On the Mathematical Model of the Control Problem for a Pseudo-Parabolic Equation with Involution

It is known that in recent years, due to the increasing interest in physics and mathematics, the boundary problems related to pseudo-parabolic equations with involution were widely studied. For this purpose, various boundary problems for parabolic and pseudo-parabolic equations have been widely studied by many researchers.

The pseudo-parabolic type equations arise in areas such as fluid flow [1], heat transfer [2], and the diffusion of radiation [3]. Roughly speaking, pseudo-parabolic equations account for higher order correction in the model than do parabolic equations. The boundary control of a pseudo-parabolic equation and compare the results to those of parabolic equations was studied in [4]. The stability, uniqueness, and existence of solutions of some classical problems for the considered equation are studied in [5]. In [6], the point control problems for pseudo-parabolic and parabolic type equations are considered.

The optimal control problem for the parabolic type equations was studied by Fattorini and Friedman [7, 8]. Control problems for the infinite-dimensional case were studied by Egorov [9], who generalized Pontryagin’s maximum principle to a class of equations in Banach space, and the proof of a bang-bang principle was shown in the particular conditions.

The boundary control problem for a parabolic equation with a piecewise smooth boundary in a n -dimensional domain was studied in [10] and an estimate for the minimum time required to reach a given average temperature was found. In [11], the considered the heat conduction equation with the Robin boundary condition and developed a mathematical model of the process of heating a cylindrical domain. Control problems for the heat transfer equation in the three-dimensional domain are studied in [12].

Control problems for parabolic equations in bounded one and two-dimensional domains are studied in works [13–17]. In these articles, an estimate was found for the minimum time required to heat a bounded domain was found to be an estimate of average temperature. The existence of control function is proved by Laplace transform method.

Basic information on optimal control problems is given in detail in monographs by Lions and Fursikov [18, 19]. General numerical optimization and optimal control for second-order parabolic equations have been studied in many publications such as [20]. Practical applications of optimal control problems for equations of parabolic type were presented in [21].

In [22], a boundary value problem for the parabolic equation associated with involution in a one-dimensional domain is studied. Many boundary value problems for parabolic type equations with involution were studied in works [23, 24]. Boundary control problems for pseudo-parabolic type equations were studied in works [25, 26], and it was proved that there is a control function for heating the domain to the average temperature.

In this work, the boundary control problem for the pseudo-parabolic equation with involution is considered. The main control problem is presented in Section 1. In Section 2, the boundary control problem studied in this work is reduced to the Volterra integral equation of the second kind by the Fourier method, and the existence of a solution to the integral equation is proved using the Laplace transform method.

1.    Statement of Problem

In this paper, we consider the following pseudo-parabolic equation with involution in the domain Q = (0, n) x (0,n)

u t (x, y, t) - Au(x, y, t) - Au t (x, y, t) +

+EUxx(n - X,y,t)+ EUxxt(n - X,y,t) = 0,

(x,y,t) G Qt := Q x (0,T), with boundary conditions

u(0,y,t) = ^(y) V(t), u(n,y,t) = 0, t > 0, 0 < y < n, and

u(x, 0,t) = 0, u(x,n,t) = 0, and initial condition

u(x,y, 0) = 0, 0 < x,y < n,(4)

where e is a nonzero real number such that | e | < 1 , ^(y ) is a given function, and v (t) is the control function. If the control function v (t) G W 1 ( R + ) satisfies the conditions, v (0) = 0 and | V ( t ) | <  1 on the half-line t >  0 , we say that it is admissible.

Assume that the given function ^ G W 2 ² (Q) satisfies the conditions

^(0) = ^(n) = 0,  ^n > 0, where

ψ n

π

= П /^(y) ^sin nydy,

n = 1, 2,....

Differential equations with modified arguments are equations in which the unknown function and its derivatives are evaluated with modifications of time or space variables;

such equations are called, in general, functional differential equations. Among such equations, one can single out, equations with involutions [27].

Definition 1. [28,29] A function f (x) ^ x maps bijectively a set of real numbers D, such that

f (f (x)) = x or   f-1(x) = f (x),

is called an involution on D.

It can be seen that equation (1) for e = 0 is a classical pseudo-parabolic equation. If e = 0 , equation (1) relates the values of the second derivatives at two different points and becomes a nonlocal equation. It is known that boundary control problems for the pseudo-parabolic equation in the case e = 0 were studied in detail in work [26].

We now consider the following control problem.

Control Problem. Assume that function ф(t) is given. Then, find the control function v(t) from the condition

ππ

У У u(x,y,t) dx dy = ф(t),   t >   0,                         (5)

where u(x,y,t) is a solution of the problem (1) - (4) and it depends on the control function v(t).

2.    Main Results

In this section, we consider the reduction of the given control problem to a Volterra integral equation of the second kind and the existence of a control function.

We now consider the spectral problem

Vxx(x,y) — EVxx(n - x,y) + Vyy (x,y) + AV (x,y) = 0, 0

V(0,y) = V(n,y) = 0, V(x, 0) = V(x,n) = 0, 0 < x,y< n, where |e|< 1, e E R \ {0}. It is proved in [22,24] that expressing the solution of a spectral problem in terms of the sum of even and odd functions, one finds the following eigenvalues:

A_m,_n,1 = (1 — e) (2m + 1)2 + n²,  m E N0 = N U {0},  n E N,           (6)

Am,n,2 = (1 + e)4 m² + n², m,n E N,                      (7)

and we have the following eigenfunctions

Vm,n,1(x, y) = sin(2m + 1)x sinny, m E N0, n E N, and

• Vm,_n,2(x,y) = sin2mx sinny, m,n E N.

By the solution of the problem (1) - (4) we understand the function u(x,y,t) represented in the form

u(x,y,t) =

^{n —}

-------^⁽y)^{v(t) — w(x}, y, t),

where the function w(x,y,t) E CX^MT) П C(QT) is the solution to the problem:

wt(x, y, t) - △ w (x, y, t) - Awt(x, y,t) + e wxx(n - x, y, t) + ewxxt(n - x, y, t) = = x My) V‘(t) - x M(y) V(t) - x M (y) V‘(t) πππ with initial-boundary conditions

w(x,y,t) |dQ = 0, w(x,y, 0) = 0.

We obtain the solution of the initial boundary value problem (1) – (4) using the solution of the above mixed problem w(x,y,t) and the equality (8) as follows:

π

u(x,y,t) = -

-

π

∞∞

- 2 EE m=0 n=1

x

- My) V(t)- t

(1 + n²) Mn Vm,n,1(x,y) f ⁽²m^{+ 1) (1 +}Xm,n, 1) J

-

q_m,n,i^{(t s)}V‘(s) ds

-

∞∞

- 2 EE m=0 n=1

0 t

П²Mn Vmn1(x,y)   f

⁽²m⁺ 1) ^{(1 +} Am,n,1) J 0 t

-

q^m,ⁿ,1^{(t s)}V(s) ds

-

_1 E E (1 + П’ Mn Vm■n,2(X, y’ I e-qm,n,2 (t-s)     ds_ п £1 £    m (1 + A n.) J e V 's’ ds

t

¹ V^ V^ n² M Vm,ri,2^(x,y) /

^П m=1 ~=1 m^{(1 + X}mn,2) J

q-^-t-V) V(s) ds,

where V_mnMx,y) are the eigenfunctions, q are defined by (6), (7), respectively.

It is clear that

λ

1 + ^m,n,i

(i = 1,2) and Xm,n,1, ^m,n,2

ππ

J J Vmni(x,y) dxdy =

2(1 - (-1)ⁿ) n (2m + 1) ,

ππ

J J Vmn2(x,y) dxdy = 0.

Using the condition (5) and the equalities (9), (10), we have

ππ

ππ

ф(t) = У У u(x,y,t) dydx = V(t) У ^ —

-

π

x

— My) dydx-

∞∞

• -    П EE

m=0 n=1

(1+ n²) Mn (1 - (-1)ⁿ)} e n (2m + 1)²(1 + Xm,n,i) J ^e

^qm,n,1^(t-s)V‘(s) ds

-

∞∞

• -    4 EE

m=0 n=1

n                         e

^{(2m + 1)2 (1 +}Xm,n,1) J

q^m,ⁿ,1^(t s) V(s) ds.

According to Parseval equality, we get

ππ

∞

∞

∞∞ nnx^(y) dydx = ПЕЕ

Л

m=0 n=1

^ф" (1 - (-1)") (2m + 1)²n

Set

α

^{4 (1 - e)}π

∞

∞

EE m=0 n=1

^n (1 - (-1)") n ⁽¹ + Am,",1⁾ ’

and

^Лт,"

4(1 - е) ф„ (1 - (-1)")

П n (1 + Am,",1)².

By (11) - (13) and from the properties of the function v(t), we can write

∞∞

ф(t) = a v(t) + 52 52 Лт,"

m=0 n=1

t

/ e-^-sv (s) ds, 0

Am. " 1 where q_m,",1 = ——--- ¹ + ^Am,n,1

.

Consider the function

K (t)

∞∞

= ЕЕ л m=0 n=1

m,n e

q^m'ⁿ'1 t,   t> 0.

Then, using (14), (15), we get the following Volterra integral equation

t av(t) + У K(t - s) v(s)ds = ф(t), 0

t > 0.

E W2(-^TO> +^) which satisfy

For any M > 0, we denote W(M) the set of functions ф the following conditions:

^^ф^Wl(R+) < M, ^ф(t) = ^{0 for}t < 0.

Below, we present the main result in this paper.

Theorem. (Existence of the solution) There exists M > 0 such that for any function ф E W(M) the solution v(t) of the equation (16) exists, and satisfies condition \v(t)| < 1.

We will prove this theorem step by step.

Lemma 1. Assume that, \е\ < 1. Then, the kernel K(t) of the integral equation (16) is continuous on the half-line t > 0.

Proof. Since ^ф" is non-negative and |е| < 1, it is clear that Л_т,_п is non-negative. It is easy to see that then the kernel of the integral equation (16) is the function K(t), which is positive and bounded.

О

It is known that we can write the Laplace transform of the function v(t) as follows

∞

^(p) = У e-^pt v(t) dt, 0

where p = a + iZ, a > 0, Z ^ R.

Then, applying the Laplace transform to the integral equation (16), we get

M = av(p) + K(p) ^(p), where α is defined by (12). Then we can write x        ф(P')

v (P) =, a + K (p)

and a+i^  ~ v (t) = -^ [    Ф(Р   eptdp = ;1 / a + iZ)   e(a+iZ)t dZ.(18)

²ⁿⁱ J a + К(p)        ²ⁿ J a + К (a + iZ)

σ-i∞-∞

Lemma 2. The following estimate is valid:

|a + К (a + iZ )| > a, a> 0, Z ^ R, where a = const > 0 is defined by (12).

Proof. It is clear that a defined by (12) is positive and bounded as |e| < 1.

It is known that we can write the Laplace transform of the function К(t) as follows

∞∞

^m,n

.

p⁺qm,n,1

К (p) = EE m=0 n=1

Then we can write where

and

=a

∞∞ a + К (a + iZ) = a + EE m=0 n=1

^{E E Л}т,п ^{(a +}Qm^,!) m=0 ^^(a+qm^)^{2+Z 2}

.^^

^m,n a + qm,n,1 + iZ

∞∞

•                  ^m,n

^:  m= 2=1 ^(a + qm,„,1)^{2 +}(²

-^^-

= Re(a + К (a + iZ)) + i Im(a + К (a + iZ)),

∞∞

Re(a + К (a + iZ)) = a + EE m=0 n=1

^Лт,п ^{(a + q}m,n,1) (a + qm,n,1) + Z²’

∞∞

I^{m(a + К(a +}i^Z)) = -^ZE E m=0 n=1

^m,n

(a + qm,n,1)²+ Z²'

We can see that the following inequality holds

^(a+ qm,n,1⁾²+ ^Z2< ^((a+ qm,n,1⁾²+ ¹⁾⁽¹ + ^Z2).

As a result, we get

1          _>    1            1

(a + qm,n,1)²+ Z²“ 1 + Z²(a + qm,n,1 )²+ 1 .

Then, using the inequality (19), we can obtain the following assumptions x I I         I W Лт,п (a + qm^l) \

IM* + K(, + iZ))| _ a + ££(-+—> m=0 n=1 4         , , 7

>^a+1+Z²

\ E \ л Лт,п (a + qm,n,1)         C1,a m=o n=(a+qm^y+1      1+z 2.

From the above, we can see that the following estimate is valid for |Im(a + K(a + iZ))|:

|Im(a + K(a + iZ))| >

аді

1 + z²,

where C^ and C₂,_a are defined as follows

∞∞

C- _EE m=0 n=1

^Лт,п^{(a + q}m,n,1) ^{(a +} qm,n,1^{)2 + 1}

∞∞

C- _ EE m=0 n=1

Лт,п (a + qmn^y2 + 1

By (20) and (21), we obtain the required estimate

C

|a + K(a + iZ)| > a +.       > a, where C _ min(C1jff, C2,a) is bounded for all a > 0.

Let the Laplace transform of function ф(і) satisfy the condition

У |<ж ) d^z< +^-

-∞

If we proceed to the limit as a ^ 0 in the equality (18), we have

+^

V(₍₎_± [           eiZtdZ.

2n J a + K(iZ)

-∞

□

Also, to prove the theorem, we need the following lemma.

Lemma 3. Assume that, ф({) E W(M). Then for the imaginary part of the Laplace transform of function ф({) the following estimate holds

+m у |#z ) dz < +Ж.

-∞

Proof. Using the formula for integration by parts in (17), we take

∞

(a + iZ) *(a + i

z 4

Further for a ж 0 we get

-^^-

iZФ(iZ)

Besides

^ф(iZ’

_е-(^а+< )t

*'(t) dt.

∞

У e iz^t ф'(t) dt. 0

∞

У e ^iiJ*(t) dt. 0

Thus, we can write the following inequality:

+m у । ж )|2(1+z 2) dz < C21*1^),

-∞

where C₂= const > 0.

By using Cauchy–Bunyakowskii inequalities, we have

+m          +m          +^

)Сл^   Z-2l*(iZ’K.,

J l*^(iZ)l^dZ= j y+77 ^dZ + J _{1+ z}2 di<

-∞         -∞         -∞

^             /9+

< (/*z)2dZ)  (/   ■

-∞-∞

1/2

+

■ ^                -I /9   +^

■(/.*•'•)

-∞

-∞

1/2 dζ

<

+m

< c 11Ж)|²(1 + Z²) dZ < C2 M?,.₃₍R+)■

-∞

□

Now we present the proof of the theorem.

Proof of the Theorem. Let us show that v E W₂1(R₊). Indeed, using (22) and (23) we can write

+m                  +m

*⁽i^Z ’ a + K (iZ)

(1 + |Z|²) dZ<

/|V(Z)|²(1 + | Z 1²’ dZ = У

-∞                 -∞

< 012 У |*^{(iZ)|2(1 + |Z|2)}d^Z = const II*IIw₂1(R) ■ -∞

Now, we show that the function v(t) satisfies the Lipschitz condition. Actually,

t

^|v(t)

ν

(s)| = J v‘(5) dt

<

IIV‘^L2(Q) Vt - s.

s

Using (22), (23) and Lemma 3, we have the following estimate

+m ^

|vWl< Г I I ,/L,1 dz < Г“ / l^)l dz <

2n J |a + K«)|

-∞-∞

C2         C^CM M.

• < 5---|ф|W1(R+) < 7

2n a 2V +    2n a where α is defined by (12) and

M —     , C₂ — const > 0.

C2

The theorem is proved. □

Conclusion

In this paper, the control problem for a pseudoparabolic equation involving involution in a bounded two-dimensional domain was considered. Using the Laplace transform method, the control function required to heat the given domain to an average temperature is found, and the admissibility of this control function was proven.