Algebraic systems theory and computer algebraic methods for some classes of linear control systems Eva Zerz Viktor Levandovskyy Lehrstuhl D f¨ ur Mathematik RISC RWTH Aachen University Johannes Kepler University Templergraben 64 Altenbergerstr. 69 52062 Aachen, Germany 4040 Linz, Austria MTNS 2006 Algebraic systems theory has been greatly advanced in the last 15 years. One the one hand, this is due to the behavioral approach to systems and control theory, which was introduced by J. C. Willems [17] in the 1980s, and which has proven to be particularly fruitful for algebraic approaches, as it studies solution sets rather than the representing equations, and it does not divide the system variables into differently treated classes a priori. On the other hand, already in the 1960s, B. Malgrange [10], V. Palamodov [12], and others started to study systems of lin- ear partial differential equations using algebraic tools such as module theory and homological methods. They founded what is now commonly referred to as the algebraic analysis approach. In 1990, a seminal paper by U. Oberst [11] estab- lished a link between the two approaches, leading to a deeper understanding of both. This stimulated the lively research activity in the area of multidimensional systems, and contributed to algebraic and behavioral systems theory in general. The aim of this paper is to give a tutorial introduction to algebraic systems theory, focussing in particular on linear • multidimensional shift-invariant systems (PDE with constant coefficients); • one-dimensional time-varying systems (ODE with variable coefficients in the field of rational or meromorphic functions); • one-dimensional parameter-dependent systems (ODE whose coefficients are polynomial or rational functions of several parameters). Moreover, we describe a recent implementation of related algorithms in the Sin- gular [5] library control.lib [20]. 1