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Many significant real world challenges arise as optimization problems on different classes of control systems. In particular, ordinary differential equations with symmetries. The purpose of this review article is twofold. First, we give the information we have about the class of Linear Control Systems ΣG on a low dimension matrix Lie group G. Second, we invite the Mathematical community to consider possible applications through the Pontryagin Maximum Principle for ΣG. In addition, we challenge some theoretical open problems. The class ΣG is a perfect generalization of the classical Linear Control System on the Abelian group Rn. Let G be a Lie group of dimension two or three. Related to ΣG, this review describes the actual results about controllability, the time-optimal Hamiltonian equations and, the Pontryagin Maximum Principle. We show how to build ΣG, through several examples on low dimensional matrix groups.

Toward Applications of Linear Control Systems on the Real World and Theoretical Challenges