On the definiteness of quadratic functionals (Q1570412)

Various characterizations of strong observability (or observability with unknown inputs) of time-dependent linear systems \(\dot x= Ax+ Bu\), \(y= C_0x\), with matrices \(A\), \(B\), \(C_0\) smooth enough, are derived. It is shown that strong observability is equivalent to the definiteness of a corresponding quadratic functional, and particularly to a certain asymptotic behaviour of solutions of associated Riccati matrix differential equations. A characterization of strong observability is given via a rank condition on certain matrices that are built up from derivatives of the matrices \(A\), \(B\), and \(C_0\).