Observation of parametrically perturbed dynamical systems (Q579177)

Assume \[ \dot x=Ax+\sum p_ i(t)q_ ic_ i^ T x,\quad y=Hx, \tag{1} \] where the summation is over \(i=1,2,\ldots,m\), \(A\) and \(H\) are constant \(n\times n\) and \(k\times n\)-matrices, resp., \(q_ i\in\mathbb R^ n\), \(c_ i\in\mathbb R^ n\), and \(0\leq p_ i(t)\leq 1\). The columns of \(H\) can be considered orthonormal. Then there is an \(n\times (n-k)\)-matrix \(L\) with orthonormal rows such that \(HL=0\) and \(x=H^ Ty+Lz\), where \(z=L^ Tx\). Let the matrix \(Q\) with columns \(q_ 1,\ldots,q_ m\) have rank \(r\). Then we can write \(Q=Q_ 0 N\), where \(Q_ 0\) and \(N\) have sizes \(n\times r\) and \(r\times m\), resp. If rank \(HQ_ 0=r\) the author presents necessary and sufficient conditions for the existence of a system of linear differential equations \[ F(D)\xi =G(D)y \tag{2} \] such that for any solution \(x\) of (1) the corresponding solution \(\xi\) of (2), with the initial conditions for \(\xi\) chosen arbitrarily, exponentially approaches \(z\) as \(t\to \infty\). In case rank \(HQ_ 0<r\) such asymptotic observation is not possible.