Algebraic Theory of Time-Varying Linear Systems: A Survey

The development of the algebraic theory of time-varying linear systems is described. The class of systems considered consists of differential-algebraic equation in kernel presentation. This class encompasses time-varying state space, descriptor systems as well as Rosenbrock systems, and time-invariant systems in the behavioural approach.One difference between time-varying and time-invariant systems is that, since the coefficients of the differential equations are time-varying function, the differential operator does not commute with the coefficients. However, the main difficulty is that solutions may exhibit a finite escape time. Hence there is a conflict between the class of time-varying coefficients and the class of admissible solution spaces. All contributions to time-varying systems have to cope with this.As an efficient tool in linear, time-invariant system theory, Kalman introduced in the 1960s elementary module theory over principal ideal rings. This tool proved efficient also for time-varying systems. Although from then on, the field of time-varying linear systems has never been a ``hot topic" in systems theory, there has been an ongoing evolution which led to a rather substantial theory. Not surprisingly, the theory is mainly restricted to linear systems and most results are on such properties as controllability, and not on stability. Recent results use successfully tools from module theory and homological algebra.