Abstract
We review recent results on control sets of linear control systems on matrix groups. We also mention some applications of systems with algebraic structures.
Highlights
This review is about control system on matrix groups and its applications
Given a matrix group G with matrix algebra g we introduce the notion of normalizer η of g, and we mention some classes of control systems inside of the normalizer which have relevant applications in many areas
We concentrate the review in the class of linear control systems on Euclidean spaces and matrix groups
Summary
This review is about control system on matrix groups and its applications. Given a matrix group G with matrix algebra g we introduce the notion of normalizer η of g, and we mention some classes of control systems inside of the normalizer which have relevant applications in many areas. Mathematical Problems in Engineering with the bracket relationships [X, Y] = Z, [L, X] = X, and [L, Y] = −Y and Z belongs to the center of g These two vectors are invariant vector fields on the space state which is a matrix group G with matrix Lie algebra g. The model of controlling the attitude of a satellite in orbit is given by the matrix group G = SO(3)⊗sR3 which is the semidirect product between the rotation group SO(3) of dimension 3 with the Euclidian space R3 [4, 5] In this case, the dynamic of the system is determined by two invariant vector fields on G, i.e., elements of the matrix Lie algebra g = so(3)⊗sR3 defined by the semidirect product between so(3) of skew-symmetric matrices of order three and R3. For specific theoretical results on linear control system on groups we refer the reader to [7] and the references therein
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