Completely controllable pairs of matrices with prescribed positions

Let F be a field and let n, p1, p2, p3 be positive integers such that n = p1 + p2 + p3. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C_{1},C_{2}) = \left( \left[\begin{array}{cccccccccc}C_{1,1} & C_{1,2}\\C_{2,1} & C_{2,2}\end{array}\right] , \left[\begin{array}{c}C_{1,3}\\C_{2,3}\end{array}\right] \right),$$\end{document}where the blocks Ci,j are of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p_{i}\times p_{j},i\in\{1,2\},j\in \{1,2,3\}}$$\end{document}. In this paper, we describe necessary and sufficient conditions under which the pair (C1, C2) is completely controllable, when three arbitrary positions are prescribed and the others are free. Furthermore, we still provide some particular solutions for the prescription of the characteristic polynomial of a partitioned matrix of the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[\begin{array}{cccccccccccc}C_{1,1} & C_{1,2} & C_{1,3}\\ C_{2,1} & C_{2,2} & C_{2,3}\\ C_{3,1} & C_{3,2} & C_{3,3}\end{array}\right] \in F^{n\times n},$$\end{document}where the blocks Ci,j are of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p_{i}\times p_{j},i,j\in\{1,2,3\}}$$\end{document}, when some of its positions are prescribed and the others vary.