Adv Syst Sci Appl 2022; 03:156–162
Published online at https://ijassa.ipu.ru.

Solvability of Control Systems with Antiperiodic Boundary
Constraints

Aram V. Arutyunov1, Zuhra T. Zhukovskaya2, Sergey E. Zhukovskiy2

1V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia
2 Institute of Information Transmission Problems (Kharkevich Institute), Moscow, Russia

Abstract: In this paper, we consider control systems with implicit dynamics and antiperiodic
boundary constraints ont the fixed time interval. We derive sufficient conditions for the existence
of admissible control for the systems of this type and obtain estimates of the admissible controls.
These results remain considerable in the partial case when the considered dynamics is explicit.

Keywords: Control system, implicit dynamics, antiperiodic boundary constraints, implicit
function

1. INTRODUCTION

Before proceeding to the formulation of the problem, we introduce the notation used.
Let Rn stand for the n-dimensional real aritheoremetic space with the norm | · | and the

inner product ⟨·, ·⟩. For τ > 0, for an absolutely continuous function x : [0, τ ] → Rn and for a
point t ∈ [0, τ ] at which x is differentiable, we denote by ẋ(t) the derivative of the function x
at the point t. Denote by ẋ the Lebesgue integrable function t 7→ ẋ(t), t ∈ [0, τ ]. Everywhere
below Ln

∞[0, τ ] stands for the set of all Lebesgue measurable essentially bounded functions
v : [0, τ ] → Rn and ACn

∞[0, τ ] stands for the set of all absolutely continuous functions
x : [0, τ ] → Rn such that ẋ ∈ Ln

∞[0, τ ]. Here and below ∀̇ stands for “for almost all”.
Let a mapping f : R× Rn × Rn × Rm → Rk and a nonempty set U ⊂ Rm be given. For

every τ > 0 consider the control system

f(t, x, ẋ, u(t)) = 0, u(t) ∈ U ∀̇ t ∈ [0, τ ], (1.1)

x(0) + x(τ) = 0. (1.2)

We will say that a pair of functions (x̄, ū) ∈ ACn
∞[0, τ ]× Lm

∞[0, τ ] is an admissible
process to the problem (1.1), (1.2), if ū(t) ∈ U for almost all t ∈ [0, τ ], the functions x̄(·)
is a solution to the differential equation f(t, x, ẋ, ū(t)) = 0 on the segment [0, τ ] and the
equality x̄(0) + x̄(τ) = 0 takes place. If a pair of functions (x̄, ū) ∈ ACn

∞[0, τ ]× Lm
∞[0, τ ]

is an admissible process to the problem (1.1), (1.2), then we say that the function x̄ is an
admissible trajectory and the function ū is an admissible control.

The main goal of our paper is to derive sufficient conditions for the existence of an
admissible process to the problem (1.1), (1.2). The proof of the main result is based on
the following idea. We pass from the control problem with implicit dynamics to the control
problem with explicit dynamics using a nonlocal implicit function theorem; for a function

∗Corresponding author: s-e-zhuk@yandex.ru



SOLVABILITY OF CONTROL SYSTEMS WITH ANTIPERIODIC BOUNDARY CONSTRAINTS 157

ū we consider a mapping φ that corresponds to the initial point x0 the endpoint φ(x0)
of a trajectory starting at x0; than we use an analog of Brouwer’s fixed point theorem to
show the existence of x0 such that x0 = −φ(x0); and prove that the corresponding trajectory
is admissible. The idea of reduction of implicit ODEs (ordinary differential equations) to
explicit using implicit function theorems is well known. The usage of nonlocal implicit
function theorems to control systems and ODEs was studied in [1, 3]. The usage of Brower’s
fixed point theorem and similar results for studying boundary value problems is a standard
tool (see, for example, [4]). In this paper, we have combined and modified these known
approaches and applied the for studying the problem (1.1), (1.2).

2. MAIN RESULTS

Denote byBn(r) the closed ball in Rn centered at zero with the radius r ≥ 0. For an arbitrary
linear operator A : Rn → Rk, denote

covA := max{α ≥ 0 : Bk(α) ⊂ ABn(1)}.

It is a straightforward task to ensure that covA > 0 if and only if A is surjective. As is
known (see, for example, [2]), that the function cov is continuous.
Theorem 2.1:
Let

(i) the mapping f be twice continuously differentiable;

(ii) ᾱ := inf
{
covf ′

v(t, x, v, u) : (t, x, v, u) ∈ R1 × Rn × Rn × U
}
> 0.

Given numbers R > 0, α ∈ (0, ᾱ), τ > 0 and functions ū ∈ Lm
∞[0, τ ] and

w̄(t) := max
x∈Bn(R)

∣∣∣f(t, x, 0, ū(t))∣∣∣, t ∈ [0, τ ], (2.3)

assume that
3

2α

τ∫
0

w̄(s) ds ≤ R. (2.4)

Then there exists a function x̄ ∈ ACn
∞[0, τ ] such that the pair (x̄, ū) is an admissible process

to the problem (1.1), (1.2) and

|x̄(t)| ≤ R ∀ t ∈ [0, τ ] and |x̄(0)| ≤ 1

2α

τ∫
0

w̄(s) ds.

Note that the function w̄ in Theorem 2.1 is Lebesgue integrable. Indeed, since f is
continuous, then the function

(t, u) 7→ max
x∈Bn(R)

∣∣∣f(t, x, 0, u)∣∣∣, (t, u) ∈ [0, τ ]× U

is continuous. Moreover, the function ū is Lebesgue measurable and essentially bounded.
Therefore, the function w̄, defined by formula (2.3), is measurable and essentially bounded,
hence w̄ is integrable.

Note that the assumptions (i) and (ii) of Theorem 2.1 imply that there exist numbers
R > 0, α ∈ (0, ᾱ), τ > 0 and a function ū ∈ Lm

∞[0, τ ] such that the inequality (2.4) holds for

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



158 ARAM V. ARUTYUNOV, ZUHRA T. ZHUKOVSKAYA, SERGEY E. ZHUKOVSKIY

the function w̄ defined by formula (2.3). Indeed, take arbitrary numbers R > 0, α ∈ (0, ᾱ)
and an arbitrary function ũ ∈ Lm

∞([0,+∞)) such that ũ(t) ∈ U for almost all t ≥ 0. Since
the function

τ 7→
τ∫

0

max
x∈Bn(R)

∣∣∣f(s, x, 0, ũ(s))∣∣∣ ds, τ ≥ 0

is continuous and vanishes zero at the point τ = 0, then there exists τ > 0 such that

3

2α

τ∫
0

max
x∈Bn(R)

∣∣∣f(s, x, 0, ũ(s))∣∣∣ ds < R.

Hence, (2.4) holds for the point τ, for the reduction ū ∈ Lm
∞[0, τ ] of ũ to the segment [0, τ ]

and for the function w̄ defined by formula (2.3).
The regularity assumption (ii) is essential. Consider the corresponding example. Let

n = m = k = 1, f(t, x, ẋ, u) = u(u− 1)ẋ+ x− t2, U = {0, 1}, τ > 0. Then ᾱ = 0 and
therefore the assumption (ii) fails. Moreover, for every measurable essentially bounded
function u : [0, τ ] → U, the only solution to the ODE f(t, x, ẋ, u(t)) = 0 is the function
x(t) = t2. For this function x, we have x(0) + x(τ) > 0. Therefore, in this example, there
exists no admissible process.

Now let us derive the sufficient conditions for the existence of an admissible control. The
following assertion follows directly from Theorem 2.1.

Corollary 2.1:
Let the assumptions (i) and (ii) of Theorem 2.1 hold, the set U be closed. Given numbers
R > 0, α ∈ (0, ᾱ), τ > 0 and a function

w̄U(t) := min
u∈U

max
x∈Bn(R)

∣∣∣f(t, x, 0, u)∣∣∣, t ∈ [0, τ ],

assume that
3

2α

τ∫
0

w̄U(s) ds ≤ R.

Then there exist functions x̄ ∈ ACn
∞[0, τ ] and ū ∈ Ln

∞[0, τ ] such that the pair (x̄, ū) is an
admissible process to the problem (1.1), (1.2) and

|x̄(t)| ≤ R ∀ t ∈ [0, τ ] and |x̄(0)| ≤ 1

2α

τ∫
0

w̄U(s) ds.

Note that both Theorem 2.1 and Corollary 2.1 remain considerable in the partial case
when the dynamics in (1.1) is explicit, i.e. for the problem

ẋ = f̃(t, x, u(t)), u(t) ∈ U ∀̇ t ∈ [0, τ ], x(0) + x(τ) = 0. (2.5)

In this case the following assertion is valid. Let f̃ be twice continuously differentiable and
U be closed. Given numbers R > 0 and τ > 0 and a function

wU(t) := min
u∈U

max
x∈Bn(R)

∣∣∣f̃(t, x, u)∣∣∣, t ∈ [0, τ ],

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



SOLVABILITY OF CONTROL SYSTEMS WITH ANTIPERIODIC BOUNDARY CONSTRAINTS 159

assume that
3

2

τ∫
0

wU(s) ds < R.

Then there exist functions x̄ ∈ ACn
∞[0, τ ] and ū ∈ Ln

∞[0, τ ] such that the pair (x̄, ū) is an
admissible process to the problem (2.5) and

|x̄(t)| ≤ R ∀ t ∈ [0, τ ], |x̄(0)| ≤ R/3.

3. PROOF OF THE MAIN RESULT

In the proof of Theorem 2.1, along with classical theorems on the properties of solutions to
ODEs and fixed point theorems, we will use the global implicit function theorem from [2].
Let us recall it.
Theorem 3.1:
(see [2, Theorem 5]) Given a positive integer s, an open nonempty set Σ ⊂ Rs and a twice
continuously differentiable mapping F : Rn × Σ → Rk, assume that

α0 := inf{covF ′
v(v, σ) > 0 : v ∈ Rn, σ ∈ Σ} > 0.

Then there exists a continuously differentiable mapping G : Σ× Rk → Rn such that

F (G(y, σ), σ) = y, |G(y, σ)| ≤ |y − F (0, σ)|
α0

∀ y ∈ Rk, ∀σ ∈ Σ. (3.6)

In what follows, we use the following corollary of this theorem.
Corollary 3.1:
Let f satisfy the assumptions (i) and (ii) of Theorem 2.1. Then for any positive α < ᾱ there
exists a continuously differentiable mapping g : R1 × Rn × U → Rn such that

f(t, x, g(t, x, u), u) = 0, |g(t, x, u)| ≤ |f(t, x, 0, u)|
α

∀ (t, x, u) ∈ R1 × Rn × U. (3.7)

Proof
Take an arbitrary α < ᾱ. Since (i) and (ii) holds, f is sufficiently smooth and the function cov
is continuous, then there exists an open neighbourhood U0 ⊂ Rm of U such that

α ≤ inf
{
covf ′

v(t, x, v, u) : (t, x, v, u) ∈ R1 × Rn × Rn × U0

}
. (3.8)

Put Σ := R1 × Rn × U0. Define the mapping F : Rn × Σ → Rk by formula

F (v, σ) := f(t, x, v, u) σ := (t, x, u) ∈ Σ, v ∈ Rn.

It is a straightforward task to ensure that F is twice continuously differentiable and

F ′
v(v, σ) ≡ f ′

v(t, x, v, u). (3.9)

Thus,

α0 := inf{covF ′
v(v, σ) > 0 : v ∈ Rn, σ ∈ Σ} (3.9)

=

(3.9)
= inf

{
covf ′

v(t, x, v, u) : (t, x, v, u) ∈ R1 × Rn × Rn × U0

} (3.8)

≥ α > 0.

(3.10)

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



160 ARAM V. ARUTYUNOV, ZUHRA T. ZHUKOVSKAYA, SERGEY E. ZHUKOVSKIY

Therefore, the mapping F satisfies the assumptions of Corollary 3.1. Hence, there exists a
continuously differentiable mapping G : Rk × Σ → Rn such that (3.6) holds.

Put
g(t, x, u) := G(0, σ), σ = (t, x, u) ∈ R1 × Rn × U ⊂ Σ.

The mapping g is well-defined, since U ⊂ U0.
Let us show that g is the desired mapping. Obviously g is continuously differentiable,

since G is. Moreover, we have

f(t, x, g(t, x, u), u) = F (G(0, σ), σ)
(3.6)
= 0,

|g(t, x, u)| = |G(0, σ)|
(3.6)

≤ |F (0, σ)|
α0

(3.10)

≤ |F (0, σ)|
α

=
|f(t, x, 0, u)|

α

for any σ = (t, x, u) ∈ R1 × Rn × U.

Proof of Theorem 2.1
Take arbitrary numbers R > 0, α ∈ (0, ᾱ), τ > 0, a function ū ∈ Lm

∞[0, τ ] and the function
w̄ defined by the equality (2.3) such that (2.4) holds. It follows from Corollary 3.1 that there
exists a continuously differentiable mapping g : R1 × Rn × U → Rn such that (3.7) holds.

Put

r :=
1

2α

τ∫
0

w̄(s) ds.

Then (2.4) implies

0 ≤ r < R,
1

α

τ∫
0

w̄(s) ds ≤ R− r and − 2r +
1

α

τ∫
0

w̄(s) ds ≤ 0. (3.11)

For arbitrary x0 ∈ Bn(r), consider the Cauchy problem

ẋ = g(t, x, ū(t)), t ∈ [0, τ ], x(0) = x0. (3.12)

Since
τ∫

0

max
|x−x0|≤R−r

|g(s, x, ū(s))| ds ≤
τ∫

0

max
x∈Bn(R)

|g(s, x, ū(s))| ds
(3.7)

≤

(3.7)

≤ 1

α

τ∫
0

max
x∈Bn(R)

|f(s, x, 0, ū(s))| ds (2.3)
=

1

α

τ∫
0

w̄(s) ds
(3.11)

≤ R− r,

it follows from [6, Theorem II.4.1] that there exists a solution φ(·, x0) : [0, τ ] → Bn(r) to
the problem (3.12). It follows from [6, Theorem II.4.5] that this solution is unique. It follows
from [6, Theorem II.4.11] that the mapping φ(τ, ·) : Bn(r) → Bn(R) is continuous.

Define the mapping ψ : Bn(r) → Rn by formula

ψ(x0) := −φ(τ, x0), x0 ∈ Bn(r).

This mapping is continuous, since φ(τ, ·) is continuous. Moreover, the following relation
takes place

⟨ψ(x0)− x0, x0⟩ ≤ 0 ∀x0 : |x0| = r. (3.13)

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



SOLVABILITY OF CONTROL SYSTEMS WITH ANTIPERIODIC BOUNDARY CONSTRAINTS 161

Indeed,

ψ(x0) = −φ(τ, x0) = −x0 −
τ∫

0

g(s, φ(s, x0), ū(s)) ds (3.14)

and, hence, for |x0| = r we obtain

⟨ψ(x0)− x0, x0⟩
(3.14)
= −2|x0|2 −

〈 τ∫
0

g(s, φ(s, x0), ū(s)) ds, x0

〉
≤

≤ −2|x0|2 + |x0|
τ∫

0

|g(s, φ(s, x0), 0, ū(s))| ds = −2r2 + r

τ∫
0

|g(s, φ(s, x0), ū(s))| ds
(3.7)

≤

(3.7)

≤ −2r2 +
r

α

τ∫
0

|f(s, φ(s, x0), 0, ū(s))| ds
(2.3)

≤ −2r2 +
r

α

τ∫
0

w̄(s) ds
(3.11)

≤ 0.

So, it follows from (3.13) that the mapping ψ has a fixed point x̄0 ∈ Bn(r), i.e. x̄0 =
ψ(x̄0) (see, for example, [5, §1.6]).

Put x̄(t) := φ(t, x̄0), t ∈ [0, τ ]. Let us show that the function x̄ is a desired admissible
trajectory. We have

f(t, x̄(t), ˙̄x(t), ū(t)) = f(t, x̄(t), φ̇(t, x̄0), ū(t)) = f(t, x̄(t), g(t, x, ū(t)), ū(t))
(3.7)
= 0

for almost all t ∈ [0, τ ]. Moreover,

x̄(0) + x̄(τ) = φ(0, x̄0) + φ(τ, x̄0) = x̄0 − ψ(x̄0) = 0.

Thus, (x̄, ū) is an admissible process. Finally,

|x̄(t)| = |φ(t, x̄0)| ≤ R ∀ t ∈ [0, τ ];

|x̄(0)| = |φ̄(0, x̄0)| = |x̄0| ≤ r.

Therefore, the function x̄ is a desired admissible trajectory.

ACKNOWLEDGEMENTS

The research is supported by the Russian Science Foundation No 22-11-00042,
https://rscf.ru/project/22-11-00042/.

REFERENCES

1. Arutyunov, A.V., Zhukovskaya, Z.T., & Zhukovskiy, S.E. (2022) On the Existence
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2. Arutyunov, A.V. & Zhukovskiy, S.E. (2019) Application of Methods of Ordinary
Differential Equations to Global Inverse Function Theorems. Differ. Equations, 55, 437–
448.

Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)



162 ARAM V. ARUTYUNOV, ZUHRA T. ZHUKOVSKAYA, SERGEY E. ZHUKOVSKIY

3. Avakov, E.R., Arutyunov, A.V., & Zhukovskiy, E.S. (2009) Covering mappings and
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4. Krasnosel’skii, M.A. & Zabreiko, P.P. (1984) Geometrical Methods of Nonlinear
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5. Nirenberg, L. (2001) Topics in Nonlinear Functional Analysis, Courant Institute of
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6. Warga, J. (2014) Optimal Control of Differential and Functional Equations, Academic
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Copyright © 2022 ASSA. Adv Syst Sci Appl (2022)


	Introduction
	Main results
	Proof of the main result