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In this paper, we study the dynamical behavior of a linear control system on $\R^2$ when the associated matrix has real eigenvalues. Different from the complex case, we show that the position of the control zero relative to the control range can have a strong interference in such dynamics if the matrix is not invertible. In the invertible case, we explicitly construct the unique control set with a nonempty interior.

The controllability issue of control-affine systems on smooth manifolds is one of the main problems in the theory, and it is recently known [Jouan P. Equivalence of control systems with linear systems on Lie groups and homogeneous spaces. ESAIM: Control Optim. Calc. Var. 2010, 16, 956-973] that it might be connected to that of a particular class of systems called linear control systems on (a homogeneous manifold of) a Lie group. Note that it may become very complicated to establish the controllability property of systems evolving on homogeneous spaces of Lie groups whose dynamics are induced by those of systems in the Lie group under consideration. In fact, even in low-dimensional certain homogeneous spaces, this is quite a challenging task, and for this reason, we have classified in [Da Silva, A., Kizil, E., Duman, O. Linear Control Systems on Homogeneous Spaces of the Heisenberg Group. J. Dyn. Control Syst. 2023, 29, 2065-2086] as a first goal all linear control systems on the homogeneous spaces of the 3-dimensional Heisenberg group $\mathbb{H}$ through its closed subgroups $L$ and, in particular, the controllability and the control sets have been studied for one of the homogeneous spaces $L\setminus \mathbb{H}$. In this paper, we study the controllability and control sets of the induced linear control systems in the homogeneous spaces left. In particular, we focus on the singularity of the induced drift vector fields that results in many cases and subcases to reveal control sets after quite a technical analysis. We give some nice illustrations to better understand what is going on geometrically., Comment: 19 pages, 6 figures

In this paper, we show that for a linear control system on a nilpotent Lie group, the Lie algebra rank condition is enough to assure the existence of a control set with a nonempty interior, as soon as the set of singularities of the drift is compact. Moreover, this control set is unique and contains the singularities of the drift in its closure.

In this article, we completely describe the control sets of one-input linear control systems on solvable, nonnilpotent 3D Lie groups. We show that, if the restriction of the associate derivation to the nilradical is nontrivial, the Lie algebra rank condition is enough to assure the existence of a control set with a nonempty interior. Moreover, such a control set is unique and, up to conjugations, given as a cylinder of the state space. On the other hand, if such a restriction is trivial, one can obtain an infinite number of control sets with empty interiors or even controllability, depending on the group considered.

In this paper we prove that automorphisms are the only isometries between rank two Almost-Riemannian Structures on the class of nonnilpotent, solvable, connected 3D Lie groups. As a consequence, a classification result for rank two ARSs on the groups in question is obtained., Comment: arXiv admin note: substantial text overlap with arXiv:2201.06414

In this paper, we analyze the chain control sets of linear control systems on connected Lie groups. Our main result shows that the compactness of the central subgroup associated with the drift is a necessary and sufficient condition to assure the uniqueness and compactness of the chain control set.

Through the Pontryagin maximum principle, we solve a minimal-time problem for a linear control system on a cylinder, considered as a homogeneous space of the solvable Lie group of dimension two. The main result explicitly shows the existence of an optimal trajectory connecting every couple of arbitrary states on the manifold. It also gives a way to calculate the corresponding minimal time. Finally, the system admits points with two distinct minimal-time trajectories connecting them.

Let H denote the 3-dimensional Heisenberg Lie group. The present paper classify all possible linear control systems on the homogeneous spaces of H through its closed subgroups and expose a detailed study on the control behavior (controllability property and control sets) of a particular dynamics evolving on a non simply connected homogeneous (state) space of dimension two.

This article describes the control behavior of any linear control systems on the group of proper motions $SE(2)$. It characterizes the controllability property and the control sets of the system.

This paper explicitly computes the unique control set $D$ with non-empty interior of a linear control system on $\mathbb{R}^2$, when the associated matrix has complex eigenvalues. It turns out that the closure of $D$ coincides with the the region delimited by a computable periodic orbit $\mathcal{O}$ of the system.

In this paper we study Almost-Riemannian Structures (ARS) on the class of nonnilpotent, solvable, conneted 3D Lie groups. The nice structures present in such groups allow us to show that the singular locus of ARSs on such groups are always embedded submanifolds.

In this paper we show that any linear vector field $\mathcal{X}$ on a connected Lie group $G$ admits a Jordan decomposition and the recurrent set of the associated ow of automorphisms is given as the intersection of the fixed points of the hyperbolic and nilpotent components of its Jordan decomposition.

The present paper shows that the bounded control set of a linear system on a connected Lie group $G$ contains all the bounded orbits of the system. As a consequence, its closure is the continuous image of the cartesian product of the set of control functions by the central subgroup associated with the drift of the system.

In this paper we introduce a weak version of level and epigraph convergence for level functions on topological spaces. In the particular case of topological groups we are able to define convolutions in the set of level functions and show that any such function is the limit in level and epigraph of robust functions.

In this paper, we introduce the concept of central periodic points of a linear system as points which lies on orbits starting and ending at the central subgroup of the system. We show that this set is bounded if and only if the central subgroup is compact. Moreover, if the system admits a control set containing the identity element of $G$ then, the set of central periodic points, coincides with its interior.