Doubly structured mapping problems of the form $\mathrm{\Delta }x=y$ and ${\mathrm{\Delta }}^{\ast }z=w$

For a given class of structured matrices $\mathbb{S}$, we find necessary and sufficient conditions on vectors $x,w\in {\text{C}}^{n+m}$ and $y,z\in {\text{C}}^n$ for which there exists $\mathrm{\Delta }=[{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2]$ with ${\mathrm{\Delta }}_1\in \mathbb{S}$ and ${\mathrm{\Delta }}_2\in {\text{C}}^{n,m}$ such that $\mathrm{\Delta }x=y$ and ${\mathrm{\Delta }}^{\ast }z=w$. We also characterize the set of all such mappings $\mathrm{\Delta }$ and provide sufficient conditions on vectors $x,y,z$, and $w$ to investigate a $\mathrm{\Delta }$ with minimal Frobenius norm. The structured classes $\mathbb{S}$ we consider include (skew)-Hermitian, (skew)-symmetric, pseudo(skew)-symmetric, $J$-(skew)-symmetric, pseudo(skew)-Hermitian, positive (semi)definite, and dissipative matrices. These mappings are then used in computing the structured eigenvalue/eigenpair backward errors of matrix pencils arising in optimal control.