Full Symmetry Algebra for ODEs and Control Systems

摘要

A description of the full symmetry algebra (i.e., including higher symmetries) for a general nonlinear system of ordinary differential equations is given in terms of its general solution and differential constants. More precisely, the full symmetry algebra of a system is a module over the ring of its differential constants; the module is generated by partial derivatives of the general solution w.r.t. independent constants. Given a general solution, this description is both effective and explicit. Special solutions, such as an envelope of a family of solutions, are described naturally in this context. These results are extended to control systems; in this case, differential constants become operators on controls. Examples are provided.