Optimal Time-sampling Problem in a Statistical Control

with a Quadratic Cost Functional

Analytical and Numerical Approaches

Valery Y. Glizer and Vladimir Turetsky

Department of Applied Mathematics, ORT Braude College of Engineering, P. O .B. 78, Karmiel 2161002, Israel

Keywords:

Statistical Control, Statistical Information, Quadratic Cost Functional, Optimal Time-sampling, Pontryagin’s

Maximum Principle, Quadratic Optimization.

Abstract:

We consider the problem of constructing an optimal time-sampling for a Statistical Process Control (or, brieﬂy,

Statistical Control (SC)). The aim of this time-sampling is to minimize the expected loss, caused by a delay

in the detection of an undesirable process change. We study the case where this loss is a quadratic functional

of the sampling time-interval. This problem is modeled by a nonstandard calculus of variations problem.

We propose two approaches to the solution of this calculus of variations problem. The ﬁrst approach is

based on its equivalent transformation to an optimal control problem. The latter is solved by application

of the Pontryagin’s Maximum Principle, yielding an analytical expression for the optimal time-sampling in

the SC. The second approach uses a discretization of the calculus of variations problem, resulting in a ﬁnite

dimensional quadratic optimization problem. Solution of the latter provides a suboptimal time-sampling in

the SC. The time-samplings, obtained by these two approaches, are compared to each other in numerical

examples.

1 INTRODUCTION

The Statistical Control is a quality control method

(see, e.g., (Qiu, 2013) and references therein). It con-

sists in a monitoring of a process state using a statisti-

cal information on samples of its characteristic index

in some time-intervals. The SC is applied in indu-

stry, medicine, veterinary, environment control, etc.

Its objective is to minimize losses which can be cau-

sed by delay in the detection of undesirable process

changes, subject to reasonable inspection expenses.

For many years, the traditional SC practice was to

take samples of the process characteristic index with

a ﬁxed time-sampling. The idea of using a variable

time-sampling (VTS) in the SC was suggested for the

ﬁrst time in the work (Reynolds et al., 1988). Then,

this idea was developed in a number of works (see,

e.g., (Amin and Hemasinha, 1993); (Amin and Miller,

1993); (Bashkansky and Glizer, 2012); (Chew et al.,

2015); (Costa, 1994); (Costa, 1997); (Costa, 1998);

(Costa, 1999b); (Costa, 1999a); (Costa and Magal-

haes, 2007); (Glizer et al., 2015); (Hatjimihail, 2009);

(Sultana et al., 2014); (Li and Qiu, 2014); (Prabhu

et al., 1994); (Reynolds, 1995)).

In (Reynolds et al., 1988), the delay in the de-

tection of a process change was considered as a cri-

terion for optimality of the variable sampling time-

interval in the SC. Another possible criterion, propo-

sed in (Taguchi et al., 2007). is the expected loss due

to such a delay. The latter criterion is more general

and, therefore, more suitable for various applications.

In many processes, the relation between the ex-

pected loss and the delay in the detection of a process

change is non-linear. Among such processes, we can

mention: (i) ﬁres propagation (Babrauskas, 2008), (ii)

oil spills spreading (Sebasti˜ao and Soares, 1995), (iii)

cholesterol plaque growth (Bulelzai and Dubbeldam,

2012), (iv) epidemics propagation (Carpenter et al.,

2011), (v) fatigue crack growth in ship hull structures

(Kim and Frangopol, 2011).

Genichi Taguchi (see e.g. (Taguchi et al., 2007))

proposed a quadratic dependence of the expected loss

on some critical performance parameter of a process.

In modern industry, medicine, veterinary, natural en-

vironment protection, etc, the statistical control of a

process becomes its indispensable part. Therefore,

the delay in the detection of a process change can be

considered as a critical performance parameter of the

process. This observation yields a quadratic depen-

dence of the expected loss on the detection delay.

Glizer, V. and Turetsky, V.

Optimal Time-sampling Problem in a Statistical Control with a Quadratic Cost Functional - Analytical and Numerical Approaches.

DOI: 10.5220/0006829000210032

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 21-32

ISBN: 978-989-758-321-6

We model the problem of the SC time-sampling

optimization by some calculus of variations problem.

This problem consists of a cost functional (the ex-

pected loss) and two types of constraints (geome-

tric and integral inequality constraints). The geo-

metric constraint gives the lower and upper bounds

of each sampling time-interval. The integral inequa-

lity constraint means that the average of the sampling

time-interval is not prescribed but it belongs to a gi-

ven interval. This model for the SC time-sampling

optimization is more general than those studied in

((Bashkansky and Glizer, 2012); (Glizer et al., 2015)).

Moreover, both types of the constraints are not studied

in the classical calculus of variations theory. Thus, the

considered extremal problem is nonstandard. We pro-

pose two methods of its solution, which are not based

on a preliminary approximate decomposition of this

problem. The ﬁrst method converts equivalently the

original extremal problem into an optimal control pro-

blem. This optimal control problem is solved using

the Pontryagin’s Maximum Principle (PMP), which

yields an exact analytical solution to the original cal-

culus of variations problem. This solution constitutes

the optimal time-sampling of the SC. In the second

method, the original calculus of variations problem is

replaced approximately with a ﬁnite-dimensional op-

timization problem. The latter is solved using cor-

responding mathematical programming tools, which

yields an approximate solution of the original extre-

mal problem. This solution constitutes the su boptimal

time-sampling of the SC.

It is important to note, that the SC time-sampling,

designed in this paper, depends on the current state of

the process. It is not designed in advance for an en-

tire period of the process control. Applying the termi-

nology of control engineering, this SC time-sampling

can be called a state-feedback time-sampling.

Also, it should be noted that in most VTS sche-

mes, described in the literature, the sampling time-

interval of only two different lengths is considered. In

the present paper, as well as in (Li and Qiu, 2014) and

(Glizer et al., 2015), more than two different lengths

of the sampling time-interval are proposed for the SC.

In (Li and Qiu, 2014), the multiple lengths sampling

time-interval is related to the p-value of the charting

statistic, while in (Glizer et al., 2015) and the present

paper, such sampling time-intervals are derived from

solutions (exact and approximate) of the optimization

problems.

2 PROBLEM STATEMENT

We analyze the SC case where the monitoring of a

characteristic index x of the process state is carried out

based on the information about its sample mean. Na-

mely, at some prescribed/precalculated time instance

t a batch of n observations x

, (j =

1,n) of the value x

is obtained, and the sample mean ¯x of these observati-

ons is derived. We assume that the sample size n is in-

dependent of t. Let µ and σ be the mean value and the

standard deviation of the random value x. Then, the

mean value and the standard deviation of the random

value ¯x are µ and σ/

√

n. In this paper, we deal with

the case where the random value ¯x is normally distri-

buted, i.e., ¯x ∼ N(µ, σ/

√

n). This occurs when either

the random value x is normally distributed, or the

sample size n is considerably large (n ≥ 30). In the

latter case, by virtue of the Central Limit Theorem,

the normal distribution N(µ, σ/

√

n) provides a good

approximation of ¯x even if x does not strictly ﬁt a nor-

mal distribution (see, e.g., (Qiu, 2013) and references

therein). Thus, the normalized sample mean, called

the standard score, is z =



¯x−µ





σ/

√



∼N(0,1).

The upper and lower limits of the standard Shewhart

control chart for z are z

min

= −3 and z

max

= 3, re-

spectively, (see (Qiu, 2013)). Therefore, the false

alarm probability α (type I error), i.e., the probability

of the event z /∈ [−3, 3], is α ≈ 0.0027.

Let the mean value of the index x is shifted by ∆,

i.e., a new mean value is µ

′

= µ+∆, while the standard

deviation σ remains unchanged. Then the distribu-

tion of z becomes z ∼ N(δ,1), where the normalized

shift δ = ∆/



σ/

√



is the so-called signal-to-noise

ratio. The probability of discovering the shift (recei-

ving the signal) by a single sample is the probability

of the event z /∈ [−3, 3] for z ∼ N(δ,1):

1 −β = 1 −

√

2π

−3

exp



−(z −δ)



= 1 −[Φ(3 −δ)−Φ(−3 −δ)], (1)

where β = β(δ) is the probability of a type II error

(not discovering the shift).

Consider the SC with a variable sampling time-

interval u(z), depending on the standard score z =



¯x −µ





σ/

√



. Since the value of the sampling

time-interval should depend only on |z|, the function

u(z) is even



u(−z) = u(z)



. Therefore, in what fol-

lows, we consider the function u(z) in the interval

[0,3]. Also, for the sake of simplicity, we assume that

δ ≥ 0. The case δ ≤0 is treated similarly. Further, we

assume that the function u(z) is bounded as:

0 < u

min

≤ u(z) ≤ u

max

, z ∈ [0,3]. (2)

The inequality (2) is a geometric constraint, imposed

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

on the function u(z). Now, let us consider the follo-

wing integral constraint, imposed on u(z):

min

≤

exp(−z

/2)u(z)dz ≤ aT

max

, (3)

where 0 < T

min

< T

max

, and

△

exp(−z

/2)dz > 0. (4)

Remark 1. The inequality (3) means that the ex-

pected sampling time-interval in the case of u nshif-

ted z (δ = 0) belongs to a prescribed nominal interval

min

max

]. We assume that

min

> u

min

, T

max

< u

max

. (5)

If the shift of the mean in the process characteris-

tic index remains constant, the time t

, required for

discovering this shift (so-called time to signal), is the

sum of a random amount K

of random independent

and identically distributed sampling time-intervals u

conditionally independent of K

: t

∑

i=1

. The va-

lue K

is distributed geometrically with the success

probability 1 −β, given by (1). Its mathematical ex-

pectation and variance are (Ross, 2009): E(K

) =

1/(1 −β), Var(K

) = β/(1 −β)

The cost functional, to be minimized by a proper

choice of the sampling time-interval u(z), is the mat-

hematical expectation E(L) of the loss L, caused by

the delay in the detection of the shift. Here, we con-

sider the loss L as a quadratic function of the delay

, i.e., L = kt

, where k = k(δ) ≥ 0 is an increasing

function of δ ≥ 0 and k(0) = 0. Thus, the expected

loss is

E(L) = k(δ)E(t

). (6)

By routine calculations, we have

E(t

) = A(δ)



ψ(z,δ)u

(z)dz

+B(δ)



ψ(z,δ)u(z)dz





, (7)

where

A(δ)

△

exp(−δ

/2)

(1 −β)β

√

2π

> 0, B(δ)

△

2 exp(−δ

/2)

(1 −β)

√

2π

> 0,

(8)

ψ(z,δ)

△

= 2exp



−z



cosh(δz) > 0, z ∈[0,3]. (9)

Since k(δ) > 0 for all δ > 0, then due to (7) – (8),

the minimization of the cost functional (6) for any gi-

ven δ > 0 is equivalent to the minimization of the fol-

lowing cost functional:



u(z)



△

ψ(z,δ)u

(z)dz

+B(δ)



ψ(z,δ)u(z)dz



. (10)

Remark 2. Two cases can be distinguished with re-

spect to the information on the value of δ: (i) the value

of δ is known; (ii) the value of δ is unknown. In the

ﬁrst case, one should minimize with respect to u(z) the

cost functio nal (10), calculated for the known δ. In

the second case, one should minimize with respect to

u(z) the cost functional (6) – (7) robustly in δ ≥0, i.e.,

one should minimize with respect to u(z) the new cost

functional J

new



u(z)



= max

δ≥0

k(δ)E(t

). In this pa-

per, we restrict our analysis with the ﬁrst case.

Thus, we can formulate the following extremal

problem.

Extremal Problem (EP): for a known δ ≥ 0, to ﬁnd

the function u(z), z ∈ [0,3], which minimizes the cost

functional (10) subject to the constraints (2), (3) and

the inequality (5).

In subsequent sections, we solve the EP in a closed

analytical form and numerically, thus designing the

optimal and suboptimal SC time-sampling.

3 ANALYTICAL SOLUTION OF

THE EP

The EP is a nonstandard calculus of variations pro-

blem with two types of constraints, the geometric con-

straint (2) and the integral inequality constraint (3),

imposed on the minimizing function. The classical

calculus of variations theory does not study extremal

problems with such types of constraints (see, e.g.,

(Gelfand and Fomin, 1963)). We propose another ap-

proach to the solution of this problem, which consists

in an equivalent transformation of the EP into an opti-

mal control problem. The latter is analyzed by appli-

cation of the control optimality necessary condition –

the Pontryagin’s Maximum Principle (PMP) (Pontry-

agin et al., 1962).

3.1 Transformation of the EP

Let us introduce the auxiliary vector-valued function

w(z) =



(z),w

(z)



, z ∈ [0,3], where

(z) =

ψ(ζ,δ)u

(ζ)dζ, (11)

(z) =

ψ(ζ,δ)u(ζ)dζ, (12)

(z) =

exp



−ζ



u(ζ)dζ. (13)

The functions w

(z), (i = 1, 2, 3), satisfy the differen-

tial equations

/dz = ψ(z,δ)u

(z), (14)

Optimal Time-sampling Problem in a Statistical Control with a Quadratic Cost Functional - Analytical and Numerical Approaches

/dz = ψ(z,δ)u(z), (15)

/dz = exp



−z



u(z), (16)

and the initial conditions

(0) = 0, w

(0) = 0. (17)

Based on (13), the integral inequality (3) of the EP

becomes

min

≤ w

(3) ≤ aT

max

. (18)

This inequality can be rewritten equivalently as the set

of two inequalities



w(3)



△

= −w

(3) + aT

min

≤ 0, (19)



w(3)



△

= w

(3) −aT

max

≤ 0. (20)

Using (11) – (12), the cost functional (10) beco-

mes



u(z)



= w

(3) + B(δ)



(3)



. (21)

Thus, we have transformed the EP into the equi-

valent optimal control problem: to ﬁnd the control

function u(z), transferring the system (14) – (16) from

the initial position (17) to the set of terminal positions

(19) – (20) and minimizing the cost functional (21),

subject to the geometric constraint (2) and the inequa-

lity (5). This optimal control problem is non-linear

with respect to u(z), and in what follows, it is cal-

led the Non-linear Optimal Control Problem (NOCP).

Due to (Ioffe and Tihomirov, 1979) (see Section 9.2,

Theorem 3), the NOCP has a solution (optimal cont-

rol).

3.2 Solution of the NOCP by

Application of the PMP

The Variational Hamiltonian of the NOCP is

H = H(w,u,λ,z)

= λ

ψ(δ,z)u

+ λ

ψ(δ,z)u + λ

exp



−z



u, (22)

where λ = λ(z) =



(z),λ

(z)



, and λ

(z), (i = 1, 2, 3) are the costate variables. These co-

state variables satisfy the differential equations

dλ

/dz = −∂H/∂w

= 0, z ∈ [0,3], (23)

dλ

/dz = −∂H/∂w

= 0, z ∈ [0,3], (24)

dλ

/dz = −∂H/∂w

= 0, z ∈ [0,3], (25)

and the terminal conditions

(3) = −C



∂J/∂w

(3)



= −C

, (26)

(3) = −C



∂J/∂w

(3)



= −2C

B(δ)γ, γ

△

= w

(3),

(27)

(3) = −C



∂J/∂w

(3)



−C



∂g



w(3)



/∂w

(3)



−C



∂g



w(3)



/∂w

(3)



= C

−C

. (28)

In these terminal conditions,

≥ 0, C

≥ 0 (29)

are some constants, such that

> 0, (30)

and



w(3)



= 0, C



w(3)



= 0. (31)

Denote I

△



u : u

min

≤ u ≤ u

max



. Due to the

PMP, an optimal control u

∗

(z) of the NOCP necessa-

rily satisﬁes the following condition for all z ∈ [0,3]:

max

u(z)∈I



w(z),u(z),λ(z),z





w(z),u

∗

(z),λ(z),z



. (32)

Thus, any control u(z), satisfying the equations (32),

(14) – (17) and (23) – (28), the conditions (19) –

(20) and (29) – (31), is an optimal control candi-

date in the NOCP. To obtain such a control, ﬁrst, we

solve the equations (23) – (28). These equations yield

the following solution for z ∈ [0,3]: λ

(z) = −C

(z) = − 2C

Bγ, λ

(z) = C

−C

. By substitution of

this solution into (22) and using (9), the Variational

Hamiltonian of the NOCP becomes

H = −exp



−z





u,z,γ,C



, (33)

where the function G



u,z,γ,C



has the form



u,z,γ,C





B(δ)γcosh(δz)−C



u +

cosh(δz)u

. (34)

Let us show that C

> 0. For this purpose, we assume

the opposite which, due to (29), is C

= 0. In this case,

the use of (33) – (34) and (30) yields

H = exp(−z

/2)(C

−C

)u (35)

and C

> 0. Note that C

6= C

. Indeed, if C

, then both constants are non-zero. In such a case,

by virtue of (31), g



w(3)



= 0 and g



w(3)



= 0.

The latter, along with (19) – (20) and the inequality

min

< T

max

(see (5)), yields a contradiction. Thus,

−C

6= 0. The unique control, satisfying (32) with

the Variational Hamiltonian of the form (35) is

∗

(z) =



min

, if C

−C

< 0,

max

, if C

−C

> 0.

(36)

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

Now, substituting (36) into (16) instead of u(z)

and solving the resulting equation subject to the initial

condition from (17), we obtain

(3) =



min

if C

−C

< 0,

max

, if C

−C

> 0.

(37)

The latter, along with the inequality (5), means that

(3) does not belong to the set of terminal positi-

ons (19) – (20). Therefore, the control (36), obtained

from (32), (35) under the assumption C

= 0, is not

admissible. This means that the assumption C

= 0 is

wrong, i.e., C

> 0.

Due to the PMP, we can set C

= 1 and rewrite the

equations (33) – (34) as:

H = −exp



−z





u,z,γ,C



, (38)



u,z,γ,C





4B(δ)γcosh(δz)

−C



u + 2 cosh(δz)u

. (39)

Thus, applying (32) to (38) – (39), we obtain the

optimal control of the NOCP in the form

∗

(z) = u

∗

(z,γ,C

) =







min

, ˜u(z,γ,C

) ≤ u

min

˜u(z,γ,C

), ˜u(z,γ,C

) ∈ (u

min

max

, ˜u(z,γ,C

) > u

max

(40)

where

˜u(z,γ,C

) =



−C





4 cosh(δz)



−Bγ (41)

is the unique solution of the following equation with

respect to u: ∂G

(u,z,γ,C

)/∂u = 0.

In order to use the equation (40), we need to know

the constants γ, C

and C

. These constants should be

chosen in such a way that the resulting control (40)

will transfer the system (14) – (16) from the initial

position (17) to the intersection of the set of termi-

nal positions (19) – (20) and the plane w

(3) −γ = 0

in the 3D-space



(3),w

(3)



. Substituting

(40) into the system (14) – (16) instead of u(z), sol-

ving the resulting system subject to the initial conditi-

ons (17), and using the above mentioned requirement

yield the following set of the inequality and the alge-

braic equation with respect to γ, C

and C

min

≤ Φ

(γ,C

) ≤ aT

max

, (42)

(γ,C

) −γ = 0, (43)

where

(γ,C

)

△

exp(−z

/2)u

∗

(z,γ,C

)dz,

(44)

(γ,C

)

△

ψ(z,δ)u

∗

(z,γ,C

)dz. (45)

Note that Φ

(γ,C

) and Φ

(γ,C

) are the va-

lues w

(3) and w

(3), generated by the control (40).

Further, due to (19) – (20) and (31), the constants

γ, C

and C

should satisfy the algebraic equations



min

−Φ

(γ,C

)



= 0, (46)



(γ,C

) −aT

max



= 0. (47)

Moreover, by (29),

≥ 0, C

≥ 0. (48)

Remark 3. Since the NOCP has the solution, the set

(42) – (48) has a solution. If this set has more th an

one solution, we choose the solution



γ = γ

∗

= C

∗



, which provides the minimum value of

the NOCP cost functional (21) in c omparison with the

other solutions.

3.3 Analysis of the Set (42) – (48)

Let (γ,C

) be a solution of this set. Since the

control (40) satisﬁes the inequality u

∗

(z,γ,C

) ≥

min

> 0 for all z ∈ [0, 3], then due to (9), (43) and

(45), γ > 0. For the further analysis, we distinguish the

following cases with respect to C

and C

: (I) C

> 0,

> 0; (II) C

= C

= 0; (III) C

= 0, C

> 0; (IV)

> 0, C

= 0.

We start with the ﬁrst case. Due to (46)-(47), this

case yields the contradictory equality aT

min

= aT

max

meaning that the ﬁrst case is impossible.

Proceed to the second case. In this case,

the function ˜u(z,γ,C

) (see (41)) becomes

˜u(z,γ,C

) = ˜u(z,γ, 0, 0) = −B(δ)γ, z ∈[0,3]. Since

B(δ) and γ are positive, then ˜u(z,γ,C

) < 0, z ∈

[0,3]. Therefore, due to the equation (40) and the

inequality u

min

> 0, we have u

∗

(z,γ,C

) = u

min

z ∈ [0,3]. Now, substitution of this control into (44),

and using (42) and the positiveness of a yield the in-

equality T

min

≤ u

min

, which contradicts the inequality

in (5). Thus, the second case also is impossible.

Now, let us treat the third case. In this

case, we have ˜u(z,γ,C

) = ˜u(z,γ,0,C

) =

−





4 cosh(δz)



+ B(δ)γ



< 0, z ∈ [0,3]. Thus, by

the same arguments as in the case (II), the third case

is impossible.

Finally, let us consider the case (IV). In this case,

by denoting C

△

= C

, the optimal control (40) becomes

∗

(z) = u

∗

(z,γ,C) =







min

, ¯u(z,γ,C) ≤ u

min

¯u(z,γ,C), u

min

< ¯u(z,γ,C) ≤ u

max

, ¯u(z,γ,C) > u

max

(49)

where

¯u(z,γ,C) = C/



4 cosh(δz)



−B(δ)γ (50)

Optimal Time-sampling Problem in a Statistical Control with a Quadratic Cost Functional - Analytical and Numerical Approaches

is the unique solution of the following equation with

respect to u: ∂G

(u,z,γ,C)/∂u = 0, G



u,z,γ,C





4B(δ)γcosh(δz) − C



u + 2 cosh(δz)u

. Moreover,

using the equation (46) and denoting T

△

= T

min

, the

set (42) – (43) becomes

(γ,C)

△

exp(−z

/2)u

∗

(z,γ,C)dz −aT = 0,

(51)

(γ,C)

△

ψ(z,δ)u

∗

(z,γ,C)dz −γ = 0. (52)

Thus, to construct the optimal control u

∗

(z,γ,C)

and to design the optimal SC sampling time-interval,

one should solve the system (51) – (52) with respect

to (γ,C) and substitute its solution into the equation

(49). Due to Remark 3, the system (51) – (52) has a

solution. Some properties of this solution, helpful for

its numerical obtaining, are presented in the next sub-

section. Examples of such an obtaining are presented

in Section 5.

Remark 4. It is important to note that in the optimal

control u

∗

(z,γ,C) of the NOCP both bounds u

min

and

max

of the geometric con straint (2) are used. At the

same time, only the lower bound T

min

of the terminal

state inequality (18), equ iv alent to the integral ine-

quality (3), is used in the optimal control of the NOCP.

Thus, the optimal SC time-sampling depends on both

bounds of the geometric constraint (2) and only on the

lower bound of the integral inequality constraint (3).

3.4 Properties of the Solution to the

System (51) – (52)

Let us introduce into the consideration the following

values:

min

(γ)

△

= 4



B(δ)γ+ u

min



max

(γ)

△

= 4 cosh(3δ)



B(δ)γ+ u

max



min

△

= u

min

ψ(z,δ)dz, γ

max

△

= u

max

ψ(z,δ)dz,

min

△

= max {γ

min

, 2aT },

max

△

= min{γ

max

, 2aT cosh(3δ)}.

Also, in the plane (γ,C), we consider the non-empty

domain

Ω

△

γ ∈



min

,Γ

max



,C ∈



min

(γ),C

max

(γ)



Based on the above introduced values and domain,

we obtain the following assertions.

Assertion 1. Let δ > 0. Let (γ,C) be a solution of the

system (51) – (52). Then, (γ,C) ∈ Ω.

Remark 5. Due to Assertion 1, in the case δ > 0, we

can look for the solution of the system (51) – (52) not

in the entire plane (γ,C), but in the bounded domain

Ω, which decreases considerably the computatio nal

effort. In the particular case of δ = 0, we can solve

the system (51) – (52) analytically.

Assertion 2. Let δ = 0. Then, the system (51) –

(52) has the unique solution



γ = 2aT,C = 8B(0)aT +



Assertion 3. For any given δ > 0 and γ ∈



min

,Γ

max



, the equation (51) has the unique solu-

tion C =

C(γ), and

C(γ) ∈



min

(γ),C

max

(γ)



. (53)

Moreover,

C(γ) is a monotonically increasing

function of γ ∈



min

,Γ

max



Assertion 4. For any given δ > 0 and γ ∈



min

,Γ

max



, the equation (52) has the unique solu-

tion C =

C(γ), and

C(γ) ∈



min

(γ),C

max

(γ)



. (54)

Moreover,

C(γ) is a monotonically increasing

function of γ ∈



min

,Γ

max



Remark 6. Based on Assertions 3 and 4, the γ-

compon ent of th e solution to the system (51) – (52)

can be obtained by solving with respect to γ ∈



min

,Γ

max



either the equation Λ



γ,

C(γ)



= 0, or

the equation Λ



γ,

C(γ)



= 0, or the equation

C(γ) =

C(γ).

The proofs of Assertions 1 – 4 are presented in

Appendix.

4 APPROXIMATE NUMERICAL

SOLUTION OF EP

Let us divide the interval [0,3] into N equal subinter-

vals by the collocation points

= i∆z, i = 0,1,.. .,N, ∆z = 3/N. (55)

Then, based on (55) and using the left rectangles for-

mula (Davis and Rabinowitz, 2007), we approximate

the integrals in the cost functional (10) and in the in-

tegral constraint (3).

Thus, the cost functional is approximated as:



u(z)



≈

(U)

△

= ∆z

N−1

∑

i=0

ψ(z

,δ)U

∆z

N−1

∑

i=0

ψ(z

,δ)U

, (56)

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

where the vector U ∈E

U =



,. . .,U

N−1





u(z

),u(z

),. .. ,u(z

N−1

)



. (57)

The constraint (3) is approximated as:

∆zT

min

N−1

∑

i=0

exp(−z

/2) ≤ ∆z

N−1

∑

i=0

exp(−z

/2)U

≤ ∆zT

max

N−1

∑

i=0

exp(−z

/2). (58)

The approximation (58) of the constraint (3) is de-

rived using the following approximation of the value

a ≈ a

∆z, a

△

N−1

∑

i=0

exp(−z

/2).

The geometric constraint (2), appearing in the EP,

is approximated as:

min

≤U

≤ u

max

, i = 0, 1, ...,N −1. (59)

Dividing the expression in the right-hand side of

(56) and the inequality (58) by ∆z, we obtain the fol-

lowing ﬁnite-dimensional cost functional and the con-

straint:

(U)

△

N−1

∑

i=0

ψ(z

,δ)U

+B∆z

N−1

∑

i=0

ψ(z

,δ)U

, (60)

min

≤

N−1

∑

i=0

exp(−z

/2)U

≤ a

max

. (61)

Now, based on (60) – (61), we can formulate the

following quadratic programming problem.

Quadratic Programming Problem (QPP):

for a known δ ≥ 0, to ﬁnd the vector

U =



,...,U

N−1



which minimizes the

cost functional (60) subject to the constraints (59),

(61) and the inequality (5).

The QPP can be solved using standard optimiza-

tion tools, for example, the MATLAB function “quad-

prog”. It is reasonable to expect that for a sufﬁciently

large N, the components U

, (i = 0, 1, ...,N −1) of the

QPP solution will be close to the corresponding va-

lues u

∗

,γ

∗

) of the optimal control in the NOCP.

In such a case, the optimal value of the cost functional

(60) in the QPP multiplied by ∆z will be close to the

optimal value of the cost functional (21) in the NOCP.

5 NUMERICAL EVALUATION OF

THE OPTIMAL AND

SUBOPTIMAL SAMPLING

TIME-INTERVALS

For the numerical evaluation, the following two sets

of parameters are chosen:

(I) u

min

= 0.5, u

max

= 3.5;

(II) u

min

= 0.1, u

max

= 2.5.

5.1 Numerical Solution of the System

(51) – (52)

To obtain the sampling time-interval u

∗

(z,γ,C), the

system (51) – (52) was solved numerically. The

value of γ was calculated by application of the bi-

section algorithm to the equation

C(γ) =

C(γ) for

γ ∈(Γ

min

,Γ

max

Using Assertions 3 and 4, the functions

C(γ) and

C(γ) also were derived by the bisection method for

(γ,C) ∈ Ω.

In Figs. 1 – 2, the functions

C(γ) and

C(γ), along

with the functions C

min

(γ) and C

max

(γ), are depicted

in the logarithmic scale for the set (I) with δ = 2.5

(Fig. 1) and for the set (II) with δ = 2.9 (Fig. 2). The

value of T = T

min

= 1. It is seen that the functions

C(γ) and

C(γ) are monotonically increasing, which

corresponds to the claims of Assertions 3 and 4. The

solution (γ,C) of the set (51) – (52), depicted by the

circle, belongs to the set Ω as it is stated in Assertion

1. Moreover, since the derivative of

C(γ) with respect

to γ is larger than the derivative of

C(γ), this solution

is unique.

20 40 60 80 100 120 140

Ω

min

max

min

(γ)

max

(γ)

C(γ)

Figure 1:

C(γ),

C(γ), C

min

(γ) and C

max

(γ): set (I).

The numerical solution (γ,C) of the set (51) –

(52), and the absolute values of the functions Λ

(γ,C)

Optimal Time-sampling Problem in a Statistical Control with a Quadratic Cost Functional - Analytical and Numerical Approaches

0 50 100 150 200

min

max

Ω

min

(γ)

max

(γ)

C(γ)

Figure 2:

C(γ),

C(γ), C

min

(γ) and C

max

(γ): set (II).

and Λ

(γ,C) are presented in Table 1. It is seen that

the obtained numerical solution provides the deviati-

ons of Λ

(γ,C) and Λ

(γ,C) from zero smaller than

−4

Table 1: Numerical solution of (51) – (52).

Set δ γ C |Λ

(γ,C)| |Λ

(γ,C)|

(I) 2.5 21.1 20.6 2.9 ·10

−5

7.0 ·10

−5

(II) 2.9 12.6 13.4 7.1 ·10

−5

7.6 ·10

−5

5.2 Optimal Sampling Time-Interval

(EP Solution) vs. Approximate

Sampling Time-Interval (QPP

Solution)

In Figs. 3 – 4, the optimal sampling time-interval

∗

(z,γ,C) (the EP solution), given by the analytical

expression (49), is compared to the approximate sam-

pling time-interval U

∗

(the QPP solution) for the set

(I) with δ = 2.5 (Fig. 3) and for the set (II) with

δ = 2.9 (Fig. 4). In both cases, T

min

= 1, T

max

= 2. It

is seen that the approximation, obtained for N = 100,

and the optimal sampling time-interval match well.

Note that in both cases, for the approximate so-

lution, the left-hand side inequality in the constraint

(61) is satisﬁed as the equality

N−1

∑

i=0

exp(−z

/2)U

= a

min

thus mimicking the corresponding property of the EP

analytical solution.

Based on the equation (49) and the above pre-

sented numerical calculations, the optimal sampling

time-interval for the set (I), depicted in Fig. 3, can be

rewritten as:

∗

(z,γ,C) =



¯u(z,γ,C), z ∈ [0,0.47),

min

, z ∈ [0.47,3].

0 0.5 1 1.5 2 2.5 3

0.5

1.5

2.5

∗

Optimal

Approximate

Figure 3: Optimal vs. approximate sampling time-interval:

set (I).

0 0.5 1 1.5 2 2.5 3

0.5

1.5

2.5

∗

Optimal

Approximate

Figure 4: Optimal vs. approximate sampling time-interval:

set (II).

The optimal sampling time-interval for the set (II), de-

picted in Fig. 4, can be rewritten in the form

∗

(z,γ,C) =







max

, z ∈ [0,0.21),

¯u(z,γ,C), z ∈ [0.21,0.95),

min

, z ∈ [0.95,3].

In Fig. 5, the optimal (minimum) value of the ex-

pected loss, given by (6)–(9), is depicted as a function

of δ for the set (I), T

min

= 1, and different coefﬁcient

functions k = k(δ). It is seen that for each considered

k(δ), the optimal expected loss admits the maximum

for some value of δ, belonging to the interval (0,1).

6 CONCLUSIONS

In this paper, the problem of constructing an opti-

mal state-feedback sampling time-interval for the sta-

tistical control was considered. The expected loss,

quadratically dependent on the delay in the detection

of a process change, was chosen as the criterion of

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

0 1 2 3

x 10

Expected loss

k =

√

k = δ

Figure 5: Optimal expected loss as a function of δ: set (I).

the optimization (minimization). This expected loss

also depends on the numerical parameter δ, characte-

rising a magnitude of the process change. The pro-

blem of the expected loss minimization was reduced

to the extremal problem in the form of a nonstan-

dard calculus of variations problem where the sam-

pling time-interval of the statistical control is a mini-

mizing function. This minimizing function depends

of the process state. Two methods of the solution of

this extremal problem were proposed. The ﬁrst met-

hod transforms the original extremal problem to an

equivalent optimal control problem. Then, the latter

was solved using the Pontryagin’s Maximium Princi-

ple, which yields the explicit analytical expression for

the optimal sampling time-interval. This expression

contains two parameters. For obtaining these para-

meters, the set of two algebraic equations was derived

and analyzed. Based on this analysis, the method of

solution of this set was proposed. The second method

of the solution of the original extremal problem uses

its discretization. This leads to a ﬁnite-dimensional

extremal problem (the quadratic programming pro-

blem), approximating the original one. This qua-

dratic programming problem was solved using the

MATLAB function quadprog, providing the subopti-

mal sampling time-interval for the statistical control.

The optimal and suboptimal sampling time-intervals

were evaluated by numerical examples. This evalua-

tion has shown a good match of the optimal analytical

sampling time-interval and the suboptimal numerical

sampling time-interval. The optimal (minimum) va-

lue of the expected loss was constructed numerically

as a function of the parameter δ. It was shown that

this function has a single maximum.

It should be noted that the results, obtained in

this paper, are rather theoretical, and these results are

strongly based on two assumptions: (a) the sample

mean ¯x of the characteristic index x in the statistical

control is normally distributed; (b) the value of the

parameter δ, characterising a magnitude of the pro-

cess change, is known. Therefore, one can ask the

following: ”What will happen if at least one of these

assumptions is violated?” To answer this question, the

following issues will be studied in a future:

(I) an evaluation (by extensive computer simulations)

of the sampling time-interval, obtained in this paper,

in the cases where either a distribution of the sample

mean ¯x differs from the normal one, or the value of δ

is unknown;

(II) a design of optimal sampling time-interval in the

case where a distribution of the sample mean ¯x is not

normal;

(III) a design of optimal sampling time-interval, ro-

bust with respect to δ, in the case where the value of

this parameter is unknown.

Results of these studies will be presented in fort-

hcoming papers.

REFERENCES

Amin, R. W. and Hemasinha, R. (1993). The switching

behavior of

X charts with variable sampling intervals.

Communication in Statistics - Theory and Methods,

22(7):2081–2102.

Amin, R. W. and Miller, R. W. (1993). A robustness study

of charts with variable sampling intervals. Journal of

Quality Technology, 25(1):36–44.

Babrauskas, V. (2008). Heat release rates. In SFPE Hand-

book of Fire Protection Engineering, pages 1–59. Na-

tional Fire Protection Association.

Bashkansky, E. and Glizer, V. Y. (2012). Novel appro-

ach to adaptive statistical process control optimiza-

tion with variable sampling interval and minimum ex-

pected loss. International Journal of Quality Engi-

neering and Technology, 3(2):91–107.

Bulelzai, M. A. K. and Dubbeldam, J. L. A. (2012). Long

time evolution of atherosclerotic plaques. Journal of

Theoretical Biology, 297:1–10.

Carpenter, T. E., OBrien, J. M., Hagerman, A., and McCarl,

B. (2011). Epidemic and economic impacts of delayed

detection of foot-and-mouth disease: a case study of a

simulated outbreak in california. Journal of Veterinary

Diagnostic Investi gation, 23(1):26–33.

Chew, X. Y., Khoo, M. B., Teh, S. Y., and Castagliola, P.

(2015). The variable sampling interval run sum

control chart. Computers & Industrial Engineering,

90:25–38.

Costa, A. F. B. (1994). X charts with variable sample size.

Journal of Quality Technology, 26(3):155–163.

Costa, A. F. B. (1997).

X charts with variable sample size

and sampling intervals. Journal of Quality Techno-

logy, 29(2):197–204.

Optimal Time-sampling Problem in a Statistical Control with a Quadratic Cost Functional - Analytical and Numerical Approaches

Costa, A. F. B. (1998). Joint

X and r control charts with va-

riable parameters. IIE Transactions, 30(6):505–514.

Costa, A. F. B. (1999a).

X charts with variable parameters.

Journal of Quality Technology, 31(4):408–416.

Costa, A. F. B. (1999b). Joint

X and r charts with varia-

ble sample sizes and sampling intervals. Journal of

Quality Technology, 31(4):387–397.

Costa, A. F. B. and Magalhaes, M. S. D. (2007). An adap-

tive chart for monitoring the process mean and vari-

ance. Quality and Rel iability Engineering Internatio-

nal, 23(7):821–831.

Davis, P. J. and Rabinowitz, P. (2007). Methods of Nume-

rical Integration. Dover Publications Inc., Mineola,

NY.

Gelfand, I. M. and Fomin, S. V. (1963). Calculus of Varia-

tions. Prentice-Hall, Englewood Cliffs, NJ.

Glizer, V. Y., Turetsky, V., and Bashkansky, E. (2015).

Statistical process control optimization with varia-

ble sampling interval and nonlinear expected loss.

Journal of Industrial and Management Optimi zati on,

11(1):105–133.

Hatjimihail, A. T. (2009). Estimation of the optimal sta-

tistical quality control sampling time intervals using a

residual risk measure. PLoS ONE, 4(6):e5770.

Ioffe, A. D. and Tihomirov, V. M. (1979). Theory of Extre-

mal Problems. North-Holland Pub. Co., Amsterdam,

Netherlands.

Kim, S. and Frangopol, D. M. (2011). Optimum inspection

planning for minimizing fatigue damage detection de-

lay of ship hull structures. International Journal of

Fatigue, 33(3):448–459.

Li, Z. and Qiu, P. (2014). Statistical process control using

dynamic sampling. Technometrics, 56(3):325–335.

Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V.,

and Mishchenko, E. F. (1962). The Mathematical The-

ory of Optimal Processes. Interscience, New York,

NY.

Prabhu, S. S., Montgomery, D. C., and Runger, G. C.

(1994). A combined adaptive sample size and sam-

pling interval x control scheme. Journal of Quality

Technology, 26(3):164–176.

Qiu, P. (2013). Introduction to Statistical Process Control.

Chapman and Hall/CRC, Boca Raton, FL.

Reynolds, M. R. (1995). Evaluating properties of variable

sampling interval control charts. Sequential Analysis,

14(1):59–97.

Reynolds, M. R., Amin, R. W., Arnold, J. C., and Nachlas,

J. (1988).

X charts with variable sampling intervals.

Technometrics, 30(2):181–192.

Ross, S. (2009). A First Course in Probability. Prentice

Hall, Upper Saddle River, NJ.

Sebasti˜ao, P. and Soares, C. G. (1995). Modeling the fate of

oil spills at sea. Spill Science and Technology Bulletin,

2(2-3):121–131.

Sultana, I., Ahmed, I., Chowdhury, A. H., and Paul, S. K.

(2014). Economic design of

X control chart using

genetic algorithm and simulated annealing algorithm.

International Journal of Productivity and Quality Ma-

nagement, 14(3):352–372.

Taguchi, G., Chowdhury, S., and Wu, Y. (2007). Taguchi’s

Quality Engineering Handbook. John Wiley and Sons

Inc., Hoboken, NJ.

APPENDIX

The proofs of Assertions 1-4 are based on the follo-

wing auxiliary propositions.

Auxiliary Propositions

Proposition 1. For any given δ ≥ 0, there are no so-

lutions of the set (51) – (52) in the half-planes

C ≤C

min

(γ) (62)

and

C ≥C

max

(γ) (63)

of the plan e (γ,C).

Proof. First, let us prove the claim of the proposition

with respect to the half-plane (62), i.e., in the case

where the pair (γ,C) satisﬁes this inequality. In this

case, since cosh(δz) ≥ 1 for z ≥ 0, then ¯u(z,γ,C) ≤

min

for z ≥0. By virtue of (49), the latter means that

∗

(z,γ,C) = u

min

. Thus, due to (4), (5), (51), and the

notation T = T

min

(γ,C) = a



min

−T



< 0, (64)

meaning that the above mentioned pair (γ,C) does not

satisfy the equation (51). The claim of the assertion

with respect to the half-plane (63) is proven similarly.

Proposition 2. For any given δ ≥ 0, there are no so-

lutions of the set (51) – (52) in the half-planes

γ ≤γ

min

(65)

and

γ ≥γ

max

(66)

of the plan e (γ,C).

Proof. First of all, let us note the following. Since

0 < u

min

< u

max

and ψ(z,δ) > 0, z ∈ [0,3], then

0 < γ

min

< γ

max

. (67)

Consider the case where the pair (γ,C) satisﬁes the

strict inequality in (65). In this case, due to (9), (49),

(52) and (65), we have

(γ,C) >

ψ(z,δ)



∗

(z,γ,C) −u

min



dz ≥ 0,

(68)

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

meaning that (γ,C) does not satisfy the equation (52).

Now, let us consider the case γ = γ

min

. In this case,

using (49) and (52), we have

(γ

min

,C) =

ψ(z,δ)



∗

(z,γ

min

,C) −u

min



dz ≥ 0. (69)

By virtue of (49), the inequality in (69) becomes

equality only if u

∗

(z,γ

min

,C) ≡ u

min

for all z ∈ [0,3],

yielding ¯u(z,γ

min

,C) ≤ u

min

for all z ∈ [0,3]. Using

(50), one directly obtains that the latter inequality for

z = 0 is equivalent to the inequality (62) with γ = γ

min

However, in this case by virtue of Proposition 1, the

equation (51) is not satisﬁed. Thus, the claim of the

assertion with respect to the half-plane (65) has been

proven. The claim of the proposition with respect to

the half-plane (66) is proven similarly.

Proposition 3. Let δ > 0. Let (γ,C) be a solution

of the set (51) – (52). Then, the component γ of this

solution satisﬁes the inequality

2aT < γ < 2aT cosh(3δ). (70)

Proof. Substitution of (9) into (52) yields after a sim-

ple rearrangement

γ = 2

exp



−z



cosh(δz)u

∗

(z,γ,C)dz. (71)

Applying the Mean Value Theorem to the integral in

the right-hand side of (71) and taking into account

the fact that cosh(δz) monotonically increases with

respect to z ∈ [0, 3] for δ > 0, we obtain

γ = 2 cosh(δ¯z)

exp



−z



∗

(z,γ,C)dz. (72)

where ¯z is some value from the interval (0, 3).

Due to (51), we can replace the integral in

(72) with aT , which leads to the equality γ =

2aT cosh(δ¯z). The latter, along with the inequality

1 < cosh(δ¯z) < cosh(3δ), directly implies the state-

ment of the proposition.

Proposition 4. For δ > 0, the following inequality

holds:

0 < Γ

min

< Γ

max

. (73)

Proof. Using the deﬁnition of γ

max

(see Subsection

3.4), the equation (4), the inequality (5), the notation

T = T

min

and the same arguments as in the proof of

Proposition 2, we obtain

max

= 2au

max

cosh(δ˜z) > 2aT, (74)

where ˜z is some value from the interval (0, 3).

Similarly, using the deﬁnition of γ

min

(see Sub-

section 3.4), the equation (4) and the inequality (5),

we have

min

= 2au

min

cosh(δ˜z) < 2aT cosh(3δ). (75)

Now, the inequalities (67), (74) and (75), along with

the inequality 2aT < 2aT cosh(3δ), yield immedia-

tely the statement of the proposition.

Proposition 5. For any γ ≥0, the following inequality

is valid:

0 < C

min

(γ) < C

max

(γ). (76)

Proof. The assertion directly follows from the deﬁni-

tions of C

min

(γ) and C

max

(γ) (see Subsection 3.4).

Proof of Assertion 1

First of all let us note that, due to Propositions 4 and

5, the domain Ω indeed is non-empty. Now, the state-

ment of the assertion directly follows from Propositi-

ons 1-3 and the deﬁnitions of Γ

min

, Γ

max

and Ω.

Proof of Assertion 2

For δ = 0, the function ¯u



z,γ,C



, given by (50), beco-

mes a constant, i.e.,

¯u(z,γ,C) ≡

−B(0)γ, z ∈ [0, 3]. (77)

Moreover, due to Proposition 1, in order to be a solu-

tion of the set (51) – (52), the pair (γ,C) should satisfy

the inequality

min

−B(0)γ < u

max

. (78)

The latter, along with (49) and (77), means that

∗

(z,γ,C) ≡

−B(0)γ, z ∈ [0,3]. (79)

Substituting (79) into the set (51) – (52) and using

(9) and the fact that δ = 0 directly yield the unique

solution of (51) – (52) in the form

−B(0)γ = T, γ = 2aT. (80)

The latter yields the unique C = 8B(0)aT +4T , which

completes the proof of the assertion.

Proof of Assertion 3

The existence and uniqueness of

C(γ) is proven simi-

larly to the work (Glizer et al., 2015) (see Lemma 5.1

and its proof where γ = 0). The inclusion (53) fol-

lows from the proof of Proposition 1. Let us prove

Optimal Time-sampling Problem in a Statistical Control with a Quadratic Cost Functional - Analytical and Numerical Approaches

the monotonicity of

C(γ), γ ∈



min

,Γ

max



. We prove

this feature of

C(γ) by contradiction. Namely, we as-

sume that the statement on the monotonic increasing

of this function is wrong. This means the existence of

∈



min

,Γ

max



and γ

∈



min

,Γ

max



such that

< γ

, (81)

while

C(γ

) ≥

C(γ

). (82)

The equation (50), and the inequalities (81) and (82)

directly yield

¯u



z,γ

C(γ

)



> ¯u



z,γ

C(γ

)



, z ∈ [0,3]. (83)

Due to the equation (49) and the inequality (83), we

immediately have

∗



z,γ

C(γ

)



≥ u

∗



z,γ

C(γ

)



∀z ∈ [0, 3]. (84)

By virtue of the inclusion (53), we obtain that for

any γ ∈



min

,Γ

max



and all z ∈ [0,3] the function

∗



z,γ,

C(γ)



is neither identical u

min

, nor identical

max

. This fact, along with (49) and (83), yields the

existence of a point ˜z ∈ [0,3], such that

∗



˜z,γ

C(γ

)



> u

∗



˜z,γ

C(γ

)



. (85)

Further, from (49) and (50), one directly con-

cludes that for any γ ∈



min

,Γ

max



the control

∗



z,γ,

C(γ)



is continuous function of z ∈[0,3]. This

observation, along with the inequality (85), yields the

existence of the interval [z

], (z

< z

), such that

] ⊂ [0,3] and

∗



˜z,γ

C(γ

)



> u

∗



˜z,γ

C(γ

)



∀z ∈[z

]. (86)

Now, the deﬁnition of Λ

(γ,C) (see (51)), along with

the inequalities (84) and (86), yields



C(γ

)



> Λ



C(γ

)



. (87)

The latter contradicts the fact that

C(γ

) and

C(γ

)

are solutions of the equation (51) with respect to C

for γ = γ

and γ = γ

. This contradiction implies

that the function

C(γ) monotonically increases for

γ ∈ (Γ

min

,Γ

max

), which completes the proof of the as-

sertion.

Proof of Assertion 4

We start the proof with the ﬁrst two statements of the

assertion. Let the pair (γ,C) be any ﬁxed satisfying

the inequalities Γ

min

< γ < Γ

max

, C ≤C

min

(γ).

Using the equation (50), the deﬁnition of C

min

(γ)

(see Subsection 3.4), the inequality (73), as well as the

positiveness of B(δ) and the inequality cosh(δz) ≥ 1,

z ∈ [0,3], one directly has the following chain of the

inequalities:

¯u(z,γ,C) ≤

min

cosh(δz)

+B(δ)γ



cosh(δz)

−1



≤ u

min

, z ∈ [0,3]. (88)

Hence, due to (49), u

∗

(z,γ,C) ≡u

min

, z ∈[0,3]. Using

the latter and (52), (65) – (66) yield

(γ,C) < 0, γ ∈



min

,Γ

max



, C ≤C

min

(γ). (89)

It is shown similarly, that

(γ,C) > 0, γ ∈



min

,Γ

max



, C ≥C

max

(γ). (90)

Further, from (49) and (50), one directly con-

cludes that for any z ∈ [0, 3] and γ ∈



min

,Γ

max



the control u

∗

(z,γ,C) is continuous and monotoni-

cally increasing function of C ∈ (−∞,+∞). There-

fore, Λ

(γ,C) (see (52)) is a continuous and mono-

tonically increasing function of C ∈ (− ∞, +∞) for

any γ ∈



min

,Γ

max



. The latter, along with (89) –

(90), implies immediately the existence of the uni-

que solution C =

C(γ) of the equation (52) for any

γ ∈



min

,Γ

max



. Moreover, the inclusion (54) is sa-

tisﬁed for any γ ∈



min

,Γ

max



The monotonicity of

C(γ), γ ∈



min

,Γ

max



is pro-

ven similarly to the same feature of the function

C(γ)

(see the proof of Assertion 3). This completes the

proof of the assertion.

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics