Observability conditions of linear time-varying systems and its computational complexity aspects

We propose necessary and sufficient observability conditions for linear time-varying systems with coefficients being time polynomials. These conditions are deduced from the Gabrielov-Khovansky theorem on multiplicity of a zero of a Noetherian function and the Wei-Norman formula for the representation of a solution of a linear time-varying system as a product of matrix exponentials. We define a Noetherian chain consisted of some finite number of usual exponentials corresponding to this system. Our results are formulated in terms of a Noetherian chain generated by these exponential functions and an upper bound of multiplicity of zero of one locally analytic function which is defined with help of the Wei-Norman formula. Relations with observability conditions of bilinear systems are discussed. The case of two-dimensional systems is examined.