On The Controllability of Some Systems on Lie Groups


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Duman O.

1st INTERNATIONAL SYMPOSIUM ON CURRENT DEVELOPMENTS IN FUNDAMENTAL AND APPLIED MATHEMATICS SCIENCES (ISCDFAMS 2022), Erzurum, Turkey, 23 - 25 May 2022, vol.1, no.1298, pp.57

  • Publication Type: Conference Paper / Summary Text
  • Volume: 1
  • City: Erzurum
  • Country: Turkey
  • Page Numbers: pp.57
  • Yıldız Technical University Affiliated: No

Abstract

Linear control systems on matrix Lie groups have an important place in terms of their application to real-life problems [1]. By the definition in the context of Lie groups a linear control system is determined by the pair Σ = (G, D), where the state space is a real finite dimensional Lie group with the Lie algebra L(G) and the dynamic is given by the family of differential equations on G

                                                                                       g(t)=X(g(t))+Σi=1ui(t)(g(t))

where is an element of normalizer of L(G) and called drift vector field, and the control vectors 1, Y 2, . . . , Y belong to L(G), where L(G) denotes the Lie algebra of left-invariant vector fields. The control function = (u1, . . . , un) is in the class of piece-wise constant functions from [0,) to Rn. The main purpose of geometric control theory is to investigate whether it is possible to reach any other state from a given specific state in a positive time via admissible trajectories. For example, from a given initial condition x0, can a new condition xbe reached by transferring xvia the admissible control in a positive time? Considering this for disease and epidemic models, is it possible to find a medical strategy to transform an initial level of disease, at another final level of health, in a positive time [2]?  In this presentation, we work on the controllability properties of some kind of control theory problems.



Keywords: Linear control systems, matrix lie groups, dynamical systems, con- trollability.
2020 Mathematics Subject Classification: 93B05, 93C05, 22E25.