Citation: Gianmaria Verzini. Foreword to the special issue 'Contemporary PDEs between theory and modeling'[J]. Mathematics in Engineering, 2021, 3(1): i-iv. doi: 10.3934/mine.2021.i
[1] | Farrukh Dekhkonov . On one boundary control problem for a pseudo-parabolic equation in a two-dimensional domain. Communications in Analysis and Mechanics, 2025, 17(1): 1-14. doi: 10.3934/cam.2025001 |
[2] | Farrukh Dekhkonov, Wenke Li, Weipeng Wu . On the time-optimal control problem for a fourth order parabolic equation in the three-dimensional space. Communications in Analysis and Mechanics, 2025, 17(2): 413-428. doi: 10.3934/cam.2025017 |
[3] | Yonghui Zou . Global regularity of solutions to the 2D steady compressible Prandtl equations. Communications in Analysis and Mechanics, 2023, 15(4): 695-715. doi: 10.3934/cam.2023034 |
[4] | Huiyang Xu . Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials. Communications in Analysis and Mechanics, 2023, 15(2): 132-161. doi: 10.3934/cam.2023008 |
[5] | Yao Sun, Pan Wang, Xinru Lu, Bo Chen . A boundary integral equation method for the fluid-solid interaction problem. Communications in Analysis and Mechanics, 2023, 15(4): 716-742. doi: 10.3934/cam.2023035 |
[6] | Shu Wang . Global well-posedness and viscosity vanishing limit of a new initial-boundary value problem on two/three-dimensional incompressible Navier-Stokes equations and/or Boussinesq equations. Communications in Analysis and Mechanics, 2025, 17(2): 582-605. doi: 10.3934/cam.2025023 |
[7] | Ho-Sik Lee, Youchan Kim . Boundary Riesz potential estimates for parabolic equations with measurable nonlinearities. Communications in Analysis and Mechanics, 2025, 17(1): 61-99. doi: 10.3934/cam.2025004 |
[8] | Yuxuan Chen . Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity. Communications in Analysis and Mechanics, 2023, 15(4): 658-694. doi: 10.3934/cam.2023033 |
[9] | Xiulan Wu, Yaxin Zhao, Xiaoxin Yang . On a singular parabolic p-Laplacian equation with logarithmic nonlinearity. Communications in Analysis and Mechanics, 2024, 16(3): 528-553. doi: 10.3934/cam.2024025 |
[10] | Yang Liu, Xiao Long, Li Zhang . Long-time dynamics for a coupled system modeling the oscillations of suspension bridges. Communications in Analysis and Mechanics, 2025, 17(1): 15-40. doi: 10.3934/cam.2025002 |
Consider the pseudo-parabolic equation in the domain Ω={(x,t):0<x<l, t>0}:
∂u∂t=∂2∂t∂x(k(x)∂u∂x)+∂∂x(k(x)∂u∂x),(x,t)∈Ω, | (1.1) |
with boundary conditions
u(0,t)= μ(t),u(l,t)=0,t>0, | (1.2) |
and initial condition
u(x,0)=0,0≤x≤l. | (1.3) |
Assume that the function k(x)∈C2([0,l]) satisfies the conditions
k(x)>0,k′(x)≤0,0≤x≤l. |
The condition (1.2) means that there is a magnitude of output given by a measurable real-valued function μ(t) (See [1,2,3] for more information).
Definition 1. If function μ(t)∈W12(R+) satisfies the conditions μ(0)=0 and |μ(t)|≤1, we say that this function is an admissible control.
Problem B. For the given function θ(t) Problem B consists looking for the admissible control μ(t) such that the solution u(x,t) of the initial-boundary problem (1.1)-(1.3) exists and for all t≥0 satisfies the equation
l∫0u(x,t)dx=θ(t). | (1.4) |
One of the models is the theory of incompressible simple fluids with decaying memory, which can be described by equation (1) (see [1]). In [2], stability, uniqueness, and availability of solutions of some classical problems for the considered equation were studied (see also [4,5]). Point control problems for parabolic and pseudo-parabolic equations were considered. Some problems with distributed parameters impulse control problems for systems were studied in [3,6]. More recent results concerned with this problem were established in [7,8,9,10,11,12,13,14,15]. Detailed information on the problems of optimal control for distributed parameter systems is given in [16] and in the monographs [17,18,19,20]. General numerical optimization and optimal boundary control have been studied in a great number of publications such as [21]. The practical approaches to optimal control of the heat conduction equation are described in publications like [22].
Control problems for parabolic type equations are considered in works [13,14] and [15]. In this work, such control problems are considered for the pseudo-parabolic equation.
Consider the following eigenvalue problem
ddx(k(x)dvk(x)dx)=−λkvk(x),0<x<l, | (1.5) |
with boundary condition
vk(0)=vk(l)=0,0≤x≤l. | (1.6) |
It is well-know that this problem is self-adjoint in L2(Ω) and there exists a sequence of eigenvalues {λk} so that 0<λ1≤λ2≤...≤λk→∞, k→∞. The corresponding eigenfuction vk form a complete orthonormal system {vk}kϵN in L2(Ω) and these function belong to C(ˉΩ), where ˉΩ=Ω∪∂Ω (see, [23,24]).
Definition 2. By the solution of the problem (1.1)–(1.3) we understand the function u(x,t) represented in the form
u(x,t)=l−xlμ(t)−v(x,t), | (2.1) |
where the function v(x,t)∈C2,1x,t(Ω)∩C(ˉΩ), vx∈C(ˉΩ) is the solution to the problem:
vt=∂2∂t∂x(k(x)∂v∂x)+∂∂x(k(x)∂v∂x)+ |
+k′(x)lμ(t)+k′(x)lμ′(t)+l−xlμ′(t), |
with boundary conditions
v(0,t)=0,v(l,t)=0, |
and initial condition
v(x,0)=0. |
Set
βk=(λkak−bk)γk, | (2.2) |
where
ak=l∫0l−xlvk(x)dx,bk=l∫0k′(x)lvk(x)dx, | (2.3) |
and
γk=l∫0vk(x)dx. | (2.4) |
Consequently, we have
v(x,t)=∞∑k=1vk(x)1+λkt∫0e−μk(t−s)(μ′(s)ak+μ′(s)bk+μ(s)bk)ds, | (2.5) |
where ak, bk defined by (2.3) and μk=λk1+λk.
From (2.1) and (2.5) we get the solution of the problem (1.1)–(1.3) (see, [23,25]):
u(x,t)=l−xlμ(t)−∞∑k=1vk(x)1+λkt∫0e−μk(t−s)(μ′(s)ak+μ′(s)bk+μ(s)bk)ds. |
According to condition (1.4) and the solution of the problem (1.1)-(1.3), we may write
θ=l∫0u(x,t)dx=μ(t)l∫0l−xldx−∞∑k=111+λk(t∫0e−μk(t−s)(μ′(s)ak+μ′(s)bk+μ(s)bk)ds)l∫0vk(x)dx=μ(t)l∫0l−xldx−∞∑k=1bkγk1+λkt∫0e−μk(t−s)μ(s)ds−∞∑k=1(ak+bk)γk1+λkt∫0e−μk(t−s)μ′(s)ds=μ(t)l∫0l−xldx−∞∑k=1bkγk1+λkt∫0e−μk(t−s)μ(s)ds−μ(t)∞∑k=1(ak+bk)γk1+λk+∞∑k=1(ak+bk)λkγk(1+λk)2t∫0e−μk(t−s)μ(s)ds. | (2.6) |
where γk defined by (2.4).
Note that
l∫0l−xldx=l∫0(∞∑k=1akvk(x))dx=∞∑k=1akγk. | (2.7) |
Thus, from (2.6) and (2.7) we get
θ(t)=μ(t)∞∑k=1βk1+λk+∞∑k=1βk(1+λk)2t∫0e−μk(t−s)μ(s)ds,t>0, | (2.8) |
where βk defined by (2.2).
Set
B(t)=∞∑k=1βk(1+λk)2e−μkt,t>0, | (2.9) |
and
δ=∞∑k=1βk1+λk. |
According to (2.8) and (2.9), we have the following integral equation
δμ(t)+t∫0B(t−s)μ(s)ds=θ(t),t>0. | (2.10) |
Proposition 1. For the cofficients {βk}∞k=1 the estimate
0≤βk≤C,k=1,2,... |
is valid.
Proof. Step 1. Now we use (1.5) and (2.3). Then consider the following equality
λkak=l∫0l−xlλkvk(x)dx=−l∫0l−xlddx(k(x)dvk(x)dx)dx |
=−(l−xlk(x)v′k(x)|x=lx=0+1ll∫0k(x)v′k(x)dx)=k(0)v′k(0)−1ll∫0k(x)v′k(x)dx |
=k(0)v′k(0)−1l(k(l)vk(l)−k(0)vk(0))+l∫0k′(x)lvk(x)dx |
=k(0)v′k(0)+bk. |
Then we have
λkak−bk=k(0)v′k(0). | (2.11) |
Step 2. Now we integrate the Eq. (1.5) from 0 to x
k(x)v′k(x)−k(0)v′k(0)=−λkx∫0vk(τ)dτ, |
and according to k(x)>0, x∈[0,l], we can write
v′k(x)−1k(x)k(0)v′k(0)=−λkk(x)x∫0vk(τ)dτ. | (2.12) |
Thus, we integrate the Eq. (2.12) from 0 to l. Then we have
vk(l)−vk(0)−k(0)v′k(0)l∫0dxk(x)=−λkl∫01k(x)(x∫0vk(τ)dτ)dx. | (2.13) |
From (1.6) and (2.13) we get
k(0)v′k(0)l∫0dxk(x)=λkl∫01k(x)(x∫0vk(τ)dτ)dx. |
Then
k(0)v′k(0)=λkl∫0G(τ)vk(τ)dτ, | (2.14) |
where
G(τ)=l∫τdxk(x)(l∫0dxk(x))−1. |
According to G(τ)>0 and from (2.14) we have (see, [24])
v′k(0)l∫0vk(τ)dτ≥0. | (2.15) |
Consequently, from (2.11) and (2.15) we get the following estimate
βk=(λkbk−ak)γk=k(0)v′k(0)⋅l∫0vk(x)dx≥0. |
Step 3. It is clear that if k(x)∈C1([0,l]), we may write the estimate (see, [24,26])
max0≤x≤l|v′k(x)|≤Cλ1/2k. |
Therefore,
|v′k(0)|≤Cλ1/2k,|v′k(l)|≤Cλ1/2k, | (2.16) |
Then from Eq. (1.5), we can write
k(l)v′k(l)−k(0)v′k(0)=−λkl∫0vk(x)dx=−λkγk. | (2.17) |
According to (2.16) and (2.17) we have the estimate
|γk|≤|1λk(k(l)v′k(l)−k(0)v′k(0))|≤Cλ−1/2k. |
Then
βk≤k(0)|v′k(0)γk|≤C. |
Proposition 2. A function B(t) is continuous on the half-line t≥0.
Proof. Indeed, according to Proposition 1 and (2.9), we can write
0<B(t)≤const∞∑k=11(1+λk)2. |
Denote by W(M) the set of function θ∈W22(−∞,+∞), θ(t)=0 for t≤0 which satisfies the condition
‖θ‖W22(R+)≤M. |
Theorem 1. There exists M>0 such that for any function θ∈W(M) the solution μ(t) of the equation (2.10) exists, and satisfies condition
|μ(t)| ≤ 1. |
We write integral equation (2.10)
δμ(t)+t∫0B(t−s)μ(s)ds=θ(t),t>0. |
By definition of the Laplace transform we have
˜μ(p)=∞∫0e−ptμ(t)dt. |
Applying the Laplace transform to the second kind Volterra integral equation (2.10) and taking into account the properties of the transform convolution we get
˜θ(p)=δ˜μ(p)+˜B(p)˜μ(p). |
Consequently, we obtain
˜μ(p)=˜θ(p)δ+˜B(p),where p=a+iξ,a>0, |
and
μ(t)=12πia+iξ∫a−iξ˜θ(p)δ+˜B(p)eptdp=12π+∞∫−∞˜θ(a+iξ)δ+˜B(a+iξ)e(a+iξ)tdξ. | (3.1) |
Then we can write
˜B(p)=∞∫0B(t)e−ptdt=∞∑k=1βk(1+λk)2∞∫0e−(p+μk)tdt=∞∑k=1ρkp+μk, |
where ρk=βk(1+λk)2≥0 and
˜B(a+iξ)=∞∑k=1ρka+μk+iξ=∞∑k=1ρk(a+μk)(a+μk)2+ξ2−iξ∞∑k=1ρk(a+μk)2+ξ2. |
It is clear that
(a+μk)2+ξ2≤[(a+μk)2+1](1+ξ2), |
and we have the inequality
1(a+μk)2+ξ2≥11+ξ21(a+μk)2+1. | (3.2) |
Consequently, according to (3.2) we can obtain the estimates
|Re(δ+˜B(a+iξ))|=δ+∞∑k=1ρk(a+μk)(a+μk)2+ξ2≥11+ξ2∞∑k=1ρk(a+μk)(a+μk)2+1=C1a1+ξ2, | (3.3) |
and
|Im(δ+˜B(a+iξ))|=|ξ|∞∑k=1ρk(a+μk)2+ξ2≥|ξ|1+ξ2∞∑k=1ρk(a+μk)2+1=C2a|ξ|1+ξ2, | (3.4) |
where C1a, C2a as follows
C1a=∞∑k=1ρk(a+μk)(a+μk)2+1, C2a=∞∑k=1ρk(a+μk)2+1. |
From (3.3) and (3.4), we have the estimate
|δ+˜B(a+iξ)|2=|Re(δ+˜B(a+iξ))|2+|Im(δ+˜B(a+iξ))|2≥min(C21a,C22a)1+ξ2, |
and
|δ+˜B(a+iξ)|≥Ca√1+ξ2,whereCa=min(C1a,C2a). | (3.5) |
Then, by passing to the limit at a→0 from (3.1), we can obtain the equality
μ(t)=12π+∞∫−∞˜θ(iξ)δ+˜B(iξ)eiξtdξ. | (3.6) |
Lemma 1. Let θ(t)∈W(M). Then for the image of the function θ(t) the following inequality
+∞∫−∞|˜θ(iξ)|√1+ξ2dξ≤C‖θ‖W22(R+), |
is valid.
Proof. We use integration by parts in the integral representing the image of the given function θ(t)
˜θ(a+iξ)=∞∫0e−(a+iξ)tθ(t)dt=−θ(t)e−(a+iξ)ta+iξ|t=∞t=0+1a+iξ∞∫0e−(a+iξ)tθ′(t)dt. |
Then using the obtained inequality and multiplying by the corresponding coefficient we get
(a+iξ)˜θ(a+iξ)=∞∫0e−(a+iξ)tθ′(t)dt, |
and for a→0 we have
iξ˜θ(iξ)=∞∫0e−iξtθ′(t)dt. |
Also, we can write the following equality
(iξ)2˜θ(iξ)=∞∫0e−iξtθ″(t)dt. |
Then we have
+∞∫−∞|˜θ(iξ)|2(1+ξ2)2dξ≤C1‖θ‖2W22(R+). | (3.7) |
Consequently, according to (3.7) we get the following estimate
+∞∫−∞|˜θ(iξ)|√1+ξ2dξ=+∞∫−∞|˜θ(iξ)|(1+ξ2)√1+ξ2 |
≤(+∞∫−∞|˜θ(iξ)|2(1+ξ2)2dξ)1/2(+∞∫−∞11+ξ2dξ)1/2≤C‖θ‖W22(R+). |
Proof of the Theorem 1. We prove that μ∈W12(R+). Indeed, according to (3.5) and (3.6), we obtain
+∞∫−∞|˜μ(ξ)|2(1+|ξ|2)dξ = +∞∫−∞|˜θ(iξ)δ+˜B(iξ)|2(1+|ξ|2)dξ |
≤C+∞∫−∞|˜θ(iξ)|2(1+|ξ|2)2dξ = C‖θ‖2W22(R). |
Further,
|μ(t)−μ(s)| = |t∫sμ′(τ)dτ| ≤ ‖μ′‖L2√t−s. |
Hence, μ∈Lipα, where α=1/2. Then from (3.5), (3.6) and (3.7), we have
|μ(t)|≤12π+∞∫−∞|˜θ(iξ)||δ+˜B(iξ)|dξ≤12πC0+∞∫−∞|˜θ(iξ)|√1+ξ2dξ |
≤C2πC0‖θ‖W22(R+)≤CM2πC0=1, |
as M we took
M=2πC0C. |
An auxiliary boundary value problem for the pseudo-parabolic equation was considered. The restriction for the admissible control is given in the integral form. By the separation variables method, the desired problem was reduced to Volterra's integral equation. The last equation was solved by the Laplace transform method. Theorem on the existence of an admissible control is proved. Later, it is also interesting to consider this problem in the n− dimensional domain. We assume that the methods used in the present problem can also be used in the n− dimensional domain.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The author declare there is no conflict of interest.
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