Control sets of linear systems on Lie groups

Bounds for invariance pressure

“…Note that the linear control systems of the form (5.1) are a special case with G = R d . Their controllability properties have been analyzed in da Silva [12], Ayala, da Silva and Zsigmond [3] and Ayala and da Silva [1]. In particular, the existence and uniqueness of control sets for general systems of the form (6.1) has been analyzed in [3].…”

“…Their controllability properties have been analyzed in da Silva [12], Ayala, da Silva and Zsigmond [3] and Ayala and da Silva [1]. In particular, the existence and uniqueness of control sets for general systems of the form (6.1) has been analyzed in [3]. If 0 is in the interior of the control range U and the reachable set O + (e) from the neutral element e is open (this holds e.g.…”

“…if e ∈ intO + (e)), then there exists a control set D containing e in the interior. For semisimple or nilpotent Lie groups G sufficient conditions for boundedness of C are given in [3,Theorem 3.9], and [3, Corollary 3.12] shows uniqueness of the control set with nonvoid interior if G is decomposable. 13 Along with system (6.1) comes an associated derivation D of the Lie algebra g of G which is given by…”

Chaos on Set-Valued Dynamics and Control Sets

“…In order to approach this problem for Σ L ,asin [27] we assume here that G has finite semisimple centre, i.e. all semisimple Lie subgroups of G have finite center.…”

“…This is true if for example, the system satisfy the ad -rank condition. About the uniqueness and boundness of control sets of a linear systems, we know few things [27].…”