9530917 Mohler The aim of this research is to develop a new global controllability and stabilization methodology for multidimensional nonhomogeneous time-invariant bilinear systems (BLS) with applications to large power transmission networks, particularly Flexible AC Transmission Systems (FACTS). A through understanding of the qualitative behavior of a dynamic system is the foundation of all the methods for controllability and stabilization studies. To this end, the basis for this proposal is formed by a new method recently introduced by the proposers to study the global qualitative behavior of BLS. Its core is a nonlinear decomposition technique converting the study into the analysis of the qualitative behavior of a finite number of the corresponding "basic" uncontrolled systems. The work assumes the unboundedness of (otherwise finite) control. One of the goals of this proposal is to refine this technique and to distinguish the conditions under which the goal controllability can be achieved (in infinite time) by bounded finite controls (which is the main practical requirement). The intent here is to focus on the 3-D case and to give its complete solution in terms of eigenvectors and eigenfunctions of the matrices involved. In turns out that the aforementioned qualitative method fits quite nicely to the analysis of some multidimensional nonlinear dynamic systems controlled via multipliers (coefficients), 9530917 including large power transmission networks. The latter systems are of great practical importance and were the subject of intensive studies for numerous researchers and practical engineers from the turn of the century. On the other hand, in relation to the topic of this proposal, mostly additive damping and local controllability were considered. The emphasis of this proposed research is given to the global aspects of reachability and controllability, and stabilization (these are closely related) via multipliers/coefficients. This type of control structure is motivated by numerous adaptive system applications such as FACTS, which play an important role in enhancing dynamic performance of modern power systems and can increase transient stability margins. To our knowledge, very little is known about the questions which this proposal addresses. The research proposed here can be split into two parts. We start, on the other hand, with the analysis making use of unbounded controls as traditionally assumed unavoidable in general global studies. On the other hand, taking into account that practical controls are normally finite and may be even of a certain sign, we also introduce these assumptions for the aforementioned unbounded controls. The idea of this approach is that if a global controllability/reachability/stabilization result holds for unbounded (but finite, say nonnegative) controls, then one can expect/show it to be true in the local aspect (not necessarily small) for the real device of restricted capability. Yet it is of particular interest, however, to distinguish, based on the proposed methodology, the cases when the global result can be achieved by bounded controls. ***
