EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 4, 2021, 1402-1414
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Necessary Conditions for the Existence of a Saddle
Point in one Optimal Control Problem for Systems of

Hyperbolic Equations

Aysel Telman qizi Ramazanova

Department of Nichtlineare Optimierung, Faculty of Mathematic, University Duisburg-Essen,
Essen, Germany,

Abstract. In this paper, we consider one optimal control problem with a multipoint quality func-
tional described by a system of nonlinear hyperbolic equations with Goursat boundary conditions.
Using a modified version of the increment method, various necessary first-order optimality con-
ditions such as the Pontryagin maximum principle and the linearized maximum condition are
proved.

2020 Mathematics Subject Classifications: 49k10, 49k20

Key Words and Phrases: Hyperbolic equation, necessary optimality condition, saddle point,
increment method.

1. Problem Statement.

Let in a given rectangle D = [t0, t1] × [x0, x1] a controlled process described by the
following system of nonlinear hyperbolic equations

ztx = f (t, x, z, zt, zx, u, v) , (t, x) ∈ D (1)

with Goursat boundary conditions

z (t0, x) = a (x) , x ∈ X = [x0, x1] ,
z (t, x0) = b (t) , t ∈ T = [t0, t1] ,

(2)

a (x0) = b (t0) .

Here a (x), b (t) – given absolutely continuous n-dimensional vector functions, f (t, x, z, zt, zx, u, v)
– a given n-dimensional vector-function continuous in the set of variables together with
partial derivatives with respect to (z, zt, zx), u (t, x) and v (t, x) - r and q -dimensional,
respectively, measurable and bounded control vector-functions satisfying type inclusion
constraints of the form

DOI: https://doi.org/10.29020/nybg.ejpam.v14i4.4135

Email address: .ramazanova@uni-due.de (A.Ramazanova)

http://www.ejpam.com 1402 © 2021 EJPAM All rights reserved.



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1403

u (t, x) ∈ U ⊂ Rr, (t, x) ∈ D ,
v (t, x) ∈ V ⊂ Rq, (t, x) ∈ D .

(3)

Control functions u (t, x), v (t, x) with the listed properties are called admissible con-
trols.

It is further assumed that for each given admissible control (uo (t, x) , vo (t, x)), the
Goursat – Darboux boundary-value problem (1) - (2) has a unique absolutely continuous
solution zo (t, x) (in the sense of [1]-[3]. Various sufficient conditions for the existence of
absolutely continuous solutions to the Goursat – Darboux problem (1) - (2) are found, for
example, in [1]-[3].

On the solutions of the boundary value problem (1) - (2) generated by all possible
admissible controls, we define a multi-point functional

S (u, v) = φ (z (T1, X1) , ..., z (Tk, Xk)) . (4)

(Ti, Xi), i = 1, k (t0 < T1 < T2 < ... < Tk ≤ t1, x0 < X1 < X2 < ... < Xk ≤ x1) are
given points, φ (z1, z2, ..., zk) is a given continuously differentiable scalar function.

Further, by ∥α∥ we denote the norm of the vector α = (α1, ..., αn)
′
in the form

∥α∥ =
n∑

i=1
|xi|, the prime (‘) for vectors is the scalar product operation, and for matrices

the transpose operation. Further, the value of o
(
α2
)
means that o

(
α2
) /
α2 → 0 for

α → 0.Obviously, functional (4) is defined for all sets (u (t, x) , v (t, x)), for which the
corresponding solution z (t, x) = z (t, x, u, v) of the boundary value problem (1) - (2) is
defined on the entire domain D.

Consider the following game problem.
Suppose that the control u (t, x) is controlled by the side , which seeks to minimize

the functional (1), and the control v (t, x), the side B, which seeks to maximize the same
functional.

Among all the sets (u (t, x) , v (t, x)) on which the functional (3) is defined, find a set
(uo (t, x) , vo (t, x)), such that

S (uo, v) ≤ S (uo, vo) ≤ S (u, vo) (5)

for any, (u (t, x) , v (t, x)) ∈ U × V , (t, x) ∈ D.
A set (pair) (uo (t, x) , vo (t, x)) satisfying condition (5) is called the saddle point of

functional (4) (see, for example, [4]-[5]).
In this paper, using the methodology, which is a generalization of the methodology [4]-

[5], the necessary conditions for the existence of a saddle point are derived. (A necessary
condition is the existence of a saddle point such as the Pontryagin maximum principle.)

2. A necessary optimality condition such as the Pontryagin maximum
principle for the existence of a saddle point.

Suppose that zo (t, x) is a solution to the boundary value problem (1) - (2) corre-
sponding to the pair (uo (t, x) , vo (t, x)) and z̄ (t, x) = zo (t, x) + ∆z (t, x) is a solu-



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1404

tion to the problem (1) - (2) corresponding to the set(ū (t, x) = uo (t, x) + ∆u (t, x) ,
v̄ (t, x) = vo (t, x) + ∆v (t, x)).

Then it is clear that the increment ∆z (t, x) state, (zo (t, x) , yo (t, x)) satisfies the
conditions

∆zt x (t, x) = f (t, x, z̄ (t, x) , z̄t (t, x) , z̄x (t, x) , ū (t, x) , v̄ (t, x))−

−f (t, x, zo (t, x) , zot (t, x) , zox (t, x) , uo (t, x) , vo (t, x)) , (t, x) ∈ D , (6)

∆z (t0, x) = ∆a (x) , x ∈ [x0, x1]
∆z (t, x0) = 0 , t ∈ [t0, t1] .

(7)

Let ψo (t, x) be an arbitrary n-dimensional vector function. Multiplying both sides of
relation (6) scalarly on the left by ψ (t, x), and then integrating both sides of the resulting
relation over the region D

t1∫
t0

x1∫
x0

ψo′ (t, x) ∆zt x (t, x) dx dt

=

t1∫
t0

x1∫
x0

ψo′ (t, x) [f (t, x, z (t, x) , zt (t, x) , zx (t, x) , ū (t, x) , v̄ (t, x))−

−f (t, x, zo (t, x) , zot (t, x) , zox (t, x) , uo (t, x) , vo (t, x))] dx dt . (8)

Hereinafter, the prime for vectors is the scalar product operation, and for matrices it
means the transpose operation.

We introduce an analog of the Hamilton – Pontryagin function in the form

H (t, x, z, zt, zx, u, v, ψ
o) = ψo′ · f (t, x, z, zt, zx, u, v) .

It is easy to see that

t1∫
t0

x1∫
x0

ψo′ (t, x) ∆zt x (t, x) dx dt =

=

t1∫
t0

x1∫
x0

[(H (t, x, z (t, x) , zt (t, x) , zx (t, x) , u (t, x) , v (t, x) , ψ
o (t, x))−

−H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u (t, x) , v (t, x) , ψ

o (t, x)))+

+ (H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u (t, x) , v (t, x) , ψ

o (t, x))−

−H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)))] . (9)



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1405

We write the increment of functional

∆S (uo, vo) = S (u, v)− S (uo, vo) =

= φ (z (T1, X1) , ..., z (Tk, Xk))− φ (zo (T1, X1) , ..., z
o (Tk, Xk)) . (10)

Given identity (9) from (10), we have

∆S (uo, vo) = S (u, v)− S (uo, vo) =

= φ (z (T1, X1) , ..., z (Tk, Xk))− φ (zo (T1, X1) , ..., z
o (Tk, Xk))+

+

t1∫
t0

x1∫
x0

ψo′ (t, x) ∆zt x (t, x) dx dt−

−
t1∫

t0

x1∫
x0

[(H (t, x, z (t, x) , zt (t, x) , zx (t, x) , u (t, x) , v (t, x) , ψ
o (t, x))−

−H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u (t, x) , v (t, x) , ψ

o (t, x)))+

+ (H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u (t, x) , v (t, x) , ψ

o (t, x))−

−H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)))] dx dt . (11)

We introduce the notation

∆uvH [t, x] ≡ H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u (t, x) , v (t, x) , ψ

o (t, x))−

−H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ,

∆uH [t, x] ≡ H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u (t, x) , v

o (t, x) , ψo (t, x))−

−H (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ,

fu [t, x] ≡ fz
(
t, x, zo (t, x) , zot (t, x) , z

o
x (t, x) , u

0 (t, x) , vo (t, x)
)
,

Hu [t, x] ≡ Hu (t, x, z
o (t, x) , zot (t, x) , z

o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ,

∆ū v̄f [t] ≡ f (t, x, zo (t, x) , zot (t, x) , z
o
x (t, x) , ū (t, x) , v̄ (t, x))−

−f (t, x, zo (t, x) , zot (t, x) , zox (t, x) , uo (t, x) , vo (t, x)) .

Taking into account the introduced notation and using the Taylor formula the incre-
ment formula (11) of the quality functional (4) is represented as:

∆S (uo, vo) =
k∑

i=1

∂φ′ (z̄ (T1, X1) , ..., z̄ (Tk, Xk))

∂zi
∆z (Ti, Xi) + o1 (∥∆z (Ti, Xi)∥)+

+

t1∫
t0

x1∫
x0

ψo′ (t, x) ∆zt x (t, x) dx dt−
t1∫

t0

x1∫
x0

∆ū(t, x) v̄(t, x)H [t, x] dx dt−



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1406

−
t1∫

t0

x1∫
x0

H ′
z (t, x, z

o (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ∆z (t, x) dx dt−

−
t1∫

t0

x1∫
x0

H ′
zt (t, x, z

o (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ∆zt (t, x) dx dt−

−
t1∫

t0

x1∫
x0

H ′
zx (t, x, z

o (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ∆zx (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆ū(t, x) v̄(t, x)H
′
z [t, x] ∆z (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆u(t, x) v(t, x)H
′
zt [t, x] ∆zt (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆ū(t, x) v̄(t, x)H
′
zx [t, x] ∆zx (t, x) dx dt−

−
t1∫

t0

x1∫
x0

o2 (∥∆z (t, x)∥+ ∥∆zt (t, x)∥+ ∥∆zx (t, x)∥) dx dt . (12)

We will deal with the transformation of individual terms in the increment formula (12).
Taking into account the boundary conditions (2), we can write that

∆z (t, x) =

t∫
t0

x∫
x0

∆zτs (τ, s) ds dτ, (13)

∆zt (t, x) =
x∫

x0

∆zts (t, s) ds ,

∆zx (t, x) =
t∫

t0

∆zτx (τ, x) dτ .
(14)

From (13)is obtain

∆z (Ti, Xi) =

t1∫
t0

x1∫
x0

αi (t, x) ∆ztx (t, x) dx dt, (15)

αi (t, x), i = 1, k characteristic function of the domain [t0, Ti]× [x0, Xi], i = 1, k.



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1407

Using (15) we obtain

k∑
i=1

∂φ′
1

(
z0 (T1, X1) , ..., z

0 (Tk, Xk)
)

∂zi
∆z (Ti, Xi) =

=

t1∫
t0

x1∫
x0

k∑
i=1

αi (t, x)
∂φ′ (z0 (T1, X1) , ..., z

0 (Tk, Xk)
)

∂zi
∆ztx (t, x) dx dt . (16)

Further, using identities (12), applying the two-dimensional analogue of the Fubini for-
mula, we arrive at the following relations:

t1∫
t0

x1∫
x0

H ′
z (t, x, z

o (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ∆z (t, x) dx dt =

=

t1∫
t0

x1∫
x0

 t1∫
t0

x1∫
x0

H ′
z (τ, s, z

o (τ, s) , zot (τ, s) , z
o
x (τ, s) , u

o (τ, s) , vo (τ, s) , ψo (τ, s)) ds dτ

 ∆ztx (t, x) dx dt ,

t1∫
t0

x1∫
x0

H ′
zt (t, x, z

o (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ∆zt (t, x) dx dt =

=

t1∫
t0

x1∫
x0

 x1∫
x0

H ′
zt (t, s, z

o (t, s) , zot (t, s) , z
o
x (t, s) , u

o (t, s) , vo (t, s) , ψo (t, s)) ds

 ∆ztx (t, x) dx dt ,

t1∫
t0

x1∫
x0

H ′
zx (t, x, z

o (t, x) , zot (t, x) , z
o
x (t, x) , u

o (t, x) , vo (t, x) , ψo (t, x)) ∆zx (t, x) dx dt =

=

t1∫
t0

x1∫
x0

 t1∫
t0

H ′
zx (τ, x, z

o (τ, x) , zot (τ, x) , z
o
x (τ, x) , u

o (τ, x) , vo (τ, x) , ψo (τ, x)) dτ

 ∆ztx (t, x) dx dt .

Taking into account the proven identities in the increment formula (12), we obtain the
increment formula in the form:

∆S

uo, vot0 t1
x1∫

x0

[
k∑

i=1

∂φ′ (z (T1, X1) , ..., z (Tk, Xk))

∂zi
αi (t, x)

]
∆zt x (t, x) dx dt+

+

t1∫
t0

x1∫
x0

ψo′ (t, x) ∆zt x (t, x) dx dt−



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1408

−
t1∫

t0

x1∫
x0

 t1∫
t0

x1∫
x0

H ′
z (τ, s, z

o (τ, s) , zot (τ, s) , z
o
x (τ, s) , u

o (τ, s) , vo (τ, s) , ψo (τ, s)) ds dτ

 ∆ztx (t, x) dx dt−

−
t1∫

t0

x1∫
x0

 x1∫
x0

H ′
zt (t, s, z

o (t, s) , zot (t, s) , z
o
x (t, s) , u

o (t, s) , vo (t, s) , ψo (t, s)) ds

 ∆ztx (t, x) dx dt−

−
t1∫

t0

x1∫
x0

 t1∫
t0

H ′
zx (τ, x, z

o (τ, x) , zot (τ, x) , z
o
x (τ, x) , u

o (τ, x) , vo (τ, x) , ψo (τ, x)) dτ

 ∆ztx (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆u(t, x) v(t, x)H (t, x) dx dt−
t1∫

t0

x1∫
x0

∆u(t, x) v(t, x)H
′
z (t, x) ∆z (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆u(t, x) v(t, x)H
′
zt (t, x) ∆zt (t, x) dx dt−

t1∫
t0

x1∫
x0

∆u(t, x) v(t, x)H
′
zx (t, x) ∆zx (t, x) dx dt+

+o1

(
k∑

i=1

∥∆z (Ti, Xi)∥

)
−

t1∫
t0

x1∫
x0

o2 (∥∆z (t, x)∥+ ∥∆zt (t, x)∥+ ∥∆zx (t, x)∥) dx dt .

Assuming that

ψ (t, x) =
t1∫
t0

x1∫
x0

H ′
z (τ, s, z

o (τ, s) , zot (τ, s) , z
o
x (τ, s) , u

o (τ, s) , vo (τ, s) , ψo (τ, s)) ds dτ+

+
x1∫
x0

H ′
zt (t, s, z

o (t, s) , zot (t, s) , z
o
x (t, s) , u

o (t, s) , vo (t, s) , ψo (t, s)) ds+

+
t1∫
t0

H ′
zx (τ, x, z

o (τ, x) , zot (τ, x) , z
o
x (τ, x) , u

o (τ, x) , vo (τ, x) , ψo (τ, x)) dτ ,

(17)
then the increment formula (17)takes the form

∆S (uo, vo) =

= −
t1∫

t0

x1∫
x0

∆ū(t, x) v̄(t, x)H [t, x] dx dt−
t1∫

t0

x1∫
x0

∆ū(t, x) v̄(t, x)H
′
z [t, x] ∆z (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆ū(t, x) v̄(t, x)H
′
zt [t, x] ∆zt (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆ū(t, x) v̄(t, x)H
′
zx [t, x] ∆zx (t, x) dx dt+ o1

(
k∑

i=1

∥∆z (Ti, Xi)∥

)
−



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1409

−
t1∫

t0

x1∫
x0

o2 (∥∆z (t, x)∥+ ∥∆zt (t, x)∥+ ∥∆zx (t, x)∥) dx dt . (18)

Relation (17) is called the adjoint system in the problem under consideration and is
a linear two-dimensional Volterra-type integral equation with one-dimensional terms. It
can be shown (for example, by the method of successive approximations) the existence
of the uniqueness of a unique solution of the adjoint system in the class of measurable
and bounded vector functions. The system of equations (18) is called the adjoint system.
From the estimates established in [1]-[3] and others it follows that

∥∆z (t, x)∥ ≤ L1

t∫
t0

x∫
x0

∥∆ūv̄f [τ, s]∥ ds dτ,

∥∆zt (t, x)∥ ≤ L2

 t∫
t0

x∫
x0

∥∆ūv̄f [τ, s]∥ ds dτ +
x∫

x0

∥∆ūv̄f [t, s]∥ ds

 , (19)

∥∆zx (t, x)∥ ≤ L3

 t∫
t0

x∫
x0

∥∆ūv̄f [τ, s]∥ ds dτ +
t∫

t0

∥∆ūv̄f [τ, x]∥ dτ

 ,
Li = const > 0, i = 1, 3 some constants.
If we assume that (uo, vo) is a saddle point, then from (5) we obtain that

S (uo +∆u, vo)− S (uo, vo) ≥ 0, (20)

S (uo, vo +∆v)− S (uo, vo) ≤ 0. (21)

Using relation (20), (21) and using formula (18) we arrive at the relations:

S (uo +∆u, vo)− S (uo, vo) =

= −
t1∫

t0

x1∫
x0

∆u(t, x)H (t, x) dx dt−
t1∫

t0

x1∫
x0

∆u(t, x)H
′
z (t, x) ∆z (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆u(t, x)H
′
zt (t, x) ∆zt (t, x) dx dt−

t1∫
t0

x1∫
x0

∆u(t, x)H
′
zx (t, x) ∆zx (t, x) dx dt+

+o1

(
k∑

i=1

∥∆z (Ti, Xi)∥

)
−

t1∫
t0

x1∫
x0

o2 (∥∆z (t, x)∥+ ∥∆zt (t, x)∥+ ∥∆zx (t, x)∥) dx dt ,

(22)
S (uo, vo +∆v)− S (uo, vo) =



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1410

= −
t1∫

t0

x1∫
x0

∆v(t, x)H (t, x) dx dt−
t1∫

t0

x1∫
x0

∆v(t, x)H
′
z (t, x) ∆z (t, x) dx dt−

−
t1∫

t0

x1∫
x0

∆v(t, x)H
′
zt (t, x) ∆zt (t, x) dx dt−

t1∫
t0

x1∫
x0

∆v(t, x)H
′
zx (t, x) ∆zx (t, x) dx dt+

+o1

(
k∑

i=1

∥∆z (Ti, Xi)∥

)
−

t1∫
t0

x1∫
x0

o2 (∥∆z (t, x)∥+ ∥∆zt (t, x)∥+ ∥∆zx (t, x)∥) dx dt .

(23)
The obtained partial increment formulas (22), (23) of the quality functional allow us

to prove the necessary optimality condition for the existence of a saddle point.
Let (θ, ξ) ∈ [t0, t1) × [x0, x1) be an arbitrary regular point (Lebesgue point) (see, for

example, [1],[2]; [8]) of the control uo (t, x), ε > 0 be a sufficiently small arbitrary number
such that, θ + ε < t1, ξ + ε < x1, and u ∈ U and an arbitrary vector. The special control
increment uo (t, x) is determined by the formula

∆u (t, x; ε) =

{
u− uo (t, x) , (t, x) ∈ Dε = (θ, θ + ε)× (ξ, ξ + ε) ,
0 , (t, x) ∈ D\Dε .

(24)

By ∆z (t, x; ε) we denote the special statezo (t, x) corresponding to the increment (24)
of the uo (t, x) control.

Taking into account estimates (19), formula (24) for a special control increment, and
also applying the mean value theorem from (22), we obtain

−ε2∆uH (θ, ξ) + o
(
ε2
)
≥ 0.

Consequently
∆uH (θ, ξ) ≤ 0.

Further, considering µ > 0 as an arbitrary sufficiently small number, if the special
control increment vo (t, x) is determined by the formula

∆v (t, x; µ) =

{
v − vo (t, x) , (t, x) ∈ Dµ = (θ, θ + µ)× (ξ, ξ + µ) ,
0 , (t, x) ∈ D\Dµ ,

where θ is an arbitrary vector, then from the increment formula (23) we similarly obtain
that along the saddle point

−µ2∆vH (θ, ξ) + o
(
µ2
)
≤ 0, (25)

Therefore, from inequality (25) it follows that

∆vH (θ, ξ) ≥ 0.

Given the arbitrariness of u ∈ U , v ∈ V , (θ, ξ) ∈ [t0, t1)× [x0, x1) formulate the result.



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1411

Theorem 1. If the set (uo (t, x) , vo (t, x)) is a saddle point of functional (4) under con-
straints (1) - (3). Then, with the necessity of the following conditions:

max
u∈U

H (θ, ξ, zo (θ, ξ) , zot (θ, ξ) , z
o
x (θ, ξ) , u, v

o (θ, ξ) , ψo (θ, ξ)) =

= H (θ, ξ, zo (θ, ξ) , zot (θ, ξ) , z
o
x (θ, ξ) , u

o (θ, ξ) , vo (θ, ξ) , ψo (θ, ξ)) ,

for all u ∈ U and (θ, ξ) ∈ [t0, t1)× [x0, x1),

min
v∈V

H (θ, ξ, zo (θ, ξ) , zot (θ, ξ) , z
o
x (θ, ξ) , u

o (θ, ξ) , v, ψo (θ, ξ)) =

= H (θ, ξ, zo (θ, ξ) , zot (θ, ξ) , z
o
x (θ, ξ) , u

o (θ, ξ) , vo (θ, ξ) , ψo (θ, ξ)) ,

for allv ∈ V , (θ, ξ) ∈ [t0, t1)× [x0, x1).

(1) is an analogue of the Pontryagin maximum principle for the problem under con-
sideration.

3. The case of convex control domains.

Suppose that the sets U and V are convex, and the vector- function f (t, x, z, zt, zx, u, v)
is continuous in the set of variables along with partial derivatives in (z, zt, zx, u, v). Then,
with arguments similar to those from n (2), we can prove the validity of the increment
formula:

∆S (uo, vo) = S (uo +∆u, vo)− S (uo, vo) = −
t1∫

t0

x1∫
x0

H ′
u (t, x) ∆u (t, x) dx dt−

−
t1∫

t0

x1∫
x0

H ′
v (t, x) ∆v (t, x) dx dt+o1

(
k∑

i=1

∥∆z (Ti, Xi)∥

)
−

−
t1∫

t0

x1∫
x0

o3 (∥∆z (t, x)∥+ ∥∆zt (t, x)∥+ ∥∆zx (t, x)∥+ ∥∆u (t, x)∥+ ∥∆v (t, x)∥) dx dt .

(26)
Sincef (t, x, z, zt, zx, u, v) is continuously differentiable with respect to (u, v), by

analogy with (19) we can prove the validity of the estimates

∥∆z (t, x)∥ ≤ L4

t1∫
t0

x1∫
x0

[∥∆u (τ, s)∥+ ∥∆v (τ, s)∥] dτ ds,



A. T. Ramazanova / Eur. J. Pure Appl. Math, 14 (4) (2021), 1402-1414 1412

∥∆zt (t, x)∥ ≤ L5

 t1∫
t0

x1∫
x0

[∥∆u (τ, s)∥+ ∥∆v (τ, s)∥] dτ ds+
x1∫

x0

[∥∆u (t, s)∥+ ∥∆v (t, s)∥] ds

 ,
∥∆zx (t, x)∥ ≤ L6

 t1∫
t0

x1∫
x0

[∥∆u (τ, s)∥+ ∥∆v (τ, s)∥] dτ ds+
t1∫

t0

[∥∆u (τ, x)∥+ ∥∆v (τ, x)∥] dτ

 ,
(27)

L4, L5, L6 some positive constants.
Let ε ∈ [0, 1] an arbitrary number, and u (t, x) ∈ U , (t, x) ∈ D arbitrary admissible

control.
Then the special increment of the admissible control uo (t, x) can be determined by

the formula
∆u (t, x; ε) = ε [u (t, x)− uo (t, x)] . (28)

Thus given the estimate (27), (28) increments of formula (26) we find that

S (uo (t, x) + ∆u (t, x) , vo (t, x))− S (uo (t, x) , vo (t, x)) =

= −ε
t1∫

t0

x1∫
x0

H ′
u [t, x] (u (t, x)− uo (t, x)) dx dt+ o (ε) . (29)

Now the special increment of the admissible control vo (t, x) is determined by the
formula

∆v (t, x;µ) = µ
(
v (t, x)− v0 (t, x)

)
, (30)

where v (t, x) arbitrary admissible control, andµ ∈ [0, 1] arbitrary number.
Moreover, taking into account estimates (27)from (26), we obtain

S (uo (t, x) , vo (t, x) + ∆v (t, x;µ))− S (uo (t, x) , vo (t, x)) =

= −µ
t1∫

t0

x1∫
x0

H ′
v [t, x] (v (t, x)− vo (t, x)) dx dt+ o (µ) . (31)

It follows from expansions (26), (31) that if
(
u0 (t, x) , v0 (t, x)

)
is a saddle point in the

problem under consideration, then

t1∫
t0

x1∫
x0

H ′
u [t, x] (u (t, x)− uo (t, x)) dx dt ≤ 0, (32)

t1∫
t0

x1∫
x0

H ′
v [t, x] (v (t, x)− vo (t, x)) dx dt ≥ 0. (33)

Thus proved



REFERENCES 1413

Theorem 2. If f (t, x, z, zt, zx, u, v) is continuously differentiable with respect to (z, zt, zx, u, v),
and the sets U and V are convex, then for the admissible control (uo (t, x) , vo (t, x)) to be
the saddle point of the problem under consideration, it is necessary that relations (32),
(33) hold for all u (t, x) ∈ U, (t, x) ∈ D, v (t, x) ∈ V, (t, x) ∈ D, respectively.

Inequalities (32), (33) are an analogue of the linearized integral maximum principle.
Using the results of the work, for example, [6], we can show that this result is equivalent

to the following.

Theorem 3. Under the assumptions made, the saddle point (uo (t, x) , vo (t, x)) in the
problem under consideration satisfies the relations

max
u∈U

H ′
u [θ, ξ]u = H ′

u [θ, ξ]u
0 (θ, ξ) ,min

v∈V
H ′

v [θ, ξ]u = H ′
v [θ, ξ] v

0 (θ, ξ)

for all (θ, ξ) ∈ [t0, t1)× [x0, x1).

This result is an analogue of the pointwise linearized maximum principle. Note that, in
the case of a nonsmooth quality functional, analogues of (2) and (3)will not be equivalent
(see, for example, [9]). Note that in the case of openness of the control domain by similar
reasoning, we can calculate the first and second variations of the quality functional and
establish an analog of the Euler equation, as well as an analog of the Legendre – Clebsch
condition.

References

[1] Plotnikov V.I., Sumin V.I, A ground-breaking achievement, The stability problem for
nonlinear Goursat-Darboux systems // Differ. equations. (1972), No. 5, 845–856.

[2] Novozhenov M.M., Sumin V.I., Sumin M.I. A ground-breaking achievement, Methods
of optimal control of mathematical physics systems. Gorky, Publishing House of the
State University, (1986), 87 pp.

[3] Suryanarayana M.B. A ground-breaking achievement, Necessary optimality conditions
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[4] Vasiliev F.P. Another fascinating Book, On the conditions for the existence of a sad-
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[5] Vasiliev F.P. A very interesting paper, On the conditions for the existence of a saddle
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type // Automation and Telemechanics. (1972), No. 2, p. 40–50.

[6] Urgent V.A, Another fascinating Book, Computational methods of optimal control.
Irkutsk, ISU Publishing House, (1982), 110 pp.



REFERENCES 1414

[7] Gabasov R., Kirillova F.M. et al. Another fascinating Book, Optimization methods.
Mn Four Quarter Publishing House. (2011), 472 p.

[8] Pontryagin L.S., Boltyansky V.G., Gamkrelidze R.V., Mishchenko E.F. Another fas-
cinating Book, The mathematical theory of optimal processes. M. Science, (1986).
386 p.

[9] Demyanov V.F. Rubinov A.M. A ground-breaking achievement, Fundamentals of non-
smooth analysis and quasidifferential calculus. M. Science, (1990).

[10] Ramazanova A.T. A new theorem, On determining initial conditions of equations
flexural-torsional vibrations of a bar. European Journal of Pure and Applied Mathe-
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