Lie groups and Lie algebras can play a crucial role in nonlinear control theory, in particular in the analysis of systems with symmetries or the global analysis of nonlinear control systems. If a nonlinear problem can be framed as a control problem on a Lie group or on a homogeneous space, then one can often bring to bear on the problem the vast and highly developed arsenal of Lie theory. Indeed via the Lie algebra one can sometimes ÿnd e ective linear methods that can make signÿcant contributions to the study of the problem. In this context Lie theory also provides a unifying language for nonlinear control.
Important advances over the past years include the further development of a systematic theory of Lie semigroups, originating from the early work of Loewner, as well as establishing a controllability theory of systems on Lie groups and homogeneous spaces. Today, in addition to further advances in the foundations of Lie theory, new applications in diverse areas such as nonlinear mechanics, numerical analysis and optimization, completely integrable Hamiltonian systems and even geometric function theory have emerged as well. These applications often involve a novel and sometimes nontrivial reformulation of and approach to a problem from the perspective of Lie theory.
The purpose of this special issue is to collect a number of recent and original research contributions that (a) demonstrate the depth and the breadth of Lie theory research in various application areas, and (b) present short state-of-the-art articles. All papers have been presented at the International Workshop on Lie Theory and its Applications, W urzburg University, August 2-4, 1999, that was organized by the guest editors (see http://www3.mathematik.uni-wuerzburg.de/ ∼ hueper=Lie.html).
This issue contains six papers that present a cross section of current research in Lie theory and its applications. The ÿrst paper by L. Faybusovich and R. Arana outlines a Jordan-algebraic approach to interior-point algorithms. The second by U. Helmke and F. Wirth considers the inverse power iteration from a control theory point of view. In the third paper by V. Jurdjevic that relates classical mechanics to Lie theory the focus is on the geometry of homogeneous spaces and its relation with complex Lie groups, as seen from a Hamiltonian point of view. In the paper by G.A. Kantor and P.S. Krishnaprasad Lie theory is applied to attack a problem in distributed control networks. In the ÿfth paper by L.A.B. San Martin some interesting and natural examples of a family of maximal noncontrollable Lie wedges with empty interior are provided. Finally, A.V. Sarychev studies stability of time-varying systems using tools from Lie theory and the chronological calculus.
Thanks are due to all authors for their contributions as well as to the reviewers for their help and constructive comments. Special thanks go to Iven Mareels for his willingness and support to make this special issue a reality.