Applications of Lie Algebras to Hyperbolic and Stochastic Differential Equations, Softcover reprint of the original 1st ed. 1999 Mathematics and Its Applications Series, Vol. 466

The main part of the book is based on a one semester graduate course for students in mathematics. I have attempted to develop the theory of hyperbolic systems of differen­ tial equations in a systematic way, making as much use as possible ofgradient systems and their algebraic representation. However, despite the strong sim­ ilarities between the development of ideas here and that found in a Lie alge­ bras course this is not a book on Lie algebras. The order of presentation has been determined mainly by taking into account that algebraic representation and homomorphism correspondence with a full rank Lie algebra are the basic tools which require a detailed presentation. I am aware that the inclusion of the material on algebraic and homomorphism correspondence with a full rank Lie algebra is not standard in courses on the application of Lie algebras to hyperbolic equations. I think it should be. Moreover, the Lie algebraic structure plays an important role in integral representation for solutions of nonlinear control systems and stochastic differential equations yelding results that look quite different in their original setting. Finite-dimensional nonlin­ ear filters for stochastic differential equations and, say, decomposability of a nonlinear control system receive a common understanding in this framework.

1 Gradient Systems in a Lie Algebra.- 1.1 Preliminaries.- 1.2 Gradient systems in Fn and Der (Rn).- 1.3 Gradient Systems Determined by a Lie Algebra.- 2 Representation of a Gradient System.- 2.1 Finite-Dimensional Lie Algebra.- 2.2 The Maximal Rank Lie Algebra.- 2.3 Integral Manifolds.- 2.4 Some applications.- 3 F. G. O. Lie Algebras.- 3.1 Lie algebras finitely generated over orbits.- 3.2 Nonsingularity of the gradient system.- 3.3 Some Applications.- 4 Applications.- 4.1 Systems of Semiliniar Equations.- 4.2 Stochastic Differential Equations.- 4.3 Systems of Hyperbolic equations.- 4.4 Finite-Dimensional Nonlinear Filters.- 4.5 Affine Control Systems.- 4.6 Integral Representation of Solutions.- 4.7 Decomposition of affine control systems.- 5 Stabilization and Related Problems.- 5.1 Equivalent Controllable Systems.- 5.2 Approximations, Small Controls.- 5.3 Nonlinear Control Systems.- 5.4 Stabilization of Affine Control Systems.- 5.5 Controlled Invariant Lie Algebras.- 5.6 Stochastic differential equations.