C.2. Calculus of Variations

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C.2. Calculus of Variations

In the above definition of the optimal control problem, the notion of a functional was employed. In this section, we will provide basic definitions on functionals and present several properties of interest. A functional is a more general notion of a function, which can be used to treat similar types of functions cumulatively. More specifically, the following definition describes formally a functional:

Definition C.2

A functional

J

$J$

is a rule of correspondence (function) that assigns to each function

x

$x$

Ω

$Ω$

a unique real number. Class

Ω

$Ω$

is called the domain of

J

$J$

and the set of real numbers associated with the functions in

Ω

$Ω$

is called the range of

J

$J$

A functional may be considered as a function of a function. A functional

J

$J$

is linear, if is satisfies the homogeneity and additivity properties:

J (a x) = a J (x)

$J (a x) = a J (x)$

, for every

x \in Ω, a \in R

$x \in Ω, a \in R$

, such that

a x \in Ω

$a x \in Ω$

(homogeneity).

J (x^{(1)} + x^{(2)}) = J (x^{(1)}) + J (x^{(2)})

$J (x^{(1)} + x^{(2)}) = J (x^{(1)}) + J (x^{(2)})$

, for every

x^{(1)}, x^{(2)}, x^{(1)} + x^{(2)} \in Ω

$x^{(1)}, x^{(2)}, x^{(1)} + x^{(2)} \in Ω$

(additivity).

The norm of a function is a rule of correspondence (functional) that assigns a real number to each function

x \in Ω

$x \in Ω$

defined for

t \in [t_{0}, t_{f}]

$t \in [t_{0}, t_{f}]$

. The norm of

x

$x$

, denoted by

‖ x ‖

$‖ x ‖$

, satisfies the following properties:

•

‖ x ‖ \geq 0

$‖ x ‖ \geq 0$

and

‖ x ‖ = 0

$‖ x ‖ = 0$

if and only if

x (t) = 0

$x (t) = 0$

for every

t \in [t_{0}, t_{f}]

$t \in [t_{0}, t_{f}]$

•

‖ a x ‖ = ∣ a ∣ \cdot ‖ x ‖

$‖ a x ‖ = ∣ a ∣ \cdot ‖ x ‖$

for every

a \in R

$a \in R$

•

‖ x^{(1)} + x^{(2)} ‖ \leq ‖ x^{(1)} ‖ + ‖ x^{(2)} ‖

$‖ x^{(1)} + x^{(2)} ‖ \leq ‖ x^{(1)} ‖ + ‖ x^{(2)} ‖$

(triangle inequality).

q

$q$

and

q + Δ q

$q + Δ q$

are elements in the domain of function

f

$f$

, then the increment of

f

$f$

is defined as

$Δ f = Δ f (q, Δ q) = f (q + Δ q) - f (q) .$ $Δ f = Δ f (q, Δ q) = f (q + Δ q) - f (q) .$

(C.9)

Similarly, if

x

$x$

and

x + δ x

$x + δ x$

are functions in the domain of functional

J

$J$

, then the increment of

J

$J$

is defined as

$Δ J = Δ J (x, δ x) = J (x + δ x) - J (x) .$ $Δ J = Δ J (x, δ x) = J (x + δ x) - J (x) .$

(C.10)

Quantity

δ x

$δ x$

is called the variation of function

x

$x$

and it may be considered analogous to the differential for a function of a single variable. The increment of a function of

n

$n$

variables, which will be of more interest in optimal control problems, can be written as

$Δ f (q, Δ q) = d f (q, Δ q) + g (q, Δ q) \cdot ‖ Δ q ‖,$ $Δ f (q, Δ q) = d f (q, Δ q) + g (q, Δ q) \cdot ‖ Δ q ‖,$

(C.11)

where

d f

$d f$

is a linear function of

Δ q

$Δ q$

. If

{lim}_{‖ Δ q ‖ \to 0} g (q, Δ q)) = 0

${lim}_{‖ Δ q ‖ \to 0} g (q, Δ q)) = 0$

f

$f$

is said to be differentiable at

q

$q$

. The differential is given by

$d f = \frac{\partial f}{\partial q_{1}} Δ q_{1} + \frac{\partial f}{\partial q_{2}} Δ q_{2} + . . . + \frac{\partial f}{\partial q_{n}} Δ q_{n}$ $d f = \frac{\partial f}{\partial q_{1}} Δ q_{1} + \frac{\partial f}{\partial q_{2}} Δ q_{2} + . . . + \frac{\partial f}{\partial q_{n}} Δ q_{n}$

(C.12)

when

f

$f$

differentiable. Thus, the increment of a functional can be eventually written as

$Δ J (x, δ x) = δ J (x, δ x) + g (x, δ x) \cdot ‖ δ x ‖,$ $Δ J (x, δ x) = δ J (x, δ x) + g (x, δ x) \cdot ‖ δ x ‖,$

(C.13)

where

δ J

$δ J$

is linear in

δ x

$δ x$

. If

{lim}_{‖ δ x ‖ \to 0} g (x, δ x)) = 0

${lim}_{‖ δ x ‖ \to 0} g (x, δ x)) = 0$

, then

J

$J$

is said to be differentiable in

x

$x$

and

δ J

$δ J$

is the variation of

J

$J$

evaluated for the function

x

$x$

A functional

J

$J$

with domain

Ω

$Ω$

has a relative extremum at

x^{*}

$x^{*}$

if there is an

ϵ > 0

$ϵ > 0$

such that for all functions

x \in Ω

$x \in Ω$

which satisfy

‖ x - x^{*} ‖ < ϵ

$‖ x - x^{*} ‖ < ϵ$

the increment of

J

$J$

has the same sign:

• If

Δ J = J (x) - J (x^{*}) \geq 0

$Δ J = J (x) - J (x^{*}) \geq 0$

J (x^{*})

$J (x^{*})$

is a relative minimum.

• If

Δ J = J (x) - J (x^{*}) \leq 0

$Δ J = J (x) - J (x^{*}) \leq 0$

J (x^{*})

$J (x^{*})$

is a relative maximum.

If the above inequalities are satisfied for arbitrarily large

ϵ

$ϵ$

, then

J (x^{*})

$J (x^{*})$

is a global (absolute) minimum/maximum and

x^{*}

$x^{*}$

is called an extremal , while

J (x^{*})

$J (x^{*})$

is referred to as an extremum .

Theorem C.1

Fundamental Theorem of Calculus of Variations

Let

x

$x$

be a vector function of

t

$t$

Ω

$Ω$

, and

J (x)

$J (x)$

be a differentiable functional of

x

$x$

. Assume that the functions in

Ω

$Ω$

are not constrained by any boundaries. Then if

x^{*}

$x^{*}$

is an extremal, the variation of

J

$J$

must vanish at

x^{*}

$x^{*}$

, i.e.

$δ J (x^{*}, δ x^{*}) = 0$ $δ J (x^{*}, δ x^{*}) = 0$

for all admissible

δ x

$δ x$

Another useful and important result associated with the Fundamental Theorem of Calculus of Variations is the following:

Lemma C.1

Fundamental Lemma of Calculus of Variations

If a function

h

$h$

is continuous and

\int_{t_{0}}^{t_{f}} h (t) δ x (t) d t = 0

$\int_{t_{0}}^{t_{f}} h (t) δ x (t) d t = 0$

for every function

δ x

$δ x$

that is continuous in the interval

[t_{0}, t_{f}]

$[t_{0}, t_{f}]$

, then

h

$h$

must be zero everywhere in

[t_{0}, t_{f}]

$[t_{0}, t_{f}]$

Both the Fundamental Theorem of Calculus of Variations, as well as the Fundamental Lemma of Calculus of Variations are frequently employed to ensure optimality of solutions in various optimal control problems. Their proofs can be found in [137] and references therein. Fig. C.1 highlights the correspondence between functions and functionals as “functions of functions” and the correspondence between extrema of functions-extreme values of functionals.

Fig. C.1 — FIGURE C.1 Analogy between functions, functionals, and extreme values.

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C.2. Calculus of Variations

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C.2. Calculus of Variations