Controllability of some control systems on Lie groups

Via this thesis, by obtaining controllability results of affine control systems on some Lie groups, a contribution to Mathematical Control Systems has been done. This work consists of 5 chapters. In the first chapter, literature summary of problem of our thesis, the purpose of which this thesis is created and hypothesis of the problem are mentioned. In the second chapter; Lie algebra, Lie group, exponential map and adjoint representation which are base of Lie theory, are elaborately explained; and as fundamental theorems and some examples are given, basic notations of our thesis are expressed. In the third chapter, basic notations of general control systems are described in details. Variety of common control systems on Lie groups are briefly given. Then affine control system which is very significant piece for our work is approached and a characterization, which is illuminating our study, on Euclidean space is referred. Fourth chapter contains the first original part of this thesis. Derivation algebras of all 3 dimensional real matrix Lie algebras and Lie brackets of the elements from the dynamics of affine control systems on these Lie groups have been calculated. Besides, 3-dimensional Lucky Guess Lie group has been obtained and; topological and algebraic properties of this group has been examined. In addition, 3n dimensional general form of Lucky Guess Lie group has been given. Finally, classification of characterization of controllability of affine control systems has been given by using linear, invariant and bilinear control systems. Fifth chapter also contains orginal part of this thesis. Affine control system on free nilpotent Lir goup has been examined. Automorphism orbit of the affine control system on free nilpotent Lie group has been found and the controllability has been characterized realted to its associated bilinear part. Keywords: Affine control system, controllability, 3-dimensional real matrix Lie groups, free nilpotent Lie algebra, free nilpotent Lie group.