Geometric Methods in System Theory : Proceedings of the NATO Advanced Study Institute Held at London, England, August 27-September 7, 1973

Geometric Methods in System Theory In automatic control there are a large number of applications of a fairly simple type for which the motion of the state variables is not free to evolve in a vector space but rather must satisfy some constraints. Examples are numerous; in a switched, lossless electrical network energy is conserved and the state evolves on an ellipsoid surface defined by x'Qx equals a constant; in the control of finite state, continuous time, Markov processes the state evolves on the set x'x = 1, xi ~ O. The control of rigid body motions and trajectory control leads to problems of this type. There has been under way now for some time an effort to build up enough control theory to enable one to treat these problems in a more or less routine way. It is important to emphasise that the ordinary vector space-linear theory often gives the wrong insight and thus should not be relied upon. Front Matter....Pages I-VII Dynamical Polysystems and Control Theory....Pages 1-42 Lie Algebras and Lie Groups in Control Theory....Pages 43-82 Realization Theory of Bilinear Systems....Pages 83-130 An Introduction to Stochastic Differential Equations on Manifolds....Pages 131-149 General Theory of Global Differential Dynamics....Pages 150-158 Two Proofs of Chow’s Theorem....Pages 159-164 On Necessary and Sufficient Conditions for Local Controllability Along a Reference Trajectory....Pages 165-173 The High Order Maximal Principle....Pages 174-184 Optimal Control on Manifolds....Pages 185-193 Problems in Geodesic Control....Pages 194-204 Controllability in Nonlinear Systems....Pages 205-214 Control Theory in Transformation Semigroups....Pages 215-226 The Imbedding Problem for Finite Markov Chains....Pages 227-236 Some remarks on the geometry of systems....Pages 237-242 Minimal Realizations of Nonlinear Systems....Pages 243-252 Causal Dynamical Systems: Irreducible Realizations....Pages 253-262 On the Internal Structure of Bilinear Input-Output Maps....Pages 263-274 Optimal Control of Discrete Bilinear Systems....Pages 275-284 Diffusions on Manifolds Arising from Controllable Systems....Pages 285-294 Signal Detection on Lie Groups....Pages 295-304 Some Estimation Problems on Lie Groups....Pages 305-314