C.3. Finding Trajectories that Minimize Performance Measures

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C.3. Finding Trajectories that Minimize Performance Measures

The problem classes presented in this section provide characteristic methodologies that can be exploited in broader optimal control problems, some of which have been exploited in this book. The rest can be used in various extensions to the study of intelligent attack strategies. In general, the Fundamental Theorem of Calculus of Variations can be exploited in optimal control theory for finding trajectories that minimize performance measures expressed as functionals. This is also the case in the corresponding modeling of malware diffusion in Chapter 6 and potential extensions to attack strategies.

C.3.1. Functionals of a Single Function

When the performance metric is a function of a single scalar function

x

$x$

in the class of continuous first derivatives, the functional can be written in the form

$J (x) = \int_{t_{0}}^{t_{f}} g (x (t), \dot{x} (t), t) d t .$ $J (x) = \int_{t_{0}}^{t_{f}} g (x (t), \dot{x} (t), t) d t .$

(C.14)

The objective is to find

x^{*}

$x^{*}$

for which

J (x)

$J (x)$

has a relative extremum. Functions (curves) in

Ω

$Ω$

that satisfy the end conditions are called admissible.

When

x (t_{0})

$x (t_{0})$

x (t_{f})

$x (t_{f})$

are free and

t_{f}

$t_{f}$

specified, the functional increment can be written as

$Δ J (x, δ x) = \int_{t_{0}}^{t_{f}} [g (x (t) + δ x (t), \dot{x} (t) + δ \dot{x} (t), t) - g (x (t), \dot{x} (t), t)] d t,$ $Δ J (x, δ x) = \int_{t_{0}}^{t_{f}} [g (x (t) + δ x (t), \dot{x} (t) + δ \dot{x} (t), t) - g (x (t), \dot{x} (t), t)] d t,$

(C.15)

which using a Taylor series expansion about

(x (t), \dot{x} (t))

$(x (t), \dot{x} (t))$

can be expressed as

$δ J (x, δ x) = \int_{t_{0}}^{t_{f}} {[\frac{\partial g}{\partial x} (x (t), \dot{x} (t), t)] δ x (t) + [\frac{\partial g}{\partial \dot{x}} (x (t), \dot{x} (t), t)] δ \dot{x} (t)} d t .$ $δ J (x, δ x) = \int_{t_{0}}^{t_{f}} {[\frac{\partial g}{\partial x} (x (t), \dot{x} (t), t)] δ x (t) + [\frac{\partial g}{\partial \dot{x}} (x (t), \dot{x} (t), t)] δ \dot{x} (t)} d t .$

(C.16)

By the fundamental theorem of calculus of variations, a necessary condition for

x^{*}

$x^{*}$

to be an extremal is

$\frac{\partial g}{\partial x} (x^{*} (t), {\dot{x}}^{*} (t), t) - \frac{d}{d t} [\frac{\partial g}{\partial \dot{x}} (x^{*} (t), {\dot{x}}^{*} (t), t)] = 0, \forall t \in [t_{0}, t_{f}] .$ $\frac{\partial g}{\partial x} (x^{*} (t), {\dot{x}}^{*} (t), t) - \frac{d}{d t} [\frac{\partial g}{\partial \dot{x}} (x^{*} (t), {\dot{x}}^{*} (t), t)] = 0, \forall t \in [t_{0}, t_{f}] .$

(C.17)

The latter is an Euler differential equation. More specifically, it is a second order nonlinear ordinary and time-varying differential equation, which is hard to solve since it yields a two-point boundary-value problem with split boundary conditions.

When an extremal for the metric given in (C.15) with

t_{0}, t_{f}, x (t_{0})

$t_{0}, t_{f}, x (t_{0})$

specified and

x (t_{f})

$x (t_{f})$

free (which corresponds to the case where one boundary condition is given and the second is obtained from Euler equation-differential boundary condition) is required, the same approach as before applies as well, where now the metric functional is integrated by parts. It turns out that the necessary condition for

x^{*}

$x^{*}$

to be an extremal is Eq. (C.17), and we may also obtain a natural boundary condition

$\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) = 0 .$ $\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) = 0 .$

(C.18)

Thus, an extremal for a free endpoint problem is also an extremal for the fixed endpoint problem with the same endpoints and the same functional.

If one is given metric (C.15) and

t_{0}, x (t_{0}) = x_{0}, x (t_{f}) = x_{f}

$t_{0}, x (t_{0}) = x_{0}, x (t_{f}) = x_{f}$

are specified while

t_{f}

$t_{f}$

is free, Euler condition (C.17) is satisfied, along with the boundary condition

$g (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) - [\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}))] {\dot{x}}^{*} (t_{f}) = 0 .$ $g (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) - [\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}))] {\dot{x}}^{*} (t_{f}) = 0 .$

Finally, when an extremal is sought for metric (C.15) with

t_{0}, x (t_{0}) = x_{0}

$t_{0}, x (t_{0}) = x_{0}$

specified and

t_{f}, x (t_{f}) = x_{f}

$t_{f}, x (t_{f}) = x_{f}$

is free, again Euler condition (C.17) is satisfied, and when

t_{f}

$t_{f}$

and

x (t_{f})

$x (t_{f})$

are unrelated,

δ x_{f}, δ t_{f}

$δ x_{f}, δ t_{f}$

are independent and arbitrary yielding

g (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) = 0

$g (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) = 0$

, while if

t_{f} - x (t_{f})

$t_{f} - x (t_{f})$

are related, e.g. in the form

x (t_{f}) = θ (t_{f})

$x (t_{f}) = θ (t_{f})$

, then

δ x_{f} = \frac{\partial θ}{\partial t} (t_{f}) δ t_{f}

$δ x_{f} = \frac{\partial θ}{\partial t} (t_{f}) δ t_{f}$

and the transversality condition

$[\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f})] [\frac{d θ}{d t} (t_{f}) - {\dot{x}}^{*} (t_{f})] + g (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) = 0$ $[\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f})] [\frac{d θ}{d t} (t_{f}) - {\dot{x}}^{*} (t_{f})] + g (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) = 0$

(C.19)

is obtained for solving the problem.

C.3.2. Functionals of Several Independent Functions

When the problem involves several independent functions, the performance metric can be written in the form

$J (x) = \int_{t_{0}}^{t_{f}} g (x (t), \dot{x} (t), t) d t,$ $J (x) = \int_{t_{0}}^{t_{f}} g (x (t), \dot{x} (t), t) d t,$

(C.20)

with

t_{0}, t_{f}, x (t_{0}) = x_{0}, x (t_{f}) = x_{f}

$t_{0}, t_{f}, x (t_{0}) = x_{0}, x (t_{f}) = x_{f}$

all specified (fixed endpoints). This leads to a more convenient matrix form of the Euler equation

(C.21)

When the metric is given by (C.20) and

t_{0}, x (t_{0}) = x_{0}

$t_{0}, x (t_{0}) = x_{0}$

are specified, but

t_{f}, x (t_{f}) = x_{f}

$t_{f}, x (t_{f}) = x_{f}$

are free, then Euler condition (C.21) is satisfied along with boundary conditions at the final time of the form

${[\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f})]}^{T} δ x_{f} + [g (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) - {[\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t)]}^{T} {\dot{x}}^{*} (t_{f})] δ t_{f} = 0 .$ ${[\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f})]}^{T} δ x_{f} + [g (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t_{f}) - {[\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{f}), {\dot{x}}^{*} (t_{f}), t)]}^{T} {\dot{x}}^{*} (t_{f})] δ t_{f} = 0 .$

(C.22)

C.3.3. Piecewise-smooth Extremals

When a function has piecewise continuous first derivatives except for a finite number of times (corner points), where

\dot{x}

$\dot{x}$

is discontinuous, then the functional of the performance metric can be defined in intervals that do not include the corner points. In this case, necessary conditions for

x^{*}

$x^{*}$

to be an extremal for

J

$J$

in each of the corresponding time intervals excluding the corner points of the function are the Weierstrass-Erdmann corner conditions, which for functions of many variables, can be written as

$\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{-}), t_{1}) = \frac{\partial g}{\partial \dot{x}} (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{+}), t_{1}) and$ $\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{-}), t_{1}) = \frac{\partial g}{\partial \dot{x}} (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{+}), t_{1}) and$

$g (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{-}), t_{1}) - [\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{-}), t_{1})] {\dot{x}}^{*} (t_{1}^{-}) = g (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{+}), t_{1}) - [\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{+}), t_{1})] {\dot{x}}^{*} (t_{1}^{+}),$ $g (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{-}), t_{1}) - [\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{-}), t_{1})] {\dot{x}}^{*} (t_{1}^{-}) = g (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{+}), t_{1}) - [\frac{\partial g}{\partial \dot{x}} (x^{*} (t_{1}), {\dot{x}}^{*} (t_{1}^{+}), t_{1})] {\dot{x}}^{*} (t_{1}^{+}),$

where

t_{1}

$t_{1}$

is corner point of

g

$g$

C.3.4. Constrained Extrema

State-control equations involve finding extrema of functionals of

n + m

$n + m$

functions, namely, the state

x

$x$

and controls

u

$u$

, where only

m

$m$

of the functions are independent, i.e. the controls

u

$u$

. For the constrained minimization of functions, two alternative methodologies can be used, namely, the elimination method and the Lagrange multipliers method. The first can be employed if the dependence of variables/functions can be solved with respect to one of them and substituted in the differential

d f (y^{*}) = 0

$d f (y^{*}) = 0$

More specifically, assume the problem of finding the extreme values for a function of

n + m

$n + m$

variables

y_{1}, . . ., y_{n + m}

$y_{1}, . . ., y_{n + m}$

f (y_{1}, . . ., y_{n + m})

$f (y_{1}, . . ., y_{n + m})$

, given

n

$n$

constraints of the form

$a_{i} (y_{1}, . . ., y_{n + m}) = 0, i = 1, 2, \dots, n,$ $a_{i} (y_{1}, . . ., y_{n + m}) = 0, i = 1, 2, \dots, n,$

where only

m

$m$

variables out of the

n + m

$n + m$

are independent.

Under the elimination method and the above constraints,

$y_{i} = e_{i} (y_{1}, . . ., y_{n + m}) = 0, i = 1, 2, \dots . n,$ $y_{i} = e_{i} (y_{1}, . . ., y_{n + m}) = 0, i = 1, 2, \dots . n,$

and then by substituting into

f

$f$

, a function

f (y_{n + 1}, . . ., y_{n + m})

$f (y_{n + 1}, . . ., y_{n + m})$

m

$m$

independent variables is obtained. Then, this satisfies the following system of equations:

$\frac{\partial f}{\partial y_{n + 1}} (y_{n + 1}^{*}, . . ., y_{n + m}^{*}) = 0,$ $\frac{\partial f}{\partial y_{n + 1}} (y_{n + 1}^{*}, . . ., y_{n + m}^{*}) = 0,$

$. . . .$ $. . . .$

$\frac{\partial f}{\partial y_{n + m}} (y_{n + 1}^{*}, . . ., y_{n + m}^{*}) = 0,$ $\frac{\partial f}{\partial y_{n + m}} (y_{n + 1}^{*}, . . ., y_{n + m}^{*}) = 0,$

which can be solved for

y_{n + 1}^{*}, . . ., y_{n + m}^{*}

$y_{n + 1}^{*}, . . ., y_{n + m}^{*}$

and substituted back in

y_{1}, . . ., y_{n}

$y_{1}, . . ., y_{n}$

to obtain

y_{1}^{*}, . . ., y_{n}^{*}

$y_{1}^{*}, . . ., y_{n}^{*}$

. Then

f (y_{1}^{*}, . . ., y_{n + m}^{*})

$f (y_{1}^{*}, . . ., y_{n + m}^{*})$

can be obtained.

In the Lagrange multipliers method, the augmented function

$f_{a} (y_{1}, . . ., y_{n + m}, p_{1}, . . ., p_{n}) = f (y_{1}, . . ., y_{n + m}) + p_{1} [a_{1} (y_{1}, . . ., y_{n + m})] + . . . + p_{n} [a_{n} (y_{1}, . . ., y_{n + m})]$ $f_{a} (y_{1}, . . ., y_{n + m}, p_{1}, . . ., p_{n}) = f (y_{1}, . . ., y_{n + m}) + p_{1} [a_{1} (y_{1}, . . ., y_{n + m})] + . . . + p_{n} [a_{n} (y_{1}, . . ., y_{n + m})]$

(C.23)

is defined and then the differential of the augmented function is obtained

$d f_{a} = \frac{\partial f_{a}}{\partial y_{1}} Δ y_{1} + . . . \frac{\partial f_{a}}{\partial y_{n + m}} Δ y_{n + m} + \frac{\partial f_{a}}{\partial p_{1}} Δ p_{1} + . . . + \frac{\partial f_{a}}{\partial p_{1}} Δ p_{n} = \frac{\partial f_{a}}{\partial y_{1}} Δ y_{1} + . . . \frac{\partial f_{a}}{\partial y_{n + m}} Δ y_{n + m} + a_{1} Δ p_{1} + . . . a_{n} Δ p_{n},$ $d f_{a} = \frac{\partial f_{a}}{\partial y_{1}} Δ y_{1} + . . . \frac{\partial f_{a}}{\partial y_{n + m}} Δ y_{n + m} + \frac{\partial f_{a}}{\partial p_{1}} Δ p_{1} + . . . + \frac{\partial f_{a}}{\partial p_{1}} Δ p_{n} = \frac{\partial f_{a}}{\partial y_{1}} Δ y_{1} + . . . \frac{\partial f_{a}}{\partial y_{n + m}} Δ y_{n + m} + a_{1} Δ p_{1} + . . . a_{n} Δ p_{n},$

from which

2 n + m

$2 n + m$

equations from the KKT conditions ([27,42])

$a_{i} (y_{1}^{*}, . . ., y_{n + m}^{*}) = 0, i = 1, 2, . . ., n,$ $a_{i} (y_{1}^{*}, . . ., y_{n + m}^{*}) = 0, i = 1, 2, . . ., n,$

$\frac{\partial f_{a}}{\partial y_{j}} (y_{1}^{*}, . . . ., y_{n + m}^{*}, p_{1}^{*}, . . . p_{n}^{*}) = 0, j = 1, 2, . . ., n + m$ $\frac{\partial f_{a}}{\partial y_{j}} (y_{1}^{*}, . . . ., y_{n + m}^{*}, p_{1}^{*}, . . . p_{n}^{*}) = 0, j = 1, 2, . . ., n + m$

can be obtained, leading to

(y_{1}^{*}, . . ., y_{n + m}^{*})

$(y_{1}^{*}, . . ., y_{n + m}^{*})$

The above are valid for functions of single or multiple variables. In case of minimization of functionals, however, additional constraints in the form of differential equation or point constraints are required, both of which lead to the necessary condition

$\frac{\partial g_{a}}{\partial w} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), t) - \frac{d}{d t} [\frac{\partial g_{a}}{\partial \dot{w}} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), t)] = 0,$ $\frac{\partial g_{a}}{\partial w} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), t) - \frac{d}{d t} [\frac{\partial g_{a}}{\partial \dot{w}} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), t)] = 0,$

(C.24)

where

g_{a} (w (t), \dot{w} (t), p (t), t) = g (w (t), \dot{w} (t), t) + p^{T} (t) [f (w (t), \dot{w} (t), t)]

$g_{a} (w (t), \dot{w} (t), p (t), t) = g (w (t), \dot{w} (t), t) + p^{T} (t) [f (w (t), \dot{w} (t), t)]$

Then the general form of the problem is to minimize the functional

J (w) = \int_{t_{0}}^{t_{f}} g (w (t), \dot{w} (t), t) d t

$J (w) = \int_{t_{0}}^{t_{f}} g (w (t), \dot{w} (t), t) d t$

and the problem can be eventually solved using either the elimination or the Lagrange multipliers (augmented functional) method.

For the case of point constraints, the optimal

w^{*}

$w^{*}$

is required to satisfy constraints of the form

f_{i} (w (t), t) = 0, i = 1, 2, . . ., n

$f_{i} (w (t), t) = 0, i = 1, 2, . . ., n$

, where only

m

$m$

out of

n + m

$n + m$

components of

w

$w$

are independent. The augmented functional is

$J_{a} (w, p) = \int_{t_{0}}^{t_{f}} {g (w (t), \dot{w} (t), t) + p_{1} (t) [f_{1} (w (t), t)] + . . . + p_{n} (t) [f_{n} (w (t), t)]} d t = \int_{t_{0}}^{t_{f}} [g (w, \dot{w} (t), t) + p^{T} (t) [f (w (t), t)] d t .$ $J_{a} (w, p) = \int_{t_{0}}^{t_{f}} {g (w (t), \dot{w} (t), t) + p_{1} (t) [f_{1} (w (t), t)] + . . . + p_{n} (t) [f_{n} (w (t), t)]} d t = \int_{t_{0}}^{t_{f}} [g (w, \dot{w} (t), t) + p^{T} (t) [f (w (t), t)] d t .$

On an extremal, the point constraints must be satisfied

f (w (t), t) = 0

$f (w (t), t) = 0$

, as well as

δ J_{a} (w^{*}, p) = 0

$δ J_{a} (w^{*}, p) = 0$

, where

$δ J_{a} (w, δ w, p, δ p) = \int_{t_{0}}^{t_{f}} {[\frac{\partial g^{T}}{\partial w} (w (t), \dot{w} (t), t) + p^{T} (t) [\frac{\partial f}{\partial w} (w (t), t)]] δ w (t) + [\frac{\partial g^{T}}{\partial w} (w (t), \dot{w} (t), t)] δ \dot{w} (t) + [f^{T} (w (t), t)] δ p (t)} d t,$ $δ J_{a} (w, δ w, p, δ p) = \int_{t_{0}}^{t_{f}} {[\frac{\partial g^{T}}{\partial w} (w (t), \dot{w} (t), t) + p^{T} (t) [\frac{\partial f}{\partial w} (w (t), t)]] δ w (t) + [\frac{\partial g^{T}}{\partial w} (w (t), \dot{w} (t), t)] δ \dot{w} (t) + [f^{T} (w (t), t)] δ p (t)} d t,$

leading to aforementioned condition (C.24).

In the cases when the constraints are given in the form of differential equations

f_{i} (w (t), \dot{w} (t), t) = 0, i = 1, 2, . . ., n

$f_{i} (w (t), \dot{w} (t), t) = 0, i = 1, 2, . . ., n$

, where again only

m

$m$

out of

n + m

$n + m$

components of

w

$w$

are independent. In this case, the elimination approach is almost impossible, and Lagrange multipliers have to be adopted. The augmented functional is as before and the conditions on the extremals

f (w^{*} (t), {\dot{w}}^{*} (t), t) = 0

$f (w^{*} (t), {\dot{w}}^{*} (t), t) = 0$

and

δ J_{a} (w^{*}, p) = 0

$δ J_{a} (w^{*}, p) = 0$

again yield condition (C.24), where

$δ J_{a} (w, δ w, p, δ p) = \int_{t_{0}}^{t_{f}} {[\frac{\partial g^{T}}{\partial w} (w (t), \dot{w} (t), t) + p^{T} (t) [\frac{\partial f}{\partial w} (w (t), \dot{w} (t), t)]] δ w (t) + [\frac{\partial g^{T}}{\partial \dot{w}} (w (t), \dot{w} (t), t) + p^{T} (t) [\frac{\partial f}{\partial \dot{w}} (w (t), \dot{w} (t), t)]] δ \dot{w} (t) + [f^{T} (w (t), \dot{w} (t), t)] δ p (t)} d t .$ $δ J_{a} (w, δ w, p, δ p) = \int_{t_{0}}^{t_{f}} {[\frac{\partial g^{T}}{\partial w} (w (t), \dot{w} (t), t) + p^{T} (t) [\frac{\partial f}{\partial w} (w (t), \dot{w} (t), t)]] δ w (t) + [\frac{\partial g^{T}}{\partial \dot{w}} (w (t), \dot{w} (t), t) + p^{T} (t) [\frac{\partial f}{\partial \dot{w}} (w (t), \dot{w} (t), t)]] δ \dot{w} (t) + [f^{T} (w (t), \dot{w} (t), t)] δ p (t)} d t .$

Finally, the extremal

w^{*}

$w^{*}$

for the functional could be subject to isoperimetric⁴ constraints of the form

\int_{t_{0}}^{t_{f}} e_{i} (w (t), \dot{w} (t), t) d t = c_{i}, i = 1, 2, . . ., n

$\int_{t_{0}}^{t_{f}} e_{i} (w (t), \dot{w} (t), t) d t = c_{i}, i = 1, 2, . . ., n$

. In this case, the auxiliary variable

z_{i} = \int_{t_{0}}^{t_{f}} e_{i} (w (t), \dot{w} (t), t) d t = c_{i}

$z_{i} = \int_{t_{0}}^{t_{f}} e_{i} (w (t), \dot{w} (t), t) d t = c_{i}$

is defined, so that in vector form

z (t) = e (w (t), \dot{w} (t), t)

$z (t) = e (w (t), \dot{w} (t), t)$

with boundary values

z_{i} (t_{0}) = 0

$z_{i} (t_{0}) = 0$

and

z_{i} (t_{f}) = c_{i}

$z_{i} (t_{f}) = c_{i}$

. In this case, the augmented functional becomes

$g_{a} (w (t), \dot{w} (t), p (t), t) = g (w (t), \dot{w} (t), t) + p^{T} (t) [e (w (t), \dot{w} (t), t) - \dot{z} (t)] .$ $g_{a} (w (t), \dot{w} (t), p (t), t) = g (w (t), \dot{w} (t), t) + p^{T} (t) [e (w (t), \dot{w} (t), t) - \dot{z} (t)] .$

Then, the overall system can be solved by using the

n + m

$n + m$

available equations obtained from

$\frac{\partial g_{a}}{\partial w} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), {\dot{z}}^{*} (t), t) - \frac{d}{d t} [\frac{\partial g_{a}}{\partial \dot{w}} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), {\dot{z}}^{*} (t), t)] = 0,$ $\frac{\partial g_{a}}{\partial w} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), {\dot{z}}^{*} (t), t) - \frac{d}{d t} [\frac{\partial g_{a}}{\partial \dot{w}} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), {\dot{z}}^{*} (t), t)] = 0,$

the

r

$r$

equations available from the following:

$\frac{\partial g_{a}}{\partial z} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), {\dot{z}}^{*} (t), t) - \frac{d}{d t} [\frac{\partial g_{a}}{\partial \dot{z}} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), {\dot{z}}^{*} (t), t)] = 0$ $\frac{\partial g_{a}}{\partial z} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), {\dot{z}}^{*} (t), t) - \frac{d}{d t} [\frac{\partial g_{a}}{\partial \dot{z}} (w^{*} (t), {\dot{w}}^{*} (t), p^{*} (t), {\dot{z}}^{*} (t), t)] = 0$

and the

r

$r$

equations available from

$z (t) = e (w (t), \dot{w} (t), t),$ $z (t) = e (w (t), \dot{w} (t), t),$

while in addition it holds that

\dot{p} (t) = 0

$\dot{p} (t) = 0$

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Table of Contents for C.3. Finding Trajectories that Minimize Performance Measures

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C.3. Finding Trajectories that Minimize Performance Measures

C.3.1. Functionals of a Single Function

C.3.2. Functionals of Several Independent Functions

C.3.3. Piecewise-smooth Extremals

C.3.4. Constrained Extrema

Table of Contents for
C.3. Finding Trajectories that Minimize Performance Measures