The identification problem for a system of linear stationary differential equations (Q1281232)

Systems of linear differential equations of the form \[ \dot x = \mathcal{A} x,\quad t \in [0,T], \eqno{(1)} \] with \(x \in {\mathbb{R}}^{n}\), \(\mathcal{A} \in {\mathbb{R}}^{n\times n}\), \(T > 0\), are considered. Let \(x_{i}(t)\), \(i = 1,2,\dots,n,\) be solutions to (1) with initial data \(x_{i}(0) = e_{i}\), where \(e_{i}\) are columns of a unit matrix. Suppose a matrix \(\mathcal{G}\), \(\mathcal{G} \in \mathbb{R}^{m\times n}\), is given and the vector \[ y_{i} (t) = \mathcal{G} x_{i} (t) \eqno{(2)} \] is known. The problem of finding the matrix \(\mathcal A\) from (2) is stated. Necessary and sufficient conditions on the existence of matrix \(\mathcal A\) are formulated. The Hamilton-Cayley theorem is used to prove these conditions. A numerical method is proposed for finding the matrix \(\mathcal A\) substituting (2) by a system of linear algebraic equations. It is noted that the formulation of this problem belongs to M. S. Nikolskij.