈ U ⊆ R

is the control input, and F : D × U → R

satisﬁes

F (0, 0) = 0. We assume that the control input u(·) in (6.73) is restricted

to the class of admissible controls U consisting of measurable functions u(·)

such that u(t) ∈ U for all t ≥ 0, where the constraint set U is given with

0 ∈ U. A measurable mapping φ : D → U satisfying φ(0) = 0 is called a

control law. Furthermore, if u(t) = φ(x(t)), where φ is a control law and

x(t), t ≥ 0, satisﬁes (6.73), then u(·) is called a feedback control law. We

assume that the mapping φ : D → U satisﬁes suﬃcient regularity conditions

such that the resulting closed-loop system

˙x(t) = F (x(t), φ(x(t)), x(0) = x

, t ≥ 0, (6.74)

has a unique solution forward in time. Speciﬁcally, we assu me that F (·, ·) is

Lipschitz continuous in a neighborhood of the origin in D × U.

The following two deﬁnitions are required for s tating the results of this

section.

Deﬁnition 6.5. Let φ : D → U be a mapping on D\{0} with φ(0) = 0.

Then (6.73) is feedback asymptotically stabilizable if the zero solution x(t) ≡

0 of the closed-loop system (6.74) is asymptotically s table.

Deﬁnition 6.6. Consider the controlled nonlinear dynamical system

given by (6.73). A continuously d iﬀerentiable positive-deﬁnite function V :

D → R satisfying

inf

u∈U

′

(x)F (x, u) < 0, x ∈ D, x 6= 0, (6.75)

is called a control Lyapunov function.

Note that, if (6.75) holds, th en there exists a feedback control law

φ : D → U such that V

′

(x)F (x, φ(x)) < 0, x ∈ D, x 6= 0, and hence,

Theorem 3.1 implies that if there exists a control Lyapunov function for

the nonlinear dyn amical system (6.73), then there exists a feedback control

law φ(x) such that the zero solution x(t) ≡ 0 of the closed-loop nonlinear

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STABILITY OF FEEDBACK SYSTEMS 439

dynamical system (6.73) is asymptotically stable. Conversely, if there exists

a feedb ack control law u = φ(x) such that the zero solution x(t) ≡ 0 of the

nonlinear dynamical system (6.73) is asymptotically stable, then it follows

from Theorem 3.9 that there exists a continuously diﬀerentiable positive-

deﬁnite function V : D → R such that V

′

(x)F (x, φ(x)) < 0, x ∈ D, x 6= 0,

or, equivalently, there exists a control Lyapunov function for the nonlinear

dynamical system (6.73). Hence, a given nonlinear dynamical system of

the form (6.73) is feedback asymptotically stabilizable if and only if there

exists a control Lyapun ov function satisfying (6.75). Finally, in the case

where D = R

and U = R

the zero solution x(t) ≡ 0 to (6.73) is globally

asymptotically stabilizable if and only if V (x) → ∞ as kxk → ∞.

Next, we consider the special case of nonlinear aﬃne systems in the

control and construct state feedback controllers that globally asymptotically

stabilize the zero solution of the nonlinear dynamical system under the

assumption that the sys tem has a radially unbou nded control Lyapunov

function. Speciﬁcally, we consider nonlinear aﬃne systems of the f orm

˙x(t) = f (x(t)) + G(x(t))u(t), x(t

) = x

, t ≥ 0, (6.76)

where f : R

→ R

satisﬁes f(0) = 0 and G : R

→ R

n×m

, and f(·) and

G(·) are smooth functions (at least continuous ly diﬀerentiable mappings).

Theorem 6.7. Consider the controlled nonlinear system given by

(6.76). Then a continuous ly diﬀerentiable positive-deﬁnite, radially un-

bounded function V : R

→ R is a control Lyapunov fun ction of (6.76)

if and only if

′

(x)f(x) < 0, x ∈ R, (6.77)

where R

△

= {x ∈ R

, x 6= 0 : V

′

(x)G(x) = 0}.

Proof. The proof is a direct consequence of the deﬁnition of a control

Lyapunov function by noting that for systems of the form (6.76),

inf

u∈R

′

(x)[f(x) + G(x)u] = −∞, x 6∈ R, x 6= 0.

Hence, (6.75) is equivalent to (6.77), which proves the result.

Example 6.5. Consider the nonlinear controlled system

˙x

(t) = −x

(t) + x

(t)e

(t)

cos(x

(t)), x

(0) = x

, t ≥ 0, (6.78)

(t) = x

(t) sin(x

(t)) + u(t), x

(0) = x

. (6.79)

To show that V (x

, x

) =

is a control Lyapunov function for

(6.78) and (6.79) note that

inf

u∈U

′

, x

)F (x

, x

, u) = inf

u∈U

[−x

+ x

cos(x

)

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440 CHAPTER 6

u + x

sin(x

)]



−x

, x

= 0,

−∞, x

6= 0,

where F (x

, x

, u) denotes the right-hand-side of (6.78) and (6.79). Thus,

V (x

, x

) =

is a control Lyapunov function, and hence, (6.78)

and (6.79) is globally asymptotically stabilizable. In particular, it is easily

veriﬁable that the feedback control law

u(t) = −x

(t) − x

(t)e

(t)

cos(x

(t)) − x

(t) sin(x

(t)) (6.80)

globally stabilizes (6.78) and (6.79). △

It follows fr om Theorem 6.7 that the zero solution x(t) ≡ 0 of a

nonlinear aﬃne system of the form (6.76) is globally feedback asymptotically

stabilizable if and only if there exists a continuously diﬀerentiable positive-

deﬁnite, radially unbounded function V : R

→ R satisfying (6.77). Hence,

Theorem 6.7 provides necessary and suﬃcient conditions for nonlinear

system stabilization.

Next, using Theorem 6.7 we construct an explicit feedback control

law that is a function of the control Lyapunov function V (·). Speciﬁcally,

consider the feedback control law given by

φ(x) =







−



α(x)+

√

(x)+(β

(x)β(x))

(x)β(x)



β(x), β(x) 6= 0,

0, β(x) = 0,

(6.81)

where α(x)

△

= V

′

(x)f(x), β(x)

△

= G

(x)V

′T

(x), and c

≥ 0. In this case,

the control Lyapunov function V (·) of (6.76) is a Lyapunov function for the

closed-loop system (6.76) with u = φ(x), where φ(x) is given by (6.81). In

particular, the control Lyapunov derivative

V (·) along th e trajectories of the

nonlinear system (6.76) w ith u = φ(x) given by (6.81) is given by

V (x)

△

= V

′

(x)[f(x) + G(x)φ(x)]

= α(x) + β

(x)φ(x)



−c

(x)β(x) −

(x) + (β

(x)β(x))

, β(x) 6= 0,

α(x), β(x) = 0,

< 0, x ∈ R

, x 6= 0, (6.82)

which implies that V (·) is a Lyapunov function for the closed-loop system

(6.76) guaranteeing global asymptotic stability with u = φ(x) given by

(6.81).

Since f(·) and G(·) are smooth it follows that α(x) and β(x), x ∈ R

are smooth functions, and hence, φ(x) given by (6.81) is smooth for all

x ∈ R

if either β(x) 6= 0 or α(x) < 0. Hence, the feedback control law

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STABILITY OF FEEDBACK SYSTEMS 441

given by (6.81) is smooth everywhere except for the origin. The following

result provides necessary and suﬃcient conditions under which the feedback

control law given by (6.81) is guaranteed to be continuous and Lipschitz

continuous at the origin in addition to being smooth everywhere else.

Theorem 6.8. Consider the n onlinear dynamical system G given by

(6.76) with a radially unbounded control Lyapunov function V : R

→ R.

Then the following statements hold:

i) The control law φ(x) given by (6.81) is continuous at x = 0 if and only

if for every ε > 0, there exists δ > 0 such that for all 0 < kxk < δ,

there exists u ∈ R

such that kuk < ε and α(x) + β

(x)u < 0.

ii) There exists a stabilizing control law

φ(x) such that α(x) + β

(x)

φ(x)

< 0, x ∈ R

, x 6= 0, and

φ(x) is Lipschitz continuous at x = 0 if and

only if the control law φ(x) given by (6.81) is Lipschitz continuous at

x = 0.

Proof. Necessity of i) is trivial with u = φ(x). Conversely, assume

that, for every ε > 0, there exists δ > 0 such that for all 0 < kxk < δ, there

exists u ∈ R

such that kuk < ε and α(x) + β

(x)u < 0. I n this case, since

kuk < ε it follows from the Cauchy-Schwarz inequality that α(x) < εkβ(x)k.

Furth ermore, since V (·) is continuously diﬀerentiable and G(·) is continuous

it follows that there exists

δ > 0 such that for all 0 < kxk <

δ, kβ(x)k < ε.

Hence, for all 0 < kxk < δ

min

, where δ

min

△

= min{δ,

δ}, it follows that

α(x) < εkβ(x)k and kβ(x)k < ε. Furthermore, if β(x) = 0, then kφ(x)k = 0,

and if β(x) 6= 0, then it follows from (6.81) that

kφ(x)k ≤ c

kβ(x)k +

|α(x) +

(x) + (β

(x)β(x))

kβ(x)k

≤

2α(x) + (c

+ 1)kβ(x)k

kβ(x)k

≤ (c

+ 3)ε, 0 < kxk < δ

min

, α(x) > 0,

and

kφ(x)k ≤ c

kβ(x)k +

α(x) +

(x) + (β

(x)β(x))

kβ(x)k

≤ c

kβ(x)k +

(x)β(x)

kβ(x)k

= (c

+ 1)kβ(x)k < (c

+ 1)ε, 0 < kxk < δ

min

, α(x) ≤ 0.

Hence, it f ollows that for every ˆε

△

= (c

+ 3)ε > 0, there exists δ

min

> 0 such

that f or all kxk < δ

min

, kφ(x)k < ˆε, which implies that φ(·) is continuous at

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442 CHAPTER 6

the origin.

Next, to s how necessity of ii) assume that there exists a stabilizing

control

φ(x) such that α(x) + β

(x)

φ(x) < 0, x ∈ R

, x 6= 0, and

φ(x) is

Lipschitz continuous at x = 0 with a Lipschitz constant

L; that is, there

exists δ > 0 such that for all x ∈ B

(0), k

φ(x)k ≤

Lkxk. Now, since V (·)

is continuous and V

′

(0) = 0, it follows that there exists K > 0 such that

kβ(x)k ≤ Kkxk, x ∈ B

(0). Hence,

kφ(x)k ≤ c

kβ(x)k +

|α(x) +

(x) + (β

(x)β(x))

kβ(x)k

≤

2α(x) + (c

+ 1)kβ(x)k

kβ(x)k

≤ 2k

φ(x)k + kβ(x)k

≤ (2

L + (c

+ 1)K)kxk, x ∈ B

(0), α(x) > 0,

and

kφ(x)k ≤ c

kβ(x)k +

α(x) +

(x) + (β

(x)β(x))

kβ(x)k

≤ c

kβ(x)k +

(x)β(x)

kβ(x)k

= (c

+ 1)kβ(x)k

< (c

+ 1)Kkxk, x ∈ B

(0), α(x) ≤ 0,

which implies that for all x ∈ B

(0), kφ(x)k ≤ Lkxk, where L

△

= 2

L +

+ 1)K, and hence, φ(·) is Lipschitz continuous. Finally, suﬃciency of ii)

follows immediately with

φ(x) = φ(x).

6.6 Optimal Control and the Hamilton-Jacobi-Bellman Equation

In this section, we consider a control problem involving a notion of optimality

with respect to a nonlinear-nonquadratic cost functional. Speciﬁcally, we

consider the following optimal control problem.

Optimal Control Problem. Consider the nonlinear controlled

system given by

˙x(t) = F (x(t), u(t), t), x(t

) = x

, x(t

) = x

, u(t) ∈ U, t ≥ t

(6.83)

where x(t) ∈ D ⊆ R

, t ≥ 0, is the state vector, D is an open set with 0 ∈ D,

u(t) ∈ U ⊆ R

, t ≥ 0, is the control input, x(t

) = x

is given, x(t

) = x

ﬁxed, and F : D ×U ×R → R

satisﬁes F (0, 0, ·) = 0. We assu me that u(·)

is restricted to the class of admissible controls U consisting of measurable

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STABILITY OF FEEDBACK SYSTEMS 443

functions u(·) such that u(t) ∈ U for all t ≥ 0, where the constraint set

U is given with 0 ∈ U . Furthermore, we assume that F (·, ·, ·) is Lipschitz

continuous in a neighborhood of the origin in D × U × R. Then determine

the control input u(t) ∈ U, t ∈ [t

, t

], such that the cost functional

J(x

, u(·), t

) =

L(x(t), u(t), t)dt, (6.84)

is minimized, where L : D × U × R → R is given.

To solve the optimal control problem we present Bellman’s principle

of optimality, which provides necessary and suﬃcient cond itions, for a given

control u(t) ∈ U, t ≥ 0, for minimizing the cost functional (6.84).

Lemma 6.1. Let u

∗

(·) ∈ U be an optimal control that generates the

trajectory x(t), t ∈ [t

, t

], with x(t

) = x

. Then the trajectory x(·) from

, x

) to (t

, x

) is optimal if and only if for all t

, t

∈ [t

, t

], the portion

of the trajectory x(·) going from (t

, x(t

)) to (t

, x(t

)) optimizes the same

cost functional over [t

, t

], where x(t

) = x

is a point on the optimal

trajectory generated by u

∗

(·).

Proof. Let u

∗

(·) ∈ U solve the optimal control problem and let x(t),

t ∈ [t

, t

], be the solution to (6.83) generated by u

∗

(·). Next, suppose, ad

absurdum, that there exist t

≥ t

, t

≤ t

, and ˆu(t), t ∈ [t

, t

], such that

L(ˆx(t), ˆu(t), t)dt <

L(x(t), u

∗

(t), t)dt,

where ˆx(t) solves (6.83) for all t ∈ [t

, t

] with u(t) = ˆu(t), ˆx(t

) = x(t

and ˆx(t

) = x(t

). Now, deﬁne

(t)

△







∗

(t), t ∈ [t

, t

ˆu(t), t ∈ [t

, t

∗

(t), t ∈ (t

, t

Then,

J(x

, u

(·), t

) =

L(x(t), u

(t), t)dt

L(x(t), u

∗

(t), t)dt +

L(ˆx(t), ˆu(t), t)dt

L(x(t), u

∗

(t), t)dt

L(x(t), u

∗

(t), t)dt +

L(x(t), u

∗

(t), t)dt

NonlinearBook10pt November 20, 2007

444 CHAPTER 6

L(x(t), u

∗

(t), t)dt

= J(x

, u

∗

(·), t

which is a contradiction.

Conversely, if u

∗

(·) min imizes J(·, ·, ·) over [t

, t

] for all t

≥ t

and

≤ t

, then it minimizes J(·, ·, ·) over [t

, t

Lemma 6.1 states that u

∗

(·) solves the optimal control problem over

the time interval [t

, t

] if and only if u

∗

(·) solves the optimal control

problem over every subset of the time interval [t

, t

]. Next, let u

∗

(·) ∈ U

solve the op timal control problem and deﬁne the optimal cost J

∗

, t

)

△

J(x

, u

∗

(·), t

). Furthermore, deﬁne, for p ∈ R

, the Hamiltonian H(x, u,

p, t)

△

= L(x, u, t) + p

F (x, u, t). With these deﬁnitions we h ave the following

result.

Theorem 6.9. Let J

∗

(x, t) denote the minimal cost for the optimal

control problem with x

= x and t

= t, and assume that J

∗

(·, ·) is

continuously diﬀerentiable in x. Then

0 =

∂J

∗

(x(t), t)

∂t

+ min

u(t)∈U

H(x(t), u(t), p(x(t), t), t), (6.85)

where p(x(t), t)

△



∂J

∗

(x(t),t)

∂x



. Furthermore, if u

∗

(·) solves th e optimal

control problem, th en

0 =

∂J

∗

(x(t), t)

∂t

+ H(x(t), u

∗

(t), p(x(t), t), t). (6.86)

Proof. It f ollows from Lemma 6.1 that, for all t

≥ t,

∗

(x(t), t) = min

u(·)∈U

L(x(s), u(s), s)ds

= min

u(·)∈U



L(x(s), u(s), s)ds +

L(x(s), u(s), s)ds



= min

u(·)∈U



L(x(s), u(s), s)ds + J

∗

(x(t

), t

)



or, equivalently,

0 = min

u(·)∈U



− t

∗

(x(t

), t

) − J

∗

(x(t), t))

−t

L(x(s), u(s), s)ds

