Ignatovych S. Method of series and free algebras in the analysis of nonlinear control systems

In the thesis, methods of analysis of nonlinear control systems are proposed and developed based on the representation of a system in the form of a series of elements of a free algebra and studying of structures in this free algebra which are generated by the system. Nonlinear real-analytic control-linear and control-affine systems are considered. For control-linear systems, the initial value problem with a fixed starting point (for convenience, it is zero) and an expansion of the end-point map into the series of iterated integrals are considered. For control-affine systems, the steering problem to a fixed equilibrium (for convenience, it is zero) and an expansion of the initial-point map into the series of nonlinear power moments are considered. In both cases, coefficients of series are constant vectors which contain all the (local) information on a concrete system while iterated integrals or nonlinear power moments do not depend on a system and form a free graded associative algebra. Coefficients of the series generate certain structures in the algebra. The central objects of the study are a core Lie subalgebra and a one-side ideal generated by the core Lie subalgebra, which are introduced in the work. It is proved that core Lie subalgebras are responsible for the homogeneous approximation, a coordinate-free definition of a homogeneous approximation is given and a complete classification of homogeneous approximations is obtained. Methods are proposed for constructing of homogeneous approximating systems and privileged coordinates. The developed approach allows solving several close problems: to find complete description of all privileged coordinates, to develop methods of construction of approximating systems, to study properties of homogeneous regular systems etc. Also, properties of core Lie subalgebras for regular and homogeneous systems are studied. A connection of a homogeneous approximation and an approximation in the sense of time optimality is studied; this allows obtaining optimal or almost optimal controls for a nonlinear system by solving much more simple time-optimal control problem for its homogeneous approximation. Besides, some problems of mappability in the class C^1 are considered. Also for systems of one special class (dual to linear systems) the set of all possible time optimal controls is completely described.