Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water

Abstract

In this paper, we investigate the inverse of the set of unknown functions $(v,g)$ of the Burgers equation in the framework of optimal theory. Firstly, we prove the existence of functional minimizers in the optimal control problem and derive the necessary conditions for the optimal solution. Subsequently, the global uniqueness of the optimal solution and its stability are explored. After completing the ill-posed analysis of the Burgers equation, we can apply it to the problem of sonic vibration velocity in water. The desired result is obtained by inverse-performing an unknown initial state with known terminal vibration velocity. This is important for solving practical problems.

1. Introduction

Partial differential Equations [1,2] have a long development history and are widely used in physics and other fields. Partial differential equations are used to model various natural phenomena, such as heat transfer processes [3,4,5,6] and fluid mechanics [7]. In real life, the partial differential equations can be used to model both direct and inverse problems. Through the specified partial differential equations and initial-boundary value conditions, we can ascertain the solution to the problem or reveal certain properties of the solution. This is the direct problem. The direct problem is deducing results from causes, and it is relatively well developed in terms of both theory and application. However, in practice, due to the existence of certain limitations, such as prohibitively high costs associated with measuring specific data and technical constraints that impede the implementation of certain methodologies, some information in the equation becomes difficult to obtain through direct measurement. Consequently, acquiring this information through indirect means becomes necessary, which is the inverse problem [8,9]. An inverse problem is a problem in which an unknown parameter or function [10,11] in a partial differential equation is determined from known boundary or initial conditions and partial observations. This type of problem usually involves reconstructing a system’s internal structure or properties from limited information, and research on it is not yet very advanced. However, many inverse problems of partial differential equations have appeared in many fields, such as aerospace engineering [12], resource exploration [13], geophysics [14], and elastic mechanics [15], which makes the study of inverse problems of partial differential equations increasingly important.

1.1. Inverse-Problem Research Descriptions

In 1923, the French mathematician Hadamard first proposed the definition of ‘well-posed problem [16,17]’: a problem is said to be well posed if it satisfies the following three conditions: (1) existence, i.e., there is a solution to the problem; (2) uniqueness, i.e., the solution to the problem is unique; and (3) stability, i.e., the solution and the data dependency relationship are continuous, and the problem is said to be ill posed if it fails to satisfy any of these conditions. Most inverse problems in many fields are ill posed [18,19]. The solutions to these problems are often non-existent, non-unique, or unstable, which renders the study of inverse problems particularly challenging. The key difficulty in resolving the inverse problem is its ill-posedness. The problem of non-uniqueness and non-existence of the solution of the inverse problem can be solved by adding some additional conditions to the solution or adjusting the definition range of the solution, but the instability of the solution is very difficult to obtain by general methods. To overcome this difficulty, A.N. Tikhonov, a Soviet scholar, proposed a new method for solving ill-posed problems in the last century: Tikhonov’s regularization method. Its main idea is to utilize the a priori estimates of the solution and data errors to confine the solution of the problem within a relatively small range. Then, the solution of the original problem is approximately obtained by using a set of problems similar to the original one. It transforms the ill-posed problem into a well-posed problem [20,21,22,23,24,25,26]. The prior estimate obtained beforehand also guarantees that the solution obtained utilizing this method is reasonable and accurate. However, this method has its limitations. For example, we can only use the Tikhonov regularization method to solve linear inverse problems, and it is less effective in solving nonlinear and large-scale problems. Commonly used regularization methods include the mollification regularization method [27], the Landweber iterative regularization method [28], the fractional order Landweber iterative regularization method [29], and the proposed inverse regularization method [30]. These methods can effectively solve problems of practical significance and, simultaneously, the study of such problems can be applied to other disciplines and various engineering fields. Therefore, it has an important research value for practical production and life.

1.2. Preliminary Knowledge

Definition 1

($L^P$ space). We suppose that Ω is a bounded open set in n-dimensional space ${\mathbb{R}}^n$, and let ${\int }_{\Omega }f\left(x\right)dx$ be the Lebesgue integral of the function $f\left(x\right)$, for $1\le p\le +\infty$, denoted by the norm

$${∥f\left(x\right)∥}_{L^p\left(\Omega \right)}={\left({\int }_{\Omega }{\left|f\left(x\right)\right|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}}$$

(1)

then, $L^p\left(\Omega \right)$ space can be defined as

$$L^p\left(\Omega \right)=\left\{\left.f\left(x\right)\right|{∥f\left(x\right)∥}_{L^p\left(\Omega \right)}<+\infty \right\}$$

(2)

For $p=+\infty$, we define the norm

$${∥f\left(x\right)∥}_{\infty ,\Omega }=ess\underset{x\in \Omega }{sup}\left|f\left(x\right)\right|$$

(3)

The $L^{\infty }\left(\Omega \right)$ space can be defined as

$$L^{\infty }\left(\Omega \right)=\left\{\left.f\left(x\right)\right|{∥f\left(x\right)∥}_{\infty ,\Omega }<+\infty \right\}$$

(4)

Lemma 1

(Sobolev Embedding Theorem). We suppose $s>{\displaystyle \frac{n}{2}}$; then, we have the continuous linear embedding

$$H^s\left({\mathbb{R}}^n\right)\hookrightarrow L^{\infty }\left({\mathbb{R}}^n\right)$$

(5)

i.e., for any $u\in H^s\left({\mathbb{R}}^n\right)$, there exists a positive $C_s$, such that the following inequality is held:

$${∥u∥}_{L^{\infty }}\le C_s{∥u∥}_{H^s}$$

(6)

Lemma 2

(Hölder inequality). If $1<p,q<+\infty$, ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, then, for any $f\in L^p\left(\Omega \right)$, $g\in L^q\left(\Omega \right)$, there is $fg\in L^1\left(\Omega \right)$, and there are

$${∥fg∥}_{L^1\left(\Omega \right)}\le {∥f∥}_{L^p\left(\Omega \right)}{∥g∥}_{L^q\left(\Omega \right)}$$

(7)

1.3. Main Work in This Paper

The Burgers equation has applications in many fields of mathematics, such as simulating the propagation of acoustic waves in water in fluid dynamics [31,32,33,34]. In this article, we consider the inverse problems of unknown functions in the Burgers equation. This type of inverse problem is usually ill posed, and traditional methods may not be able to directly obtain stable solutions. In this paper, we solve the Burgers equation using optimal control.

Optimal control [35,36] provides a systematic approach to ill-posed problems. By constructing appropriate objective functions and applying optimization algorithms, we can approximate the potential solution step by step. In this paper, we not only simply utilize the optimal theory to invert the unknown function, but also analyze the characteristics of the optimal solution. We rigorously prove the existence and uniqueness of the optimal solution under the given conditions. This means that our method can find a globally optimal solution under certain assumptions and constraints. The problem is described as follows.

Problem P: In the Burgers equation, the first term on the left-hand side, represented by the variable $\partial v/\partial x$, denotes the rate of change of velocity concerning time. The second term, $v·(\partial v/\partial t)$, represents the convection term, which describes the convection effect caused by the non-uniformity of the fluid’s velocity. This is to say that it reflects the change in the physical quantity (velocity) carried by the fluid parcels during the flow process. The term $C_2·{\partial }^{2}v/\partial t^2$ on the right side is the viscous diffusion term, which reflects the velocity diffusion due to the viscosity of the fluid. We consider the inverse problem of the Burgers equation under fluid dynamics.

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial v}{\partial x}}-C_{1}v{\displaystyle \frac{\partial v}{\partial t}}=C_2·{\displaystyle \frac{{\partial }^{2}v}{\partial t^2}},\hfill & (x,t)\in Q\hfill \\ v\left(0,t\right)=g\left(t\right),\hfill & t\in \left[0,T\right]\hfill \\ v\left(x,0\right)=0,\hfill & x\in \left[0,l\right]\hfill \end{array}\right.$$

(8)

In Equation (8), $v\left(x,t\right)$ is for the sonic vibration velocity in water, $Q=\left[0,l\right]\times \left[0,T\right]$, $C_1$ and $C_2$ are constant terms, and $g\left(t\right)$ represents the sonic vibration velocity under the initial condition and is unknown. We suppose the terminal vibration velocity is

$$v\left(l,t\right)=f\left(t\right)$$

(9)

where $f\left(t\right)$ is a known function and satisfies the homogeneous Dirichlet boundary conditions; it is necessary to invert the unknown function groups $\left(v,g\right)$ in the equation.

2. Optimal Control

Given the ill-posedness of problem P, we consider the following optimal control problem P1:

P1: To find $\overline{g}\in A$, $J\left(\overline{g}\right)$ must satisfy

$$J\left(\overline{g}\right)=\underset{g\left(t\right)\in A}{min}J\left(g\right)$$

(10)

$$J\left(g\right)={\displaystyle \frac{1}{2}}{{\int }_0^T\left|v\left(l,t;g\right)-f\left(t\right)\right|}^{2}dt+{\displaystyle \frac{N}{2}}{{\int }_0^T\left|\nabla g\right|}^{2}dt$$

(11)

$$A=\left\{\left.g\left(t\right)\right|g\left(t\right)\le M,g\left(t\right)\in H^1\left(0,T\right)\right\}$$

(12)

where M is a positive constant, and $v\left(l,t,g\right)$ is the solution of the given source term $g\left(t\right)$. The first term of $J\left(g\right)$ is a measure of the error between the model output $v\left(l,t,g\right)$ and the known objective function $f\left(t\right)$, and the second term is a regularization term that controls the smoothness of the function $g\left(t\right)$. N is a regularization parameter; we assume that $f\in L^2\left(0,T\right)$.

Theorem 1.

In the optimal control problem P1, there exists $\overline{g}\in A$, which is the minimal element of $J\left(\overline{g}\right)$, i.e.,

$$J\left(\overline{g}\right)=\underset{g\left(t\right)\in A}{min}J\left(g\right)$$

(13)

Proof of Theorem 1.

The function $J\left(g\right)$ is non-negative on A. Therefore, there exists a lower bound ${\displaystyle \underset{g\in A}{inf}}J\left(g\right)$. Let $\left\{g_n\right\}$ be a minimizing sequence that satisfies

$$\underset{g\in A}{inf}\mathrm{J}\left(g\right)\le \mathrm{J}\left(g_n\right)\le \underset{g\in A}{inf}J\left(g\right)+{\displaystyle \frac{1}{n}},n=1,2,···$$

(14)

Furthermore, given that $J\left(g_n\right)\le K$, where K is a constant, it follows that

$${∥\nabla g_n∥}_{L^2\left(0,T\right)}\le K$$

(15)

From the boundedness of the sequence $\left\{g_n\right\}$, we obtain

$${∥g_n∥}_{L^2\left(0,T\right)}\le K$$

(16)

Based on the Compactness Theorem [37], a subsequence (denoted as $\left\{g_n\right\}$) can be extracted from $\left\{g_n\right\}$, giving us

$$g_n\left(x\right)\to \overline{g}\left(x\right)\in H^1\left(0,T\right),n\to \infty$$

(17)

According to Lemma 1, we find

$${∥g_n\left(x\right)-\overline{g}\left(x\right)∥}_{L^2\left(0,T\right)}\to 0,n\to \infty$$

(18)

It is easy to see that $\left\{g_n\left(x\right)\right\}\in A$, when $n\to \infty$; from the holder inequality and the above equation, it follows that

$${{\int }_0^T\left|\nabla g\right|}^{2}dt=\underset{n\to \infty }{lim}{\int }_0^T\nabla g_n·\nabla \overline{g}dt\le \underset{n\to \infty }{lim}\sqrt{{{\int }_0^T\left|\nabla g_n\right|}^{2}dt·{{\int }_0^T\left|\nabla \overline{g}\right|}^{2}dt}$$

(19)

Furthermore, ${\displaystyle \underset{n\to \infty }{lim}}{\int }_0^T{\left|v\left(l,t;g_n\right)-f\left(t\right)\right|}^{2}dt={\int }_0^T{\left|v\left(l,t;\overline{g}\right)-f\left(t\right)\right|}^{2}dt$ and the above equation lead to

$$\begin{array}{c}J\left(\overline{g}\right)={\displaystyle \underset{n\to \infty }{lim}}{\int }_0^T{\left|v\left(l,t;g_n\right)-f\left(t\right)\right|}^{2}dt+{{\int }_0^T\left|\nabla g\right|}^{2}dt\hfill \\ \le {\displaystyle \underset{n\to \infty }{lim}}J\left(g_n\right)={\displaystyle \underset{g\in A}{inf}}J\left(g\right)\hfill \end{array}$$

(20)

We can conclude that $J\left(\overline{g}\right)={\displaystyle \underset{g\left(t\right)\in A}{min}}J\left(g\right)$. □

3. Necessary Conditions

Theorem 2.

If g is a solution to the optimal control $J\left(\overline{g}\right)={\displaystyle \underset{g\left(t\right)\in A}{min}}J\left(g\right)$, then there exists a set of ternary functions $\left(v,\xi ,g\right)$ that satisfy the following equations and inequality for $\forall h\in A$:

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial v}{\partial x}}-C_{1}v{\displaystyle \frac{\partial v}{\partial t}}=C_2·{\displaystyle \frac{{\partial }^{2}v}{\partial t^2}},\hfill & (x,t)\in Q\hfill \\ v\left(0,t\right)=g\left(t\right),\hfill & t\in \left[0,T\right]\hfill \\ v\left(x,0\right)=0,\hfill & x\in \left[0,l\right]\hfill \end{array}\right.$$

(21)

$$\left\{\begin{array}{cc}{\xi }_x-C_1\xi {\xi }_t=0,\hfill & (x,t)\in Q\hfill \\ \xi \left(0,t\right)=h\left(t\right)-g\left(t\right),\hfill & t\in \left[0,T\right]\hfill \\ \xi \left(x,0\right)=0,\hfill & x\in \left[0,l\right]\hfill \end{array}\right.$$

(22)

$${\int }_0^T\left[v\left(l,t,g\right)-f\left(t\right)\right]\xi \left(l,t\right)dt+N{\int }_0^T\nabla g·\nabla \left(h-g\right)dt\ge 0$$

(23)

Proof of Theorem 2.

For $\forall h\in A$, we define a new control function $g_{\delta }$ through the parameter $\delta$, and its form is

$$g_{\delta }\equiv \left(1-\delta \right)g+\delta h$$

(24)

where $\delta \in \left[0,1\right]$, and $g_{\delta }\in A$; then, we have

$$J_{\delta }=J\left(g_{\delta }\right)={\displaystyle \frac{1}{2}}{{\int }_0^T\left|v\left(l,t;g_{\delta }\right)-f\left(t\right)\right|}^{2}dt+{\displaystyle \frac{N}{2}}{{\int }_0^T\left|\nabla g_{\delta }\right|}^{2}dt$$

(25)

Let $g=g_{\delta }$ and $v_{\delta }$ be the solution of the following system:

$$\left\{\begin{array}{cc}{v}_{x,\delta }-C_1{v}_{\delta }{v}_{\delta ,t}=C_2·{\left(v_{\delta ,t}\right)}_t,\hfill & (x,t)\in Q\hfill \\ v_{\delta }\left(0,t\right)=g\left(t\right),\hfill & t\in \left[0,T\right]\hfill \\ v_{\delta }\left(x,0\right)=0,\hfill & x\in \left[0,l\right]\hfill \end{array}\right.$$

(26)

Since g is the optimal solution to the control problem, using the chain rule and the calculus of variations yields

$${\left.{\displaystyle \frac{d{J}_{\delta }}{d\delta }}\right|}_{\delta =0}={\int }_0^T\left[\left[v\left(l,t;g\right)-f\left(t\right)\right]{\left.{\displaystyle \frac{\partial v_{\delta }}{\partial \delta }}\right|}_{\delta =0}\right]dt+N{\int }_0^T\nabla g·\nabla \left(h-g\right)dt\ge 0$$

(27)

Let ${\tilde{v}}_{\delta }={\displaystyle \frac{\partial v_{\delta }}{\partial \delta }}$; then, from the above equation, we have

$$\left\{\begin{array}{cc}{\displaystyle \frac{\partial }{\partial x}}\left({\tilde{v}}_{\delta }\right)-C_1{\tilde{v}}_{\delta }{\displaystyle \frac{\partial }{\partial t}}\left({\tilde{v}}_{\delta }\right)=C_2·{\displaystyle \frac{{\partial }^2}{\partial t^2}}\left({\tilde{v}}_{\delta }\right),\hfill & (x,t)\in Q\hfill \\ {\tilde{v}}_{\delta }\left(0,t\right)=h-g,\hfill & t\in \left[0,T\right]\hfill \\ {\tilde{v}}_{\delta }\left(x,0\right)=0,\hfill & x\in \left[0,l\right]\hfill \end{array}\right.$$

(28)

Let $\xi ={\left.{\tilde{v}}_{\delta }\right|}_{\delta =0}$; then, $\xi$ satisfies

$$\left\{\begin{array}{cc}{\xi }_x-C_1\xi ·{\xi }_t=C_2·{\xi }_{tt},\hfill & (x,t)\in Q\hfill \\ \xi \left(0,t\right)=h\left(t\right)-g\left(t\right),\hfill & t\in \left[0,T\right]\hfill \\ \xi \left(x,0\right)=0,\hfill & x\in \left[0,l\right]\hfill \end{array}\right.$$

(29)

Combining the above equation with Equation (25) yields

$${\int }_0^T\left[v\left(l,t,g\right)-f\left(t\right)\right]\xi \left(l,t\right)dt+N{\int }_0^T\nabla g·\nabla \left(h-g\right)dt\ge 0$$

(30)

This shows that g is the optimal control, which holds for $\forall h\in A$. In the case of optimal control, the set of ternary functions $\left(v,\xi ,g\right)$ satisfying the necessary conditions is determined to exist. □

4. Stability and Uniqueness

Theorem 3.

We suppose $f_1\left(t\right)$ and $f_2\left(t\right)$ are two functions satisfying the following equations, such that $g_1\left(t\right)$ and $g_2\left(t\right)$ are solutions to the optimal control problem P corresponding to $f_1\left(t\right)$ and $f_2\left(t\right)$, respectively, and, if there exists a point $t_0\in \left(0,T\right)$ such that $g_1\left(t_0\right)=g_2\left(t_0\right)$, we assume that $f_i,g_i\in L^2\left(0,T\right)$$(i=1,2)$; then, the following estimate holds:

$$\underset{t\in \left(0,T\right)}{max}\left|g_1-g_2\right|\le C{∥f_1-f_2∥}_{L^2\left(0,T\right)}$$

(31)

where C depends on T and N.

Proof of Theorem 3.

If g takes the value of $g_1$, we select the value of h to be $g_2$, and if g takes the value of $g_2$, we select the value of h to be $g_1$. From Equation (30) it follows that

$${\int }_0^T\left[v_1\left(l,·\right)-f_1\left(t\right)\right]{\xi }_1\left(l,·\right)dt+N{\int }_0^T\nabla g_1·\nabla \left(g_2-g_1\right)dt\ge 0$$

(32)

$${\int }_0^T\left[v_2\left(l,·\right)-f_2\left(t\right)\right]{\xi }_2\left(l,·\right)dt+N{\int }_0^T\nabla g_2·\nabla \left(g_1-g_2\right)dt\ge 0$$

(33)

where $\left\{v_i,{\xi }_i\right\}\left(i=1,2\right)$ is the solution to the equation in Theorem 2 when $g=g_i\left(i=1,2\right)$.

If $v_1-v_2=V,{\xi }_1+{\xi }_2=E$, then V and E satisfy the following equation:

$$\left\{\begin{array}{cc}{E}_x-C_{1}E·E_t=C_2·E_{tt}, \hfill & (x,t)\in Q\hfill \\ E\left(0,t\right)=0,\hfill & t\in \left[0,T\right]\hfill \\ E\left(x,0\right)=0,\hfill & x\in \left[0,l\right]\hfill \end{array}\right.$$

(34)

We perform the Cole–Hopf transformation on the above equation. Then, according to the extreme value theorem, we know that equation E has only the zero solution. Then, ${\xi }_1\left(x,t\right)=-{\xi }_2\left(x,t\right)$, and ${\xi }_1\left(x,t\right)$ satisfies

$$\left\{\begin{array}{cc}{\xi }_{1x}-C_1{\xi }_1·{\xi }_{1t}=C_2·{\xi }_{1tt},\hfill & (x,t)\in Q\hfill \\ {\xi }_1\left(0,t\right)=g_2\left(t\right)-g_1\left(t\right),\hfill & t\in \left[0,T\right]\hfill \\ {\xi }_1\left(x,0\right)=0,\hfill & x\in \left[0,l\right]\hfill \end{array}\right.$$

(35)

which means that $V\left(x,t\right)=-{\xi }_1\left(x,t\right)$, and

$$\begin{array}{c}{∥g_1-g_2∥}_{L^2\left(0,T\right)}={{\int }_0^T\left|\nabla \left(g_1-g_2\right)\right|}^{2}dt\hfill \\ \le N{\int }_0^T\left[v_1\left(l,·\right)-f_1\left(t\right)\right]{\xi }_1\left(l,·\right)dt+N{\int }_0^T\left[v_2\left(l,·\right)-f_2\left(t\right)\right]{\xi }_2\left(l,·\right)dt\hfill \\ \le N{\int }_0^T\left[V\left(l,·\right)-{\xi }_1\left(l,·\right)\right]dt+N{\int }_0^T\left[f_2\left(t\right)-f_1\left(t\right)\right]{\xi }_1\left(l,·\right)dt\hfill \\ \le -N{\int }_0^T{\left|{\xi }_1\left(l,·\right)\right|}^{2}dt+{\displaystyle \frac{N}{2}}{\int }_0^T{\left|{\xi }_1\left(l,·\right)\right|}^{2}dt+{\displaystyle \frac{N}{2}}{\int }_0^T{\left|f_1-f_2\right|}^{2}dt\hfill \\ \le -{\displaystyle \frac{N}{2}}{\int }_0^T{\left|{\xi }_1\left(l,·\right)\right|}^{2}dt+{\displaystyle \frac{N}{2}}{\int }_0^T{\left|f_1-f_2\right|}^{2}dt={\displaystyle \frac{N}{2}}{∥f_1-f_2∥}_{L^2\left(0,T\right)}^2\hfill \end{array}$$

(36)

Then, by the assumptions of Theorem 3 and Lemma 2, when $0<t<T$, we have

$$\begin{array}{c}\left|\left(g_1-g_2\right)\left(x\right)\right|\le \left|\left(g_1-g_2\right)\left(x_0\right)\right|+\left|{\int }_0^T\left(g_1-g_2\right)dt\right|\hfill \\ \le {\left({\int }_0^{T}dt\right)}^2·{\left({\int }_0^T{\left|\nabla \left(g_1-g_2\right)\right|}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \le \sqrt{T}{\left({\int }_0^T{\left|\nabla \left(g_1-g_2\right)\right|}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\hfill \end{array}$$

(37)

It can be further obtained that

$$\underset{t\in \left(0,T\right)}{max}\left|g_1-g_2\right|\le \sqrt{T}{∥\left(g_1-g_2\right)∥}_{L^2\left(0,T\right)}$$

(38)

Combining Equation (37) and Equation (38), we can obtain

$$\begin{array}{c}{\displaystyle \underset{t\in \left(0,T\right)}{max}}\left|g_1-g_2\right|\le \sqrt{{\displaystyle \frac{T}{2N}}}{∥\left(f_1-f_2\right)∥}_{L^2\left(0,T\right)}\hfill \\ \le C{∥\left(f_1-f_2\right)∥}_{L^2\left(0,T\right)}\hfill \end{array}$$

(39)

Thus, Theorem 3 is proved, indicating that the equation is stable and has a unique solution. □

5. Practical Application

When studying the transmission properties of sonic vibration in water, we often express the wave properties in terms of the Burgers equation. When a finite amplitude sinusoidal plane sonic vibration with frequency $\omega$ propagates in a nonlinear medium, if the Reynolds number Re is larger, the waveform will be distorted, and high harmonics will be generated, which will eventually form a shock wave. The Burgers equation is used to describe this sonic vibration transmission characteristic:

$${\displaystyle \frac{\partial v}{\partial x}}-{\displaystyle \frac{\beta }{c_0^2}}v{\displaystyle \frac{\partial v}{\partial \tau }}={\displaystyle \frac{b}{2c_0^2{\rho }_0}}·{\displaystyle \frac{{\partial }^{2}v}{\partial {\tau }^2}}$$

(40)

In Equation (40), v represents the sonic vibration velocity in water. $\beta =1+{\displaystyle \frac{B}{2A}}$, where $B/A$ denotes the nonlinear parameter of the medium, and $B/A$ is the ratio of the quadratic coefficient to the linear coefficient in the Taylor series expansion of the equation of state, which can reflect the dynamic characteristics of the medium. Let $c_0$ be the static speed of sound. b represents the viscous coefficient of the aqueous medium. ${\rho }_0$ is the density of the aqueous medium. In addition, $\tau =t-x/c_0$ is the time delay, and x is the measurement distance.

We assume that if, at $x=0$, two sources emit sonic vibration with vibration velocities $v_1=g_1\left(\tau \right)$ and $v_2=g_2\left(\tau \right)$, and $g_1\left(\tau \right),g_2\left(\tau \right)$ is unknown, then the form of the initial vibration velocity at $x=0$ is

$${\left.v\left(x,\tau \right)\right|}_{x=0}=g_1\left(\tau \right)+g_2\left(\tau \right)$$

(41)

At the initial moment of vibration velocity $v\left(x,0\right)=0$, we now know that the sonic vibration velocity obtained at $x=l$ is

$${\left.v\left(x,\tau \right)\right|}_{x=l}=f\left(\tau \right)$$

(42)

The ill-posedness of the Burgers equation has been demonstrated above, and we can directly solve the inverse problem to obtain the initial end sonic vibration state.

6. Conclusions

In this paper, we explored the inverse problem of the Burgers equation ill-posedness analysis and its application in fluid dynamics, especially the importance of the sonic vibration propagation phenomenon. The Burgers equation, as a classical equation describing the one-dimensional nonlinear fluctuations, is effective in explaining the nonlinear effects of the sonic vibration propagation in a medium, such as the waveform distortion with the generation of high-order harmonics. We formulate problem P regarding how to invert the set of unknown functions in the equation, especially the solution of the initial state, under the condition of given terminal measurements. By introducing optimal control theory, we construct the optimization problem P1 to minimize the objective function $J\left(g\right)$. This function ensures the stability and uniqueness of the solution by introducing a regularization term. To prove the existence and uniqueness of the optimal solution, we use necessary conditions and stability theorems to ensure the existence and continuous dependence between different inputs under specific conditions. This process is an effective way to overcome the challenges of the inverse problem.

In practical applications, the Burgers equation is particularly important for describing the properties of sonic vibration propagating through water. When a sonic vibration with frequency $\omega$ propagates in a nonlinear medium with a large acoustic Reynolds number Re, its waveform is significantly distorted, leading to the formation of shock waves. Through the combination of the analysis of experimental data and optimal control, we can determine the initial vibration velocity state of the sound source, thus providing theoretical support and a practical basis for the study of sonic vibration propagation.

Overall, this paper not only provides a new perspective on nonlinear fluctuations in fluid dynamics through an in-depth discussion of the inverse problem of the Burgers equation but also provides a powerful mathematical tool for solving complex problems encountered in practical engineering. The results of this study have a wide range of potential applications in acoustics, geophysics, and other areas of applied mathematics, and lay a solid foundation for further research in the future.