Maximal and minimal iterative positive solutions for $p$-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term

1.   Introduction

The optimal control problems (OCPs) of partial differential equations have been extensively studied in numerous fields of science and engineering applications, including fluid mechanics, earth science, petroleum engineering, telecommunication, etc., (see ^[1,2,3]). In the past few decades, scholars have conducted extensive research on OCPs (see ^[4,5,6]). From the perspective of control types, there is distributed control and boundary control (see ^[7,8,9]). With respect to the types of state equations, there are elliptic equations, parabolic equations, and second-order hyperbolic equations (SOHEs) ^[10]. Among the common numerical discretization schemes, there are finite element methods (FEM) ^[11], mixed element methods (MFVM) ^[12], and finite volume methods (FVM)^[13]. In terms of handling optimization problems, there are two approaches: optimize-then-discretize and discretize-then-optimize ^[14].

The OCPs constrained by SOHEs is an active area of research, attracting significant attention from numerous scholars. Gugat et al. in ^[15] proposed a valid method based on Lavrentiev regularization and obtained a result similar to the penalty function method for second-order hyperbolic optimal control problems (SOHOCPs) with state constraint. Kröner in ^[16] used the space-time FEM to discretize SOHOCPs and derived a posteriori error estimates, which separately considers the influence of space, time, and control. In ^[17], Kröner et al. analyzed the convergence of three types of controls constrained by wave equations utilizing the semismooth Newton method, and numerically implemented the solution in conjunction with the space-time FEM. Lu et al. derived a priori error estimates using the mixed finite element method for a general SOHOCPs in ^[18]. Luo et al. in ^[19,20] studied linear SOHOCPs using the FVM and obtained priori error estimates for the Euler and Crank-Nicolson schemes. Lu et al. in ^[21] used the FVM to study nonlinear SOHOCPs and obtained optimal error estimates for a semi-discrete system. Li et al. in ^[22] used the FEM and variational discretization approach to investigate linear SOHOCPs by introducing an intermediate variable and obtained optimal priori error estimates.

Inspired by ^[21,22], we consider the following nonlinear SOHOCPs:

such that

where $\Omega\subset R^{2}$ is a bounded convex polygon domain with boundary $\partial \Omega$. $\alpha$ is a positive number. $T > 0$ is a constant. $f, y_{d}\in L^{2}(0, T; L^2(\Omega))$ are given functions. $\phi (y)$ is a nonlinear function that satisfies $\phi(\cdot)\in C^2$. For any $R > 0$ and $y \in H^{1}(\Omega)$, the function $\phi(\cdot)\in W^{2, \infty}(-R, R)$, $\phi'(\cdot)\in L^{2}(\Omega)$, and $\phi'(\cdot)\geq 0$. $A = (a_{ij}(x))_{2\times 2}\in (W^{1, \infty}(\bar{\Omega}))^{2\times 2}$ is symmetric and uniformly positive definite, i.e., for any $X\in R^2$, there exist two positive constants $C_{1}, C_{2}$ such that

$B: L^{2}(0, T; L^{2}(\Omega))\rightarrow L^{2}(0, T; L^{2}(\Omega))$ is a bounded linear operator. The space $U_{ad}$ is defined by

In this work, by introducing a new variable, we transform the hyperbolic equation into two parabolic equations. For the SOHOCPs (1.1) and (1.2), we obtain the continuous first-order necessary condition (FNC) and the second-order sufficient optimality condition (SSC). Using the discretize-then-optimize procedure, we derive the discrete optimality condition for the fully discrete scheme. Based on these, we obtain some optimal error estimates.

The paper is organized as follows. We give some notations in Section 2. In Section 3, we derive the first and second-order optimality conditions. The Crank-Nicolson finite element approximation and a priori error estimates are presented in Section 4. In Section 5, a numerical experiment is presented to confirm the validity of the proposed numerical scheme.

After this, $C$ represents different positive constants in different places, each of which is independent of $h$ and $\Delta t$.

2.   Preliminary

Here, we first introduce some notations. $W^{m, p}(\Omega)$ is a standard Sobolev space with norm $||\cdot||_{W^{m, p}(\Omega)}$. $W^{1, 2}(\Omega) = H^1(\Omega)$ with norm $||\cdot||_1$, and $H_{0}^1(\Omega) = \{ \upsilon\in H^{1}(\Omega):\upsilon|\partial\Omega = 0\}$. $L^k(\Omega)$ denotes a $k$-squared integrable function space in region $\Omega$ with norm $||\cdot||$. The norm of $L^k(\Omega)$ is denoted by $||\cdot||_{0, U}$. We denote by $L^{k}(0, T; W^{m, p}(\Omega))$ the Banach space of all $L^{k}$ integrable functions from $(0, T)$ into $W^{m, p}(\Omega)$ with norm $||\upsilon||_{L^{k}(W^{m, p})} = ||\upsilon||_{L^{k}(0, T; W^{m, p}(\Omega))} = \left(\int_{0}^{T} ||\upsilon||^{k}_{W^{m, p}(\Omega)}\right)^{\frac{1}{k}}$ for $k\in[1, \infty)$ and the standard modification for $k = \infty$.

For a positive integer $N$, define time step size $\Delta t$ by $\Delta t = \frac{T}{N}$. For $n = 0, 1, ..., N-1$, $t^{n} = n\Delta t$, $I^{n+1} = [t^{n}, t^{n+1}]$. write

and $Q = \Omega\times (0, T]$. For any given sequence $\{\zeta^{n}\}^{M}_{n = 0}$, $\zeta^{n} = \zeta(x, t^{n})$ for the function $\zeta(x, t)$ defined in $Q$. For $1\leq q\leq \infty$, a discrete time-dependent norm is given by

and the standard modification for $q = \infty$, where

The inner product is noted by

For convenience, we take $A$ to be the identity matrix and write

It is obvious that

Let $\mathcal{T}_{h}$ be a quasi-uniform triangulation of $\Omega$, $h_{K}$ denotes the diameter of element $K$, and $h = \max_{K\in\mathcal{T}_{h}} \{h_{K}\}$. The space $V_{h}$ associated with $\mathcal{T}_{h}$ is defined by

where $P_{1}(K)$ denotes the polynomials space with the degree being no more than one on $K \in \mathcal{T}_{h}$.

We consider the following piecewise constant finite element space

which is a finite dimensional subspace of $U_{ad}$.

For any $\mu\in U$, we define the orthogonal projection operator $\Pi_{h}:\; U \rightarrow U_{h}$ such that

By the definition of Eq (2.1), we have

where $|K|$ is the measure of $K$. For the operator $\Pi_{h}$ defined in Eq (2.1), we have

for all $\mu\in W^{1, p}(\Omega)$ and $1\leq p\leq \infty$ (see ^[23]).

For $t\in(0, T]$, define $L^{2}$ projection $\mathcal{R}_{h}\upsilon(t)\in V_{h}$ for $\upsilon(t)\in V$ by

As in ^[24], for $1\leq r \leq 2$, the $\mathcal{R}_{h}\upsilon(t)$ satisfies

3.   Optimality conditions

In this section, we first give the weak form of the state equation (1.2) as: We seek $y(\cdot, t)\in H^{1}_{0}(\Omega)$ such that

Next, we introduce a new variable $\omega = y_{t}$, then the problem (1.1) and (1.2) can be written as

subject to

The problem (3.2) and (3.3) is formulated in standard reduced functional form as

Since the problem (3.4) is non-convex, we cannot guarantee the global unique solutions of (3.4). Therefore, we consider the local optimal solutions (see ^[25,26,27]).

Definition 3.1. (Local solution ^[28]). The control $\bar{u}\in U_{ad}$ is called a local solution of the problem (3.4) if for each fixed $t\in [0, T]$, there exists a constant $\iota > 0$, such that for all $u\in U_{ad}$ with $||u-\bar{u}|| < \iota$, it satisfies

For the problem (3.4), the existence of the local solution can be guaranteed in ^[10]. Next, we can derive the following FNC for the local solution $\bar{u}$.

Theorem 3.1. If $\bar{u}$ is a local optimal control for the problem (3.4), then there exists a set of functions $(\omega(t), y(t), \bar{u}(t), q(t), p(t))\in \big(H^{1}(L^2)\cap L^2(H^{1})\big)\times \big(H^{2}(L^2)\cap L^2(H^{1})\big)$$\times U_{ad}\times \big(H^{1}(L^2)\cap L^2(H^{1})\big)\times \big(H^{2}(L^2)\cap L^2(H^{1})\big)$ such that

where $B^{*}$ is the adjoint operator of $B$.

Proof. Applying the variational rule, the optimal condition reads

where

Next, differentiating the state equation (3.6) at $\bar{u}$ in the direction $\varrho$, we have

Defining the co-state $(p, q)$ satisfying Eq (3.7), and letting $v_{1} = \omega'\mathcal{D}_{\bar{u}}y(\varrho), v_{2} = \mathcal{D}_{\bar{u}}y(\varrho)$ in Eq (3.7), we can obtain

Meanwhile, letting $v_{1} = q$ in Eq (3.10) and $v_{2} = p$ in Eq (3.11), we get

Since $\omega'\mathcal{D}_{\bar{u}}y(\varrho)(t = 0) = 0, \mathcal{D}_{\bar{u}}y(\varrho)(t = 0) = 0, \; q(T) = 0, \; p(T) = 0$, integrating by parts and a direct calculation, we can yield

Substituting Eqs (3.17) and (3.18) into Eqs (3.15) and (3.16), we derive

Combining Eqs (3.19) and (3.20) with Eqs (3.13) and (3.14) yields

Substituting Eq (3.21) into Eq (3.9) implies

This completes the proof of Theorem 3.1. □

According to ^[26] and ^[29], we can know that for the non-convex problem (3.4), the FNC is not sufficient. So, we need to consider the SSC:

(SSC) There exist constants $\kappa > 0$ and $\lambda > 0$ such that

for all $v\in L^{2}(0, T; L^{2}(\Omega))$ satisfying

where

and the notation $\mathcal{D}^{2}_{\bar{u}}y(v, v)$ is defined as follows: Let $\tilde{y} = \mathcal{D}_{\bar{u}}y(v)$. Then, its directional derivative in the direction $v$, denoted as $\tilde{y}'(v)$, is given by

More details about the notation $\mathcal{D}^{2}_{\bar{u}}y(v, v)$ can be found in ^[2].

Referring to ^[30], we give the coercive property of the problem (3.4) in a neighborhood of the local solution by the following lemma.

Lemma 3.1. Assume that $\bar{u}$ is a local solution of the problem (3.4) and satisfies the SSC (3.23). There exists a sufficient small constant $\iota > 0$ such that

for all $u\in U_{ad}$ with $||u-\bar{u}|| < \iota$.

4.   Crank-Nicolson scheme

In this section, we develop a fully discrete Crank-Nicolson scheme for the optimality system (3.2)–(3.3) as follows:

where $\delta = \delta(h, \Delta t)$ denotes the fully discrete in space and time. We can reformulate the problem (4.1)–(4.4) as

We also give the definition of the local solution of the discrete control problem (4.1)–(4.4).

Definition 4.1. The control $\bar{u}_{h} ^{n+\frac{1}{2}}\in U_{h}$ is called a fully discrete local solution of the problem (4.1)–(4.4) if for each fixed $t^{n+\frac{1}{2}}$, there exists a constant $\iota > 0$, such that for $\forall u_{h}^{n+\frac{1}{2}}\in U_{h}$ with $||u_{h}^{n+\frac{1}{2}}-\bar{u}_{h} ^{n+\frac{1}{2}}|| < \iota$, it satisfies

We present the first-order necessary optimality condition for problem (4.1)–(4.4) at the local solution $\bar{u}_{h} ^{n+\frac{1}{2}}$ by the following theorem.

Theorem 4.1. Assume that $\bar{u}_{h}^{n+\frac{1}{2}}, n = 0, 1, ..., N-1$ is a local solution of discrete control problem (4.1)–(4.4) , then there are state $(\omega_{h}^{n+\frac{1}{2}}, y_{h}^{n+\frac{1}{2}})\in V_h\times V_h$ and co-state $(q_{h}^{n+\frac{1}{2}}, p_{h}^{n+\frac{1}{2}})\in V_h\times V_h$, $n = 0, 1, ..., N-1$, such that the following optimality conditions hold:

Proof. Differentiate the Eq (4.5) at $\bar{u}_{h} ^{n+\frac{1}{2}}$ in the direction $v_{h}-\bar{u}_{h} ^{n+\frac{1}{2}}$, and the discrete optimal condition reads

Similarly, differentiating the Eqs (4.7) and (4.8) at $\bar{u}_{h}^{n+\frac{1}{2}}$ in the direction $\nu$, we have

where

Choosing the discrete co-state $(p_{\delta}, q_{\delta})$ satisfying Eqs (4.10)–(4.12), and selecting $v = \omega'(y_{h}^{n+\frac{1}{2}})\mathcal{D}y_{h}^{n+\frac{1}{2}}(\nu)$ in Eq (4.10) and $v = \mathcal{D}y_{h}^{n+\frac{1}{2}}(\nu)$ in Eq (4.11), we get

Taking $v = q_{h} ^{n+\frac{1}{2}}$ in Eq (4.15) and $v = p_{h} ^{n+\frac{1}{2}}$ in Eq (4.16), we have

Since $\mathcal{D}y_{h}^{0}(\nu) = 0, \; \omega'(y_{h}^{0})\mathcal{D}y_{h}^{0}(\nu) = 0, \; p_{h} ^{N} = 0, \; q_{h} ^{N} = 0$, we know that

and

Substituting Eqs (4.22) and (4.23) into Eqs (4.20) and (4.21), we obtain

Combining Eqs (4.24) and (4.25) with Eqs (4.18) and (4.19), and summing from $0$ to $N-1$ leads to

Therefore, substituting Eq (4.26) into Eq (4.14), we get

This completes the proof of Theorem 4.1. □

Similarly, we also provide the discrete SSC for the local solution $\bar{u}_{h} ^{n+\frac{1}{2}}$ as

where $\kappa > 0$, and $\bar{u}_{h} ^{n+\frac{1}{2}}$ satisfies Eq (4.13). From Eq (4.1), we can get

The following lemma shows the coercive property of the second derivative of the discrete objective function $\mathcal{J}_{\delta}$ in a neighborhood of a local solution $\bar{u}$.

Lemma 4.1. Let $\bar{u}$ be a local solution of the problem (3.4) and the SSC (3.23) is valid. There exists sufficiently small constant $\iota > 0$, and $h$, for all $u\in U_{ad}$ with $||u-\bar{u}|| < \iota$ and $v\in U_{ad}$,

holds.

4.1.   Auxiliary problems

So as to get a priori estimates, it is needed to introduce an auxiliary problem: find $(\omega_{h}^{n+\frac{1}{2}}(u), y_{h}^{n+\frac{1}{2}}(u))$ $\in V_h\times V_h$, $n = 0, 1, \ldots, N-1$, such that for all $v\in V_h$,

Another auxiliary problem: find $(q_{h}^{n+\frac{1}{2}}(u), p_{h}^{n+\frac{1}{2}}(u))\in V_h\times V_h$, $n = 0, 1, \ldots, N-1$, such that for all $v\in V_h$,

For convenience, we denote

It is clear that

Next, we describe the error caused by the control discretization using the following lemma.

Lemma 4.2. Let $(\omega_{\delta}, y_{\delta}, q_{\delta}, p_{\delta})$ and $(\omega_{\delta}(u), y_{\delta}(u), q_{\delta}(u), p_{\delta}(u))$ be the solutions of Eqs (4.7)–(4.12) and Eqs (4.31)–(4.36), respectively. Then,

Proof. To begin, we develop the inequality for $\theta _{\omega}$ and $\theta _{y}$. Subtracting Eqs (4.7) and (4.8) from Eqs (4.31) and (4.32), we derive

Choosing $v = \theta_{y} ^{n+\frac{1}{2}}$, $v = \theta_{\omega}^{n+\frac{1}{2}}$ as the test function in Eqs (4.40) and (4.41), respectively, we have

Substituting Eq (4.42) into Eq (4.43), we get

Summing up from $n = 0$ up to $N-1$, it yields

For the first term, we derive

For the second term, we get

Now, combining Eqs (4.46) and (4.47) with Eq (4.45), we arrive at

The discrete Gronwall's inequality leads to

Next, we develop the inequality for $\theta _{q}$ and $\theta _{p}$. Subtracting Eqs (4.10) and (4.11) from Eqs (4.34) and (4.35), we get

Choosing $v = -\theta_{q}^{n+\frac{1}{2}}$ as the test function in Eq (4.51), we have

Substituting Eq (4.50) into Eq (4.52), we know

By calculation, the following formula holds:

Multiplying both sides of Eq (4.53) by $2\Delta t$, then summing it over $n$ from $M$ to $N-1$, it leads to

Here, the estimates of $\Pi_{1}$, $\Pi_{2}$, $\Pi_{3}$ are similar to those of $\Upsilon_{1}$, $\Upsilon_{2}$, $\Upsilon_{3}$, and we have

Finally, inserting the above estimates of $\Pi_{1}$–$\Pi_{3}$ into Eq (4.55), we get

Thus, combine the Poincaré inequality and the discrete Gronwall's inequality, such that

Eqs (4.38) and (4.57) follows Eq (4.39), which also completes the proof of Lemma 4.2. □

Next, so as to estimate the error of the control discretization, we choose a local solution $\bar{u}$ of the continuous problem (3.6)–(3.8), and an associated approximate solution $u_{\delta}$ of the discrete problem (4.1)–(4.4). We introduce the following auxiliary problem:

where $U_{h}^{\iota}(\bar{u}) = \{u_{h} ^{n+\frac{1}{2}}\in U_{h}:||u_{\delta}-\bar{u }|| < \iota\}$. In addition, since $\mathcal{J}''_{\delta}$ satisfies Lemma 4.1 for $\bar{u}\in U_{h}^{\iota}(\bar{u}), \; v\in U_{ad}$, the existence and uniqueness of the problem (4.58) are guaranteed; see ^[28].

From the definitions of the local solution (4.6) and $U_{h}^{\iota}$, we can get

Utilizing (4.58) and (4.59), we can deduce that $\bar{u}_{h} ^{n+\frac{1}{2}}\in U_{h}^{\iota}$ is the unique solution of the problem (4.58).

Lemma 4.3. Let $\bar{u}$ be a local solution of the problem (3.4) and the SSC (3.23) is valid. Then, the discrete problem (4.58) has a unique solution $\bar{u}_{h} ^{n+\frac{1}{2}}$, and the following estimate holds:

for $\iota > 0, h > 0$ sufficiently small.

Proof. From Lemma 4.1, it is clear to infer that

By the FNC (4.13), we can get

for $h$ sufficiently small. Utilizing $\upsilon^{n+\frac{1}{2}} = \theta \bar{u}^{n+\frac{1}{2}}+(1-\theta)\bar{u}_{h}^{n+\frac{1}{2}}$ with $\theta\in [0, 1]$, we get

To start, using the Cauchy-Schwarz inequality, we obtain

For the $\mathcal{S}_{2}$ and $\mathcal{S}_{3}$, by the definition of the operator $\Pi_{h}$, we have

and

Then, for the $\mathcal{S}_{4}$ and $\mathcal{S}_{5}$, from Lemma 4.2 and the Cauchy inequality, it yields

Substituting $\mathcal{S}_{1}$–$\mathcal{S}_{5}$ into Eq (4.61), we obtain

This completes the proof of Lemma 4.3. □

4.2.   Error analysis

In this subsection, we will establish the error caused by the discretization of the Crank-Nicolson FEM scheme. To do this, we define the error function as

and introduce the following truncation errors:

For these definitions, we first present the following estimates of the truncation error.

Lemma 4.4. The following estimates hold:

Lemma 4.5. Let $(\omega, y, u, q, p)$ and $(\omega_{\delta}(u), y_{\delta}(u), q_{\delta}(u), p_{\delta}(u))$ be the solutions of Eqs (3.6)–(3.7) and Eqs (4.31)–(4.36), respectively. Assume that $\omega, \; q\in L^{2}(H^{2})\cap H^{1}(H^{2}), \; \omega_{tt}, \; q_{tt}\in L^{2}(H^{1}),$ $y_{tt}\in L^{2}(L^{2})\cap L^{2}(H^{1})\cap L^{2}(H^{2}), p_{tt}\in L^{2}(L^{2})\cap L^{2}(H^{2}),$ $\omega_{ttt}, \; q_{ttt}\in L^{2}(L^{2}), \; y_{ttt}, \; p_{ttt}\in L^{2}(H^{1}),$ $y_{0}(x), \; g(x)\in H^{2}(\Omega)$. Then, we have

Proof. From Eq (3.7), the exact solution $(\omega, y)$ satisfies

From relations Eqs (4.64) and (4.65) and Eqs (4.31) and (4.32), it holds that

Taking the discrete inner product of Eq (4.67) with $v_h = \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u) = \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega(t_{n+\frac{1}{2}}) +$$\omega(t_{n+\frac{1}{2}})-\omega^{n+\frac{1}{2}}_{h}(u) = \mathcal{R}_{h}\omega(t_{n+\frac{1}{2}})-\omega(t_{n+\frac{1}{2}})+e^{n+\frac{1}{2}}_{\omega}$ and rearranging the terms, we have

Meanwhile, choosing $v_h = e_{y}^{n+\frac{1}{2}}$ in Eq (4.66), we obtain

Through Eqs (4.68) and (4.69), and summing from $n = 0$ up to $N$–$1$, we have that

By utilizing Young's and H$\ddot{o}$lder inequalities, we have

For $\Theta_{4}$–$\Theta_{7}$, by definition of the truncation error and Lemma 4.4, it can be obtained that

Substituting $\Theta_{1}$–$\Theta_{7}$ into Eq (4.70), we get

Note that

Then, the discrete Gronwall's inequality implies that

Furthermore, from Eq (3.7), we can find that the exact solution $(p, q)$ satisfies

From relations Eqs (4.72) and (4.73) and Eqs (4.34) and (4.35), we have

Choosing $v_{h} = -(\mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u)) = q(t_{n+\frac{1}{2}})-\mathcal{R}_{h}q(t_{n+\frac{1}{2}})-e_{q}^{n+\frac{1}{2}}$ in Eq (4.75), we have

Meanwhile, letting $v_{h} = -e_{p}^{n+\frac{1}{2}}$ in Eq (4.74), we get

Combining Eqs (4.76) and (4.77), then summing from $n = M$ to $N-1$, it can conclude that

By the properties of $L^{2}$ projection, we have

A standard algebraic manipulation implies that

For the bound of $\mathcal{Y}_{3}$, it holds

Then, for $\mathcal{Y}_{4}$–$\mathcal{Y}_{9}$, noting that $\mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q_{h}^{n+\frac{1}{2}}(u) = \mathcal{R}_{h}q(t_{n+\frac{1}{2}})-q(t_{n+\frac{1}{2}})+e_{q}^{n+\frac{1}{2}}$ and applying Lemma 4.4, we have

Collecting the above bounds and using the discrete Gronwall's inequality, we deduce that

The proof of Eq (4.63) can be completed by combining Eq (4.79) with Eq (4.62). □

Above all, the error of the Crank-Nicolson scheme (4.7)–(4.13) is given by the following theorem.

Theorem 4.2. Let $(\omega, y, \bar{u}, q, p)$ and $(\omega_{\delta}, y_{\delta}, \bar{u}_{\delta}, q_{\delta}, p_{\delta})$ be the local solutions of Eqs (3.6)–(3.8) and Eqs (4.7)–(4.13), respectively. Moreover, we assume that all conditions in Lemmas 4.2–4.5 are valid. Then, we have

5.   Numerical experiments

In this section, we provide a numerical example to verify the theoretical results, which will consider the following nonlinear SOHOCPs:

We adopt the same mesh triangular partition for the state and control. Furthermore, we choose

such that the exact $(y, u, p)$ is

We show the convergence results of the Crank-Nicolson scheme in Table 1. The profile of the exact $(y, p, u)$ is drawn in Figure 1. The simulated results for the second-order scheme is presented in Figure 2.

From Table 1, it implies that the numerical results are consistent with the theoretical results. From Figure 1 and Figure 2, we can see the Crank-Nicolson scheme is efficient.

6.   Conclusion

This paper presents a second-order fully discrete scheme for nonlinear SOHOCPs and, in conjunction with auxiliary problems, derives a priori error estimates. Furthermore, a numerical experiment is conducted to confirm the convergence order of the theoretical results.

In ^[31], Li et al. established a mixed-form discrete scheme for the nonlinear stochastic wave equations (SWEs) with multiplicative noise by defining a new variable. In ^[32], Sonawane et al. studied the existence of an optimal control problem for the bilinear SWEs. In the future, we plan to consider the method based on the definition of the new variable mentioned in this paper and in ^[31]. We will apply this method to the optimization system described in ^[32] and further conduct an error analysis of the resulting discrete scheme.

Author contributions

Huanhuan Li was responsible for the methodology and writing the original draft. Meiling Ding contributed the software. Xianbing Luo handled the review, editing, and supervision. Shuwen Xiang oversaw the supervision and validation.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Granted No. 11961008) and Guizhou University Doctoral Foundation (Granted NO. 15 (2022)).

Conflict of interest

The authors declare there is no conflict of interest.