On the solutions of the dual matrix equation $A^{\mathrm{\top }}XA=B$

Let ${\mathbb{D}}^{m\times n}=\{A=A_1+\epsilon A_2|A_1,A_2\in {\mathbb{R}}^{m\times n}\}$ be the set of all $m\times n$ real dual matrices. In this paper, the following problems are considered. Problem I: Given dual matrices $A=A_1+\epsilon A_2\in {\mathbb{D}}^{m\times n}$ and $B=B_1+\epsilon B_2\in {\mathbb{D}}^{n\times n}$, find $X\in S$ such that the dual matrix equation $A^{\mathrm{\top }}XA=B$ is satisfied, where $S=\{X\in {\mathbb{D}}^{m\times m}|CX=D,C,D\in {\mathbb{D}}^{p\times m}\}$. Problem II: Given dual matrices $A=A_1+\epsilon A_2\in {\mathbb{D}}^{m\times n},B=B_1+\epsilon B_2\in {\mathbb{D}}^{n\times n}$ and $\tilde{X}={\tilde{X}}_1+\epsilon {\tilde{X}}_2\in {\mathbb{D}}^{m\times m}$, with $B_i=B_i^{\mathrm{\top }},i=1,2$, find $\hat{X}\in T$ such that $\Vert \hat{X}-\tilde{X}{\Vert }_{\mathrm{D}}=\underset{X\in T}{min}\Vert X-\tilde{X}{\Vert }_{\mathrm{D}}=\underset{X\in T}{min}\sqrt{\Vert X_1-{\tilde{X}}_1{\Vert }^2+\Vert X_2-{\tilde{X}}_2{\Vert }^2}$, where $T=\{X=X_1+\epsilon X_2\in {\mathbb{D}}^{m\times m}|A^{\mathrm{\top }}XA=B\text{s. t.}{X}_i=X_i^{\mathrm{\top }},i=1,2\}$. We derive the solvability conditions and the representation of the general solution of Problem I using the Moore-Penrose inverse. Also, we deduce the solvability conditions and the explicit formula of $T$ and the unique approximation solution $\hat{X}$ of Problem II by applying the Moore-Penrose inverse and Kronecker product of matrices. Finally, we give a numerical example to show the correctness of our result.