Adv Syst Sci Appl 2024; 04:66–81
Published online at https://ijassa.ipu.ru.

On the External Estimation of the Limit Reachable Set for the
Linear Discrete-Time System Based on Support Hyperplanes

Anastasiya Simkina*

Moscow Aviation Institute, Moscow, Russia

Abstract: A linear discrete-time system with geometric constraints on control is considered. For
the given system, the problem of constructing the limit reachable set is solved. It is proven that it is
a cylinder oriented along certain elements of the real Jordan basis. Statements about the structure
of the supporting hyperplane to the convex section of this cylinder are formulated and proven.
Necessary and sufficient conditions for the boundedness of the limit reachable set are determined.

Keywords: discrete-time system, limit reachable set, support hyperplane, convex set,
polyhedral approximation

1. INTRODUCTION

The classical controllability condition for linear dynamic systems assumes unbounded control
actions. However, when considering practical problems, constraints often arise that are
determined by various technical characteristics of the control system under consideration.
As a result, even with an infinite number of steps, it is not always possible to transfer the
system from a given initial state to a certain fixed final state. This fact makes it relevant to
study not only the issues of reachability and controllability of various dynamic systems but
also to develop methods for constructing and evaluating the limit reachable and controllable
sets for an arbitrary control system. On the other hand, these sets can be applied to solving
synthesis problems for discrete-time systems [7].

Currently, two main directions can be distinguished in this area: the study of individual
states for controllability [3–5, 12] and geometric methods for constructing reachable and
controllable sets [?, 1,15,16]. Thus, when studying nonlinear systems, it is possible to obtain
only general properties of controllable sets [3] or their estimates [15, 16]. For the case of
linear dynamic equations, it is possible to construct more constructive results for various
classes of systems: periodic [11], switchable [4], with positive control [1]. The most rigorous
results are formulated for the case of compact and convex constraints on control values [?,7],
even allowing the description of limit reachable and controllable sets [2, 5, 12]. Analogous
problems were also studied in the works of Kurzhanskiy A.B. [13–15]. In [8], for linear
discrete-time systems with scalar control, which is subject to a first-order summary constraint,
it is shown that in the case of stable systems, it is possible to explicitly find the limit reachable
set, which is a convex polyhedron symmetric about zero. For higher-order constraints, the
description of the limit reachable and null-controllable sets is obtained using supporting half-
spaces [9].

The structure of the article is as follows. Section 2 is devoted to the problem statement.
Section 3 describes the procedure for decomposing the original system into subsystems of
smaller dimensions, allowing the problem to be simplified. Section 4 develops a method

∗Corresponding author: abv1998@yandex.ru



ON THE EXTERNAL ESTIMATION OF THE LIMIT REACHABLE SET... 67

for the external evaluation of limit reachable sets. The resulting estimates are cylinders or
polyhedra, whose borders are determined by supporting hyperplanes and half-spaces. Based
on the developed methods, Section 5 constructs the limit reachable set for the system of
stabilizing the glucose level in blood plasma.

2. PROBLEM STATEMENT

Consider linear stationary system with discrete time and geometric constraints on the control
(A,U):

x(k + 1) = Ax(k) + u(k),

x(0) = 0, u(k) ∈ U , k ∈ N ∪ {0}, (2.1)

where x(k) ∈ Rn is the state vector, u(k) ∈ Rn is the control action at step k, U ⊂ Rn denotes
the set of admissible control values defining the geometric constraints, and A ∈ Rn×n is the
system matrix. It is assumed that U is a convex compact set and 0 ∈ intU .

We define the limit reachable set Y∞ of the system (2.1) as the set of states to which the
system (A,U) can be transferred from the origin in any finite number of steps:

Y∞ =

{
x ∈ Rn : x =

N−1∑
k=0

AN−k−1u(k), u(0), . . . , u(N − 1) ∈ U , N ∈ N

}
. (2.2)

It is required to investigate the properties and construct estimates of the set (2.2).
As an internal estimate of the limit reachable set, the reachable set in a finite number of

steps Y(N) can be considered, where Y(N) is the set of terminal states to which the system
(A,U) can be transferred from 0 in N ∈ N steps. A description of Y(N) is known [7, Lemma
1]. From (2.2) the following representation for Y∞ follows:

Y∞ =
∞⋃

N=0

Y(N).

It is known that the set (2.2) is convex [18, Theorem 1]. This determines the range of
problems addressed in this article. It is necessary to determine the necessary and sufficient
conditions for the boundedness of the limit reachable set Y∞, to investigate its closedness
property, and, in the case of boundedness, to construct a description in terms of supporting
hyperplanes, which is guaranteed for any convex set [17, Theorem 18.8].

In many respects, the reachable sets share properties with the null-controllable sets, which
were studied in [2]. For this reason, some of their properties will be analogous. However, there
are also fundamental differences, which will be revealed in the subsequent sections.

3. THE GENERAL STRUCTURE OF REACHABLE SETS

The article [19] demonstrates that the limit reachable and null-controllable sets for
systems with scalar control are cylinders. Their orientation is determined by the elements
of the real Jordan basis and the eigenvectors of the matrix A. A similar result was obtained
in [2, Lemmas 2–4] for 0-controllable sets for systems with vector control. We will use a
similar approach to describe the limit reachable sets (2.2) for the system (2.1).

Let h1, . . . , hn ∈ Rn denote the real Jordan basis of the matrix A, which is a set of
linearly independent eigenvectors and generalized eigenvectors, in which basis A is described
by its real Jordan form, that is the following decomposition [6] holds for the matrix S =

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



68 A.V. SIMKINA

(h1, . . . , hn) ∈ Rn×n:

A = S

J1 . . . O
... . . . ...
O . . . Jm

S−1, Ji ∈ Rni×ni , i = 1,m, n1 + . . .+ nm = n, (3.3)

where the Jordan blocks J1, . . . , Jm have one of the following two forms:

J =


λ 1 · · · 0

0 λ
. . . 0

...
... . . . 1

0 0 · · · λ

 ∈ Rñ×ñ, (3.4)

J =


rAφ I · · · O

O rAφ
. . . O

...
...

. . . I
O O · · · rAφ

 ∈ R2ñ×2ñ, Aφ =

(
cosφ sinφ
− sinφ cosφ

)
, I =

(
1 0
0 1

)
. (3.5)

Let P ∈ Rñ×n denote the projection matrix onto an ñ-dimensional subspace:

P =

0 · · · 0 1 · · · 0
...

...
... . . . ...

0 · · · 0 0 · · · 1


ñ×n

.

We describe the structure of the limit reachable set in the following theorem.

Theorem 3.1:
Let the order of h1, . . . , hn in the decomposition (3.3) be chosen such that the representation
holds

A = S

(
A⩾1 O
O A<1

)
S−1,

where A⩾1 ∈ R(n−ñ)×(n−ñ) consists of Jordan blocks of the form (3.4) and (3.5)
corresponding to eigenvalues with modulus not less than 1, and A<1 ∈ Rñ×ñ consists of
Jordan blocks corresponding to eigenvalues with modulus less than 1. Let Ỹ∞ ⊂ Rñ be the
limit reachable set (2.2) for the system (A<1, PS−1U).

Then the relation holds
Y∞ = S

(
Rn−ñ × Ỹ∞

)
.

Proof
Let us denote Ũ = PS−1U . First, we will demonstrate that the theorem holds for the case
when the matrix A coincides with its real Jordan form, i.e. S is the identity matrix.

Since U is bounded, there exists a convex set V ∈ R(n−ñ) such that

U ⊂ V × Ũ . (3.6)

Consider the dynamic equations (2.1) for the system (A,V × Ũ) and an arbitrary control
sequence {u(k)}N−1

k=0 ⊂ V × Ũ . For all N ∈ N the following representation holds:

x(N) = AN−1u(0) + AN−2u(1) + . . .+ u(N − 1) =

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



ON THE EXTERNAL ESTIMATION OF THE LIMIT REACHABLE SET... 69

=


N−1∑
k=0

AN−k−1
⩾1 v(k)

N−1∑
k=0

AN−k−1
<1 ũ(k)

 =

(
y(N)
x̃(N)

)
,

where v(0), . . . , v(N − 1) ∈ V , ũ(0), . . . , ũ(N − 1) ∈ Ũ , y(N) ∈ Rn−ñ, x̃(N) ∈ Rñ. There-
fore, if we denote by Ŷ∞ ⊂ Rn the limit reachable set for the system (A,V × Ũ), then due
to the representation (2.2), for any x(N) ∈ Ŷ∞ the inclusion x̃(N) ∈ Ỹ∞ must also hold. We
obtain

Ŷ∞ ⊂ Rn−ñ × Ỹ∞.

On the other hand, from (3.6) it follows that Y∞ ⊂ Ŷ∞. Finally, we obtain:

Y∞ ⊂ Rn−ñ × Ỹ∞. (3.7)

Let’s prove the reverse inclusion. To do this, we will choose an arbitrary x =
(
yT, x̃T

)T ∈
Rn−ñ × Ỹ∞. From the inclusion 0 ∈ intU it follows that there exists v′ > 0 such that

[−v′; v′]n−ñ × {0} ⊂ U .
Therefore, for all N ∈ N the following control will satisfy the constraints of the system (2.1):

u(k) =

(
v(k)
0

)
, v(k) ∈ [−v′; v′]n−ñ, k = 0, N − 1.

since x̃ ∈ Ỹ∞, by definition (2.2), there exist M ∈ N and ũ(N), . . . , ũ(N +M − 1) ∈ Ũ
such that

x̃ =
N+M−1∑
k=N

AN+M−k−1
<1 ũ(k).

Additionally, considering the definition of Ũ , it is possible to choose v(N), . . . , v(N +M −
1) ∈ Rn−ñ such that

u(k) =

(
v(k)
ũ(k)

)
∈ U , k = 0,M − 1.

Thus, the control sequence {u(k)}N+M−1
k=0 is admissible for the system (A,U) and, with a

zero initial state, leads to the following relations:

x(N +M) =


M+N−1∑

k=0

AM+N−k−1
⩾1 v(k)

M+N−1∑
k=0

AM+N−k−1
<1 ũ(k)

 =

=


N−1∑
k=0

AN+M−k−1
⩾1 v(k) +

N+M−1∑
k=N

AN+M−k−1
⩾1 v(k)

N−1∑
k=0

AN+M−k−1
<1 ũ(k) +

N+M−1∑
k=N

AN+M−k−1
<1 ũ(k)

 =

AM
⩾1

N−1∑
k=0

AN−k−1
⩾1 v(k) + y′

x̃

 ,

where

y′ =
N+M−1∑
k=N

AN+M−k−1
⩾1 v(k).

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



70 A.V. SIMKINA

Therefore, the equality x(N +M) = x can be obtained by demonstrating that for some
N ∈ N, it is possible to choose v(0), . . . , v(N − 1) ∈ [−v′; v′]n−ñ that satisfy the following
relations:

y = AM
⩾1

N−1∑
k=0

AN−k−1
⩾1 v(k) + y′,

A−M
⩾1 (y − y′) =

N−1∑
k=0

AN−k−1
⩾1 v(k).

Considering (2.2), is equivalent to the inclusion A−M
⩾1 (y − y′) ∈ Y ′

∞, where Y ′
∞ ⊂ Rn−ñ is

the limit reachable set for the system
(
A⩾1, [−v′; v′]n−ñ

)
.

For further reasoning, we will use known analytical representations of reachable and 0-
controllable sets for discrete-time linear systems. It is known [20, Lemma 1] that for any
N ∈ N the 0-controllable set of the system

(
A−1

⩾1, A
−1
⩾1[−v′; v′]n−ñ

)
in N steps has the form:

X ′(N) = −
N∑
k=1

(
A−1

⩾1

)−k (
A−1

⩾1[−v′; v′]n−ñ
)
= −

N∑
k=1

Ak−1
⩾1 [−v′; v′]n−ñ.

On the other hand, according to [7, Lemma 1], the reachable set in N steps for the system
(A⩾1, [−v′; v′]n1) has the form:

Y ′(N) =
N−1∑
k=0

Ak
⩾1[−v′; v′]n−ñ.

Thus, the 0-controllable set of the system
(
A−1

⩾1, A
−1
⩾1[−v′; v′]n−ñ

)
and the reachable set of

the system (A⩾1, [−v′; v′]n1) over the same number of steps coincide, and consequently,
their limit set analogs X ′

∞ and Y ′
∞ also coincide. However, according to [2, Lemma 3]

X ′
∞ = Rn−ñ, since all eigenvalues of A−1

⩾1 do not exceed 1 in magnitude, being reciprocals of
the eigenvalues of the matrix A⩾1 [6].

This trivially implies the inclusion:

A−M
⩾1 (y − y′) ∈ Y ′

∞ = Rn−ñ,

from which it follows that the equality x(N +M) = x is admissible, and as a consequence,
x ∈ Y∞, leading to

Rn−ñ × Ỹ∞ ⊂ Y∞. (3.8)
Together, (3.7) and (3.8) for the case S = I define the identity:

Y∞ = Rn−ñ × Ỹ∞. (3.9)

Let us consider the case where S ∈ Rn×n is an arbitrary non-singular matrix. From (3.9),
it follows that for the system (S−1AS, S−1U), the limit reachable set Y0

∞ has the form:

Y0
∞ = Rn−ñ × Ỹ∞.

Taking into account the decomposition (3.3) and for N ∈ N from the dynamic equations we
obtain:

x(N) =
N−1∑
k=0

AN−k−1u(k) =
N−1∑
k=0

S

(
AN−k−1

⩾1 O
O AN−k−1

<1

)
S−1u(k) =

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



ON THE EXTERNAL ESTIMATION OF THE LIMIT REACHABLE SET... 71

= S
N−1∑
k=0

(
AN−k−1

⩾1 O
O AN−k−1

<1

)
S−1u(k).

From this, considering (2.2), it follows that x ∈ Y∞ if and only if S−1x ∈ Y0
∞, i.e.

Y∞ = SY0
∞.

Theorem 3.1 is proven.

For the limit null-controllable sets, it is also guaranteed that they are open and convex [2,
Theorem 1]. However, when considering the limit reachable sets, general statements about
their openness or closedness do not seem possible.
Example 3.1:
Consider a two-dimensional system (A,U) with parameters defined as follows:

A =

(
0 1
0 0

)
, U = [−1; 1]× [−1; 1].

Given that A2 = O, for N ⩾ 2 we have

x(N) = u(N − 1) + Au(N − 2).

Therefore, by definition (2.2) the inclusions x ∈ Y∞ and x ∈ U + AU are equivalent. We can
derive the following representation from this:

Y∞ = U + AU = conv

{(
1
1

)
,

(
−1
1

)
,

(
1
−1

)
,

(
−1
−1

)}
+ conv

{(
1
0

)
,

(
−1
0

)}
=

= conv

{(
2
1

)
,

(
−2
1

)
,

(
2
−1

)
,

(
−2
−1

)}
.

Finally, Y∞ is closed.
Example 3.2:
Consider a two-dimensional system (A,U) with parameters defined as follows:

A =

(
0 0
0 1

2

)
, U = [−1; 1]× [−1; 1].

For any N ∈ N and control sequence {u(k)}N−1
k=0 ⊂ U it holds that

x(N) = u(N − 1) + Au(N − 2) + . . .+ AN−1u(0) =

 u1(N − 1)
N−1∑
k=0

1
2k
u2(N − k − 1)

 ,

|x1(N)| ⩽ 1, |x2(N)| ⩽
N−1∑
k=0

1

2k
< 2.

According to (2.2)
Y∞ ⊂ [−1; 1]× (−2; 2).

On the other hand, for any x = (x1, x2)
T ∈ [−1; 1]× (−2; 2) there exists N ∈ N such

that
|x2| ⩽ 2− 1

2N−1
.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



72 A.V. SIMKINA

Consequently, for all k = 0, N − 1 the following control actions will be admissible

u2(k) =
x2

2− 2−N+1
∈ [−1; 1], u1(k) = x1 ∈ [−1; 1].

Then

x(N) =

 x1
N−1∑
k=0

1
2k

x2

2−2−N+1

 =

 x1

x2

2−2−N+1

N−1∑
k=0

1
2k

 =

(
x1

x2

2−2−N+1

(
2− 2−N+1

)) =

(
x1

x2

)
.

We have the inclusion
[−1; 1]× (−2; 2) ⊂ Y∞.

Finally, the representation holds

Y∞ = [−1; 1]× (−2; 2),

meaning that the limit reachable set Y∞ in this example is neither closed nor open.
Example 3.3:
Let us consider the case where n = 1 and in the system (A,U)

A =
1

2
, U = [−1; 1].

By repeating the reasoning from Example 3.2 but applying it only to the single coordinate,
we obtain:

Y∞ = (−2; 2).

Thus, the limit reachable set Y∞ in this example is open.

4. CONSTRUCTION OF ESTIMATES OF LIMIT NULL-CONTROLLABLE SETS

Although Theorem 3.1 guarantees that the set (2.2) is a cylinder, it does not determine
the structure of the cross-section of this cylinder. Essentially, this statement only makes it
possible to reduce the original problem of constructing the limit set to an analogous problem
for a system of smaller dimension, but it does not solve it completely. For this reason, in
this section, we will consider a method for constructing polyhedral external estimates of
the set Y∞, based on the apparatus of supporting half-spaces and properties of convex sets.
However, Theorem 3.1 allows us to assume, without any loss of generality, that all eigenvalues
of the matrix A are strictly less than 1. Indeed, in the opposite case, the original problem is
equivalent to the problem of constructing the set Ỹ∞ for the subsystem (A<1, PS−1U), which
satisfies the eigenvalue constraint. Therefore, we will henceforth assume that the spectral
radius of the matrix A is strictly less than 1, i.e., the system (2.1) is stable.

We will use the previously proven property of the convexity of Y∞ [18, Theorem 1].
As it is known, every closed and convex set is the intersection of its supporting half-spaces
[17, Theorem 18.8]. Although, as demonstrated in Examples 3.1–3.3, the set (2.2) is not
necessarily either closed or open. By means of supporting half-spaces, it is possible to obtain
an external estimate of its closure, i.e., to reconstruct its structure up to boundary points.
Theorem 4.1:
The set Y∞ in the system (A,U) is bounded if and only if the spectral radius of the matrix
A ∈ Rn×n is less than 1. In this case, when the set is bounded, for each p ∈ Rn \ {0} the
following representation holds for the support function:

S (p,Y∞)
△
= sup

x∈Y∞

(p, x) =
∞∑
k=0

max
uk∈U

(
(Ak)Tp, uk

)
.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



ON THE EXTERNAL ESTIMATION OF THE LIMIT REACHABLE SET... 73

Proof
Let the spectral radius of A be at least 1. Then, in the condition of Theorem 3.1, the inequality
n > ñ holds, and as a consequence, Y∞ is not bounded.

Now consider the reverse situation. Note that from the condition 0 ∈ U for all k ∈
N ∪ {0} it follows that:

max
uk∈U

(
(Ak)Tp, uk

)
⩾ 0.

Then for any p ∈ Rn \ {0} due to (2.2), the following chain of equalities holds:

sup
x∈Y∞

(p, x) = sup
N∈N∪{0}

uk∈U
k=0,N−1

N∑
k=0

(
p,Akuk

)
= sup

N∈N∪{0}

N∑
k=0

max
uk∈U

(
p,Akuk

)
=

∞∑
k=0

max
uk∈U

(
p,Akuk

)
.

The convergence of the last series follows from the fact that all eigenvalues of the matrix
A are strictly less than 1 in magnitude, boundedness of the set U and classical inequality:
|(p,Akv)| ⩽ ∥p∥∥Ak∥∥v∥.

Thus, we conclude that:

S(p,Y∞) =
∞∑
k=0

sup
uk∈U

(
(Ak)Tp, uk

)
< ∞.

The boundedness of Y∞ follows from the boundedness of the support function for the basis
vectors and their opposites.

Theorem 4.1 is proven.

The challenge in applying Theorem 4.1 lies in the need to compute the exact value of
the series. For an arbitrary choice of the support vector p ∈ Rn \ {0} solving this problem is
difficult. However, if p is chosen to be an element of the real Jordan basis of A, it is possible
to obtain the exact structure of the supporting half-space.
Corollary 4.1:
Consider the system (2.1) where all eigenvalues of the matrix A have magnitudes less than 1,
and the decomposition (3.3) holds. Let J ∈ Rñ×ñ be a Jordan block of the form (3.4), located
in the rows numbered n0 + 1, . . . , n0 + ñ. Let p ∈ Rn \ {0} be chosen such that for some
i = 1, ñ, the following equality holds:

STp = (0, . . . , 0︸ ︷︷ ︸
n0+i−1

, 1, 0, . . . , 0)T.

Additionally, the following notations are introduced

vn0+j = max
v∈S−1U

vn0+j, vn0+j = min
v∈S−1U

vn0+j, j = 1, ñ.

Then for the set (2.2), the following inclusion holds

Y∞ ⊂
{
x ∈ Rn : xn0+i ⩽ (p, x) ⩽ xn0+i

}
,

where:

1. if λ ≥ 0, then

xn0+i =
ñ−i∑
j=0

vn0+i+j

(1− λ)j+1
,

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



74 A.V. SIMKINA

xn0+i =
ñ−i∑
j=0

vn0+i+j

(1− λ)j+1
;

2. if λ < 0, then

xn0+i =
ñ−i∑
j=0

(
vn0+i+j + vn0+i+j

2(1 + |λ|)j+1
−

vn0+i+j − vn0+i+j

2(1− |λ|)j+1

)
,

xn0+i =
ñ−i∑
j=0

(
vn0+i+j + vn0+i+j

2(1 + |λ|)j+1
+

vn0+i+j − vn0+i+j

2(1− |λ|)j+1

)
.

Proof
By the given condition, for the chosen vector p, it holds that STp = (0, . . . , 0, 1, 0, . . . , 0)T ∈
Rn, where 1 corresponds to the (n0 + i)-th coordinate of the vector STp.

Let k ⩾ ñ− 1. Then, as known [6, section 3.2.5]

Jk =


λk C1

kλ
k−1 C2

kλ
k−2 . . . C ñ−1

k λk−ñ+1

0 λk C1
kλ

k−1 . . . C ñ−2
k λk−ñ+2

...
...

... . . . ...
0 0 0 . . . λk

 ,

where Cj
k denotes the binomial coefficient:

Cj
k =

k!

(k − j)!j!
.

Taking into account (3.3),

(Ak)Tp = (S−1)T diag
(
Jk
1 , . . . , J

k
m

)T
STp =

= (S−1)T(0, . . . , 0︸ ︷︷ ︸
n0+i−1

, λk, C1
kλ

k−1, . . . , C ñ−i
k λk−ñ+i︸ ︷︷ ︸

ñ−i+1

, 0, . . . , 0︸ ︷︷ ︸
n−n0−ñ

)T,

(
(Ak)Tp, u

)
= λkvn0+i + C1

kλ
k−1vn0+i+1 + C2

kλ
k−2vn0+i+2 + . . .+ C ñ−i

k λk−ñ+ivn0+ñ =

=
ñ−i∑
j=0

Cj
kλ

k−jvn0+j+i,

where S−1u = v.
Let k < ñ− 1. Then

Jk =


λk C1

kλ
k−1 C2

kλ
k−2 . . . Ck

kλ
k−k 0 . . . 0

0 λk C1
kλ

k−1 . . . Ck−1
k λk−(k−1) Ck

kλ
k−k . . . 0

...
...

... . . . ...
... . . . ...

0 0 0 . . . 0 0 . . . λk

 .

Taking into account (3.3),

(Ak)Tp = (S−1)T diag
(
Jk
1 , . . . , J

k
m

)T
STp =

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



ON THE EXTERNAL ESTIMATION OF THE LIMIT REACHABLE SET... 75

= (S−1)T(0, . . . , 0︸ ︷︷ ︸
n0+i−1

, λk, C1
kλ

k−1, . . . , Ck−1
k λ, 1︸ ︷︷ ︸

k+1

, 0, . . . , 0︸ ︷︷ ︸
n−n0−i−k

)T,

(
(Ak)Tp, u

)
= λkvn0+i + C1

kλ
k−1vn0+i+1 + C2

kλ
k−2vn0+i+2 + . . .+ Ck

kλ
k−kvn0+i+k =

=
k∑

j=0

Cj
kλ

k−jvn0+j+i.

Thus, (
(Ak)Tp, u

)
=

min{k,ñ−i}∑
j=0

Cj
kλ

k−jvn0+j+i,

sup
u∈U

(
(Ak)Tp, u

)
= sup

v∈S−1U

min{k,ñ−i}∑
j=0

Cj
kλ

k−jvn0+j+i ⩽

⩽
min{k,ñ−i}∑

j=0

Cj
k sup
vn0+j+i∈[vn0+j+i;vn0+j+i]

λk−jvn0+j+i =

=



min{k,ñ−i}∑
j=0

Cj
kλ

k−jvn0+j+i, λ ⩾ 0,

min{k,ñ−i}∑
j=0

Cj
k|λ|

k−j max
{
(−1)k−jvn0+j+i, (−1)k−jvn0+j+i

}
, λ < 0.

On the other hand,

inf
u∈U

(
(Ak)Tp, u

)
= inf

v∈S−1U

min{k,ñ−i}∑
j=0

Cj
kλ

k−jvn0+j+i ⩾

⩾
min{k,ñ−i}∑

j=0

Cj
k inf
vn0+j+i∈[vn0+j+i;vn0+j+i]

λk−jvn0+j+i =

=



min{k,ñ−i}∑
j=0

Cj
kλ

k−jvn0+j+i, λ ⩾ 0,

min{k,ñ−i}∑
j=0

Cj
k|λ|

k−j min
{
(−1)k−jvn0+j+i, (−1)k−jvn0+j+i

}
, λ < 0,

Taking into account Theorem 4.1, for any x ∈ Y∞ we obtain for λ ⩾ 0

(p, x) ⩽
∞∑
k=0

min{k,ñ−i}∑
j=0

Cj
kλ

k−jvn0+j+i =
ñ−i∑
j=0

vn0+j+i

∞∑
k=j

Cj
kλ

k−j =
ñ−i∑
j=0

vn0+j+i

(1− λ)j+1
= xn0+i,

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



76 A.V. SIMKINA

and we obtain for λ < 0

(p, x) ⩽
min{k,ñ−i}∑

j=0

∞∑
k=0

max
{
(−1)k−jvn0+j+i, (−1)k−jvn0+j+i

}
Cj

k+j|λ|
k−j =

=
ñ−i∑
j=0

∞∑
k=0

max{(−1)kvn0+j+i; (−1)kvn0+j+i}Cj
k+j|λ|

k =

=
ñ−i∑
j=0

(
∞∑
k=0

vn0+j+iC
j
2k+j|λ|

2k −
∞∑
k=0

vn0+j+iC
j
2k+j+1|λ|

2k+1

)
=

=
ñ−i∑
j=0

(
vn0+j+i

∞∑
k=0

Cj
2k+j|λ|

2k − vn0+j+i

∞∑
k=0

Cj
2k+j+1|λ|

2k+1

)
=

=
ñ−i∑
j=0

(
vn0+j+i

2

(
1

(|λ|+ 1)j+1
+

1

(1− |λ|)j+1

)
+

vn0+j+i

2

(
1

(|λ|+ 1)j+1
− 1

(1− |λ|)j+1

))
=

=
ñ−i∑
j=0

(
vn0+j+i + vn0+j+i

2(1 + |λ|)j+1
+

vn0+j+i − vn0+j+i

2(1− |λ|)j+1

)
= xn0+i.

Similarly, from Theorem 4.1, it can be derived since max
uk∈U

(
−(Ak)Tp, uk

)
=

−min
uk∈U

(
(Ak)Tp, uk

)
that (p, x) ⩾ xn0+i.

Corollary 4.1 is fully proven.

For the case of a Jordan block of size 3.5, it is possible to construct supporting hyperplanes
for an infinite number of vectors p. Therefore, the corresponding external estimate of the set
Y∞ will be a cylinder.

Corollary 4.2:
Consider the system (2.1) where all eigenvalues of the matrix A have magnitudes less than
1, and the decomposition (3.3) holds. Let J ∈ R2ñ×2ñ be a Jordan block of the form (3.5),
located in the rows numbered n0 + 1, . . . , n0 + 2ñ. Let p ∈ Rn \ {0} be chosen such that for
some i = 1, ñ, the following equality holds:

STp = (0, . . . , 0︸ ︷︷ ︸
n0+2i−2

, p̃1, p̃2, 0, . . . , 0)
T, p̃11 + p̃22 = 1.

Additionally, the following notations are introduced

rn0+j = max
v∈S−1U

√
v2n0+2j−1 + v2n0+2j, j = 1, ñ.

Then the following inclusion holds for the set (2.2)

Y∞ ⊂
{
x ∈ Rn : x2

n0+2i−1 + x2
n0+2i ⩽ Rn0+j

2
}
, Rn0+j =

ñ−i∑
j=0

rn0+i+j

(1− r)j+1
.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



ON THE EXTERNAL ESTIMATION OF THE LIMIT REACHABLE SET... 77

Proof
By the given condition, for the chosen vector p, it holds that STp =
(0, . . . , 0, p̃1, p̃2, 0, . . . , 0)

T ∈ Rn, where the two-dimensional subvector p̃ corresponds
to the (n0 + 2i− 1)-th and (n0 + 2i)-th coordinates of the vector p.

Let k ⩾ ñ− 1. Then

Jk =


rkAkφ C1

kr
k−1A(k−1)φ . . . C ñ−1

k rk−ñ+1A(k−ñ+1)φ

0 rkAkφ . . . C ñ−2
k rk−ñ+2A(k−ñ+2)φ

...
... . . . ...

0 0 . . . rkAkφ

 ,

Taking into account (3.3),

(Ak)Tp = (S−1)T diag
(
Jk
1 , . . . , J

k
m

)T
STp =

= (S−1)T
(
0, . . . , 0︸ ︷︷ ︸
n0+2i−2

, C0
kr

k−0
(
AT

(k−0)φp̃
)T

, . . . , C ñ−i
k rk−ñ+i

(
AT

(k−ñ+i)φp̃
)T

, 0, . . . , 0
)T

.

In case k < ñ− 1

Jk =


rkAkφ C1

kr
k−1A(k−1)φ . . . Ck

k r
k−kA(k−k)φ 0 . . . 0

0 rkAkφ . . . Ck−1
k rk−(k−1)A(k−(k−1))φ Ck

k r
k−kA(k−k)φ . . . 0

...
...

. . .
...

...
. . .

...
0 0 . . . 0 0 . . . rkAkφ

 ,

Taking into account (3.3),

(Ak)Tp = (S−1)T diag
(
Jk
1 , . . . , J

k
m

)T
STp =

= (S−1)T
(
0, . . . , 0︸ ︷︷ ︸
n0+2i−2

, C0
kr

k−0
(
AT

(k−0)φp̃
)T

, . . . , Ck−1
k rk−(k−1)

(
AT

(k−(k−1))φp̃
)T

, p̃T, 0, . . . , 0
)T

.

Let us denote for v ∈ S−1U by vj ∈ R2, j = 1, ñ its two-dimensional subvectors by:
vj = (vn0+2j−1, vn0+2j)

T. Note that by the assumptions, ∥vj∥ ⩽ rn0+j . Also, let u = Sv ∈ U .
We obtain for k ⩾ ñ− 1(
(Ak)Tp, u

)
= rk(A−kφp̃, v

i) + C1
kr

k−1(A−(k−1)φp̃, v
i+1) + C ñ−i

k rk−ñ+i(A−(k−ñ+i)φp̃, v
ñ) ⩽

⩽ rk∥A−kφp̃∥∥vi∥+ C1
kr

k−1∥A−(k−1)φp̃∥∥vi+1∥+ . . .+ C ñ−i
k rk−ñ+i∥A−(k−ñ+i)φp̃∥∥vñ∥ ⩽

⩽ rkrn0+i + C1
kr

k−1rn0+i+1 + . . .+ C ñ−i
k rk−ñ+irn0+ñ =

ñ−i∑
j=0

Cj
kr

k−jrn0+i+j.

For k < ñ− 1(
(Ak)Tp, u

)
= rk(A−kφp̃, v

i) + C1
kr

k−1(A−(k−1)φp̃, v
i+1) + Ck

kr
k−k(A−(k−k)φp̃, v

i+k) ⩽

⩽ rk∥A−kφp̃∥∥vi∥+ C1
kr

k−1∥A−(k−1)φp̃∥∥vi+1∥+ . . .+ Ck
kr

k−k∥A−(k−k)φp̃∥∥vi+k∥ ⩽

rkrn0+i + C1
kr

k−1rn0+i+1 + . . .+ Ck
kr

k−krn0+i+k =
k∑

j=0

Cj
kr

k−jrn0+i+j.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



78 A.V. SIMKINA

Due to the arbitrariness of the choice of u ∈ U we obtain

max
u∈U

(
(Ak)Tp, u

)
⩽

min{k,ñ−i}∑
j=0

Cj
kr

k−jrn0+i+j,

According to Theorem 4.1, for any x ∈ Y∞, the following relations hold:

(p, x) ⩽
∞∑
k=0

min{k,ñ−i}∑
j=0

Cj
kr

k−jrn0+i+j =
n−i∑
j=0

∞∑
k=0

Cj
k+jr

krn0+i+j =

=
n−i∑
j=0

rn0+i+j

(1− r)j+1
= Rn0+i.

Taking into account the choice of p, we obtain the inclusion:

Y∞ ⊂
⋂

p̃21+p̃22=1

{
x ∈ Rn : (p, x) ⩽ Rn0+i

}
=
{
x ∈ Rn :

√
x2
n0+2i−1 + x2

n0+2i ⩽ Rn0+i

}
.

Corollary 4.2 is proven.

Using Corollaries 4.1 and 4.2, it is possible to estimate from above the sought set
Y∞ in certain specific directions determined by the real Jordan basis. The problem] of
constructing supporting hyperplanes is reduced to computing the spectrum of the system
matrix. Additionally, the more general Theorem 4.1 allows for refining the computed external
estimates if necessary.

5. BLOOD PLASMA GLUCOSE STABILIZATION SYSTEM

We will demonstrate the effectiveness of the developed methods using the example of
constructing the limit reachable set for a stabilization glucose and insulin levels in blood
plasma system. For clarity and the ability to visualize the computed sets, we will consider the
simplest mathematical model by Bergman [10], described by a three-dimensional system:

Ġ(t) = −q1G(t)−X(t)(G(t) +GB) + v1(t),

Ẋ(t) = −q2X(t) + q3I(t),

İ(t) = −m(I(t) + IB) +
v2(t)

VI

,

(5.10)

where G(t) and I(t) represent the deviations of the glucose and free insulin concentrations in
blood plasma from their normal values GB and IB, respectively. X(t) is a term accounting for
the delay in the metabolism processes of free insulin. The coefficient m denotes the partial
elimination (removal from the body), VI is the volume of distribution of insulin in tissues,
v1(t) and v2(t) are the rates of external glucose and insulin input, respectively, and q1, q2,
q3 are auxiliary numerical parameters. The following numerical values of the parameters,
obtained experimentally in [10], are used for the calculations,:

q1 = 0.028, q2 = 0.025, q3 = 0.000013,

VI = 12 L, m =
5

54
min−1, GB = 4.5 mmol/L, IB = 15 nmol/L.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



ON THE EXTERNAL ESTIMATION OF THE LIMIT REACHABLE SET... 79

By performing linearization and subsequent discretization with a step size of δt = 10 min
assuming the control inputs v1(t) and v2(t) are piecewise constant, we obtain the following
finite-difference relations of the form (2.1) with the parameters:

Ad =

(
0.7557 −34.5225 −0.0018

0 0.7788 0.000073
0 0 0.3961

)
,

Ud = conv

{( −30.61
−0.1418
−0.93

)
,

( −28.03
−0.1418
−0.93

)
,

( −30.62
−0.1423
2.11

)
,

( −28.04
−0.1423
2.11

)
,(−28.03

0.1421
−0.93

)
,

(
30.62
0.1421
−0.93

)
,

(
28.02
0.1416
2.11

)
,

(
30.61
0.1416
2.11

)}
.

Note that for the system (Ad,Ud) the condition 0 ∈ intUd is not satisfied because dimUd = 2.
For this reason, we will double the quantization step and transite to an equivalent system in
terms of constructing the limit reachable sets with the following parameters:

If we linearize (5.10) and put v1(t) = v1k, v2(t) = v2k for t = [kδ; (k + 1)δ] for some
δ > 0, we can solve the resulting system of linear differential equations explicitly. Then for
x(k) = (G(kδ), X(kδ), I(kδ))T the finite-difference system relations will be correct when
selecting:

A = A2
d =

(
0.5712 −52.9824 −0.0046

0 0.6065 0.0001
0 0 0.1569

)
, U = U + AdU .

The eigenvalues of the matrix A are λ1 = 0.5712, λ2 = 0.6065 and λ3 = 0.1569, which
satisfy the conditions of Corollary 4.1. We will use this result to construct an external
polyhedral estimate of the set Y∞.

We will compute the vectors p1, p2, p3 from Corollary 4.1:

p1 =
(
(1, 0, 0)S−1

)T
= (1, 1500, 0.3)T,

p2 =
(
(0, 1, 0)S−1

)T
= (0, 1500, 0.28)T,

p3 =
(
(0, 0, 1)S−1

)T
= (0, 0, 1)T.

Additionally, the following equalities hold

min
v∈S−1U

v1 = −997.88, min
v∈S−1U

v2 = −963.25, min
v∈S−1U

v3 = −1.54,

max
v∈S−1U

v1 = 998.18, max
v∈S−1U

v2 = 963.53, max
v∈S−1U

v3 = 3.506.

Ultimately, we obtain the following estimate for the limit reachable set:

Y∞ ⊂ Ŷ∞ = {x ∈ R3 : − 997.88 ⩽ (p1, x) ⩽ 998.18}∩

∩{x ∈ R3 : − 963.25 ⩽ (p2, x) ⩽ 963.53} ∩ {x ∈ R3 : − 1.54 ⩽ (p3, x) ⩽ 3.506}.

The set Ŷ∞ is graphically represented in Figure 5.1.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



80 A.V. SIMKINA

Fig. 5.1. External estimate Y∞ (by lines) and reachable set Y(11) (by surface) for discretized system (5.10).

CONCLUSION

The article investigates various properties of the limit reachable sets for linear stationary
discrete-time systems with geometric constraints on the control. Specifically, it is proven that
these sets are cylinders oriented along the elements of the real Jordan basis of the system
matrix, corresponding to eigenvalues with magnitudes not less than 1. The cross-section of
each such cylinder is convex and can be estimated from above using appropriate supporting
half-spaces, the structure of which is also described in the article. Examples demonstrate that
the limit reachable sets do not necessarily have to be either open or closed, unlike similar
limit null-controllable sets.

For future research, one could consider generalizing the obtained results to a class of
systems where the set of admissible control values U has a lower dimension than the state
space, and 0 is assumed to be only a relative interior point of the set U . Additionally, it is
equally important to develop numerical methods that allow the computation of the support
function value in Theorem 4.1 with any predetermined accuracy.

REFERENCES

1. Benvenuti, L. & Farina, L. (2006) The Geometry of the Reachability Set for Linear
Discrete-Time Systems with Positive Controls, SIAM J. Matrix Anal. Appl., 28(2), 306–
325.

2. Berendakova, A. V. & Ibragimov, D. N. (2023) About the Method for Constructing
External Estimates of the Limit 0-Controllability Set for the Linear Discrete-Time
System with Bounded Control, Autom. Remote Control, 84(2), 83–104.

3. Colonius, F., Cossich, J. A. N. & Santana, A. J. (2022) Controllability Properties and
Invariance Pressure for Linear Discrete-Time Systems, J. Dynam. Different. Equat., 34,
5–22.

4. Ge, S. S., Zhendong, S. & Lee, T. H. (2001) Reachability and controllability of switched
linear discrete-time systems, IEEE Transact. Autom. Control, 46(9), 1437–1441.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



ON THE EXTERNAL ESTIMATION OF THE LIMIT REACHABLE SET... 81

5. Heemels, W. P. M. H. & Camlibel, M. K. (2008) Null Controllability of Discrete-Time
Linear Systems with Input and State Constraints, Proc. of the 47th IEEE Conf. Decis.
Control (Cancun, Mexico), 3487–3492.

6. Horn, R. A. & Johnson, C. R. (1985) Matrix Analysis. New York, NY: Cambridge
University Press.

7. Ibragimov, D. N. (2019) On the Optimal Speed Problem for the Class of Linear
Autonomous Infinite-Dimensional Discrete-Time Systems with Bounded Control and
Degenerate Operator, Autom. Remote Control, 80(3), 393–412.

8. Ibragimov, D. N., Osokin, A. V., Sirotin, A. N. & Sypalo, K. I. (2022) On the Properties
of the Limit Control Sets for a Class of Unstable Linear Systems with Discrete Time
and l1-Restrictions, Journal of Computer and Systems Sciences International, 61(4),
467–484.

9. Ibragimov, D. N. & Sirotin, A. N. (2023) On Some Properties of Sets of Bounded
Controllability for Stationary Linear Discrete Systems with Total Control Constraints,
Journal of Computer and Systems Sciences International, 62(6), 727–756.

10. Fisher, M. E. (1991) A Semiclosed-Loop Algorithm for the Control of Blood Glucose
Levels in Diabetics, IEEE Transactions on Biomedical Engineering, 38(1), 57–61.

11. Fucheng, L., Mengyuan, S. & Usman (2019) Optimal Preview Control for Linear
Discrete-Time Periodic Systems, Mathematical Problems in Engineering, 1–11.

12. Kaba, M. D. & Camlibel, M. K. (2015) A Spectral Characterization of Controllability
for Linear Discrete-Time Systems with Conic Constraints, SIAM J. Control Optim.,
53(4), 2350–2372.

13. Kurzhanskiy, A. & Varaiya, P. (2007) Ellipsoidal Techniques for Reachability Analysis
of Discrete-Time Linear Systems, IEEE Transactions on Automatic Control, 52(1),26–
38.

14. Kurzhanskiy, A. B. & Varaiya, P. (2011) Theory and Computational Techniques
for Analysis of Discrete-Time Control Systems with Disturbancens, Optim. Method
Software, 26(4-5), 719–746.

15. Kuntsevich, V. M. & Kurzhanski, A. B. (2010) Attainability Domains for Linear and
Some Classes of Nonlinear Discrete Systems and Their Control, J. Autom. Inform. Sci.,
42(1), 1–18.

16. Tochilin, V. A. (2015) On the Construction of Nonconvex Approximations to Reach
Sets of Piecewise Linear Systems, Differential Equations, 51(11), 1499–1511.

17. Rockafellar, R. T. (1970) Convex Analysis. Princeton, NJ: Princeton University Press.
18. Simkina, A. V., Ibragimov, D. N. & Kibzun, A. I. (2024) On the Method of Numerical

Simulation of Limit Reachable Sets for Linear Discrete-Time Systems with Bounded
Control, Vestnik YuUrGU. Ser. Mat. Model. Progr., 17(3), 46–56.

19. Sirotin, A. N. & Formal’skii, A. M. (2003) Reachability and Controllability of Discrete-
Time Systems under Control Actions Bounded in Magnitude and Norm, Autom. Remote
Control, 64(12), 1844–1857.

20. Ibragimov, D. N. & Sirotin, A. N. (2017) On the Problem of Operation Speed for
the Class of Linear Infinite-Dimensional Discrete-Time Systems with Bounded Control
Autom. Remote Control, 78(10), 1731–1756.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)


	Introduction
	Problem Statement
	The general structure of reachable sets
	Construction of estimates of limit null-controllable sets
	Blood plasma glucose stabilization system