In this manuscript, the authors investigate the concept of convex linear combination of the controllability pairs of linear continuous-time finite dimensional systems.
Given a linear system \(Ax=b\), the authors use in this paper the following nonsingular transformation: \[ A^c=PAP^{-1}=\left( \begin{array} {cccccc} 0 & 0 & 0 & \cdots & 0 & -a_{11} \\ 1 & 0 & 0 & \cdots & 0 & -a_{12} \\ 0 & 1 & 0 & \cdots & 0 & -a_{13} \\ \vdots & \vdots & \vdots & &\vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & -a_{1n} \end{array} \right), \ \ b^c= \left( \begin{array} {c} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \end{array} \right). \]
For the linear control systems \((A_1,b_1)\) and \((A_2,b_2)\), their linear convex combination \((A(q),b(q))\) is defined as follows: \[ A(q)=(1-q)A_1+q A_2, \ \ b(q)=(1-q)b_1+q b_2, \ \ 0 \leq q \leq 1, \] where \(q\) is a real number.
The main result of this paper is:
Let the pairs \((A_1,b_1)\) and \((A_2,b_2)\) of linear control systems be controllable. Then their linear convex combination \((A(q),b(q))\) is also controllable for all \(0 \leq q \leq 1\), if \(P=P_1=P_2\) is a nonsingular matrix.
Some examples are presented for confirming the theoretical result.