A Review of Control Sets of Linear Control Systems on Two-Dimensional Lie Groups and Applications

Abstract

This review article explores the theory of control sets for linear control systems defined on two-dimensional Lie groups, with a focus on the plane ${\mathbb{R}}^2$ and the affine group $Af{f}_{+}\left(2\right)$. We systematically summarize recent advances, emphasizing how the geometric and algebraic structures inherent in low-dimensional Lie groups influence the formation, shape, and properties of control sets—maximal regions where controllability is maintained. Control sets with non-empty interiors are of particular interest as they characterize regions where the system can be steered between states via bounded inputs. The review highlights key results concerning the existence, uniqueness, and boundedness of these sets, including criteria based on the Ad-rank condition and orbit analysis. We also underscore the central role of the symmetry properties of Lie groups, which facilitate the systematic classification and description of control sets, linking the abstract mathematical framework to concrete, physically motivated applications. To illustrate the practical relevance of the theory, we present examples from mechanics, motion planning, and neuroscience, demonstrating how control sets naturally emerge in diverse domains. Overall, this work aims to deepen the understanding of controllability regions in low-dimensional Lie group systems and to foster future research that bridges geometric control theory with applied problems.

1. Introduction

Let M be a finite-dimensional differential manifold. A control system in M is defined by the family of ordinary differential equations (ODEs),

$$\begin{array}{ccccc}& & \hfill \dot{x}\left(t\right)=f_0\left(x\left(t\right)\right)+\sum _{j=1}^m{u}_j\left(t\right)f_j\left(x\left(t\right)\right),\hspace{1em}u\in \mathcal{U},& & \hfill \left({\mathsf{\Sigma }}_M\right)\end{array}$$

determined by $u=(u_1,\dots ,u_m)$ in $\mathcal{U}$, the set of the admissible class of piecewise constant functions, with $u\left(t\right)\in \mathsf{\Omega }$. The set $\mathsf{\Omega }\subset {\mathbb{R}}^m$ is a closed and convex subset of the m-dimensional Euclidean space with $0\in int\mathsf{\Omega }$. When $\mathsf{\Omega }={\mathbb{R}}^m$ the system is called unrestricted. We consider the restricted case, i.e., when $\mathsf{\Omega }$ is compact.

Here, $f_0,f_1,\dots ,f_m$ are smooth vector fields on M, and $f_0$ is referred to as the drift, while $f_1,\dots ,f_m$ are the control vectors that influence the drift.

For any $x\in M$ and $u\in \mathcal{U}$, the solution of ${\mathsf{\Sigma }}_M$ is the integral curve $t\mapsto \varphi (t,x,u)$ on M satisfying $\varphi (0,x,u)=x$. The positive and negative orbits of ${\mathsf{\Sigma }}_M$ at x are defined as follows,

$${\mathcal{O}}^{+}\left(x\right)=\{\varphi (t,x,u),t\ge 0,u\in \mathcal{U}\}and{\mathcal{O}}^{-}\left(x\right)=\{\varphi (-t,x,u),t\ge 0,u\in \mathcal{U}\},$$

respectively. We say that ${\mathsf{\Sigma }}_M$ satisfies the Lie Algebra Rank Condition (LARC) if the Lie algebra $\mathcal{L}$ generated by the vector fields $f_0,f_1,\dots ,f_m$ satisfy

$$\mathcal{L}\left(x\right)=T_{x}M\mathrm{for}\mathrm{all}x\in M.$$

It is well known that the Lie Algebra Rank Condition (LARC) assures that the positive and negative orbits of the system have non-empty interiors. The system ${\mathsf{\Sigma }}_M$ is said to be controllable if $M={\mathcal{O}}^{+}\left(x\right)$ for all $x\in M$. Controllability is a powerful property of a control system. It means that given an initial condition x and a desired final state y, there exists a control u such that the associated integral curve $\varphi (t,x,u)$ corresponding to the ordinary differential equation determined by u transfers the first state to the second one over a positive time interval, i.e., $\varphi (0,x,u)=x$ and $\varphi (T,x,u)=y$ for some positive time T. This property is essential, for instance, when addressing optimization problems, such as minimizing time, maximizing profit, minimizing energy cost, and minimizing collateral damage. In fact, to establish the existence of a minimum time curve connecting two states, it must first be demonstrated that at least one connecting curve exists. Regarding the Chow–Rashevskii Theorem [1,2] states that if the LARC is satisfied, then there exists a metric defined on the manifold M. Specifically, any two states on M can be connected by a curve formed through vector fields in the Lie algebra of the system. The Lie brackets take into account both positive and negative time, which cannot be considered in this context.

Achieving controllability can be quite challenging, even for specific control systems acting on analytical manifolds with additional structures, like Lie groups. In the next section, we will describe the Kalman rank condition, which characterizes controllability for classical linear control systems in Euclidean spaces. It is important to note that this condition is based on the assumption that the set $\mathsf{\Omega }={\mathbb{R}}^m$, which is often unrealistic. The problem is mathematically well-posed, allowing for the determination of the limits one can expect in the restricted process. Additionally, when the control range set is bounded, the Kalman rank condition, combined with specific requirements regarding the spectrum of the drift, can offer valuable insights into controllability.

Let G be a Lie group with Lie algebra $\mathfrak{g}$. Since the landmark paper by R. W. Brockett, titled “System Theory on Group Manifolds and Coset Spaces,” published in 1972, the concept of controllability has been explored in the context of control systems on Lie groups [3].

There are two main categories of systems defined on Lie groups, distinguished by their dynamics, which arise from Abelian, nilpotent, solvable, and semisimple Lie algebras [4]: invariant systems and linear systems. For a thorough understanding of invariant systems, where the drift and control vectors are elements of $\mathfrak{g}$ considered as left-invariant vector fields, we refer to Y. Sachkov’s survey, “Controllability of Invariant Systems on Lie Groups and Homogeneous Spaces,” which summarizes results from over 40 years of research [5].

On the other hand, when the drift is linear, i.e., when its flow is a one-parameter group of G-automorphisms, and the control vectors are element of the Lie algebra $\mathfrak{g}$, we arrive at the definition of LCSs initially established for matrix groups [6] and subsequently generalized for any Lie group in [7].

In the context of Lie groups, the dynamical behavior of LCSs has been extensively studied by utilizing the inherent geometric richness found within Lie groups. See refs. [8,9,10] and references therein. In that work, the significance of this extension is demonstrated through an equivalence theorem that, in simple terms, establishes a fundamental relationship: any control-affine system on a connected manifold as ${\mathsf{\Sigma }}_M$, whose associated vector fields are complete and generate a finite Lie algebra, is diffeomorphically equivalent to an LCS on a homogeneous space [11], which is the reason why we intend to submit a manuscript for LCSs on these kinds of special manifolds.

For more recent developments related to LCSs, please consult [8,12,13,14].

Next, we introduce the notion of a control set, which is a region of the state space where controllability holds in its interior. In the following, the term “maximal” will refer to set inclusion. Additionally, $cl\left(P\right)$ will denote the topological closure of a set P.

Definition 1. 

A set $\mathcal{C}\subset M$ is a control set of ${\mathsf{\Sigma }}_M$ if it is maximal with respect to the following properties:

1. 

For every $x\in \mathcal{C}$, there exits $u\in \mathcal{U}$ such that $\varphi ({\mathbb{R}}_{+},x,u)\subset \mathcal{C}$;

2. 

For every $x\in \mathcal{C}$, it holds that $\mathcal{C}\subset cl\left({\mathcal{O}}^{+}\left(x\right)\right)$.

For general control systems defined on manifolds, several papers have focused on the existence, uniqueness, and topological properties of control sets [9,15,16]. Specifically, assume the control system satisfies LARC and let $\mathcal{C}$ be a control set with a non-empty interior. It holds that

1.

$\mathcal{C}$ is connected and $cl\left(int\mathcal{C}\right)=cl\left(\mathcal{C}\right)$;

2.

$int\mathcal{C}\subset {\mathcal{O}}^{+}\left(x\right)$.

3.

For any $x\in int\mathcal{C}$, it follows that

$$\mathcal{C}=cl\left({\mathcal{O}}^{+}\left(x\right)\right)\cap {\mathcal{O}}^{-}\left(x\right).$$

It is revealed that the controllability property holds in $int\mathcal{C}$. Extending this framework to the class of LCSs defined on Lie groups, which naturally generalize classical linear systems, our approach considers control sets as subsets of the state space where controllability holds.

In this survey, we review the literature that explicitly exhibits control sets with and without non-empty interiors. Our focus is on the class of linear control systems on two-dimensional Lie groups and their homogeneous spaces. We provide a comprehensive overview of these control sets, which include classical linear systems on the plane, as well as a linear control system on the two-dimensional solvable Lie group G. The results are based on several papers produced by our research team, including [8,12,13,17,18]. On the other hand, we include several application models. Specific examples include a planar drivetrain with a neutral mode, a planar servo with antagonist damping, a lightly damped oscillator with complex eigenvalues, and linear control on $Af{f}_{+}\left(2\right)$ demonstrating global controllability. The study also extends to applications in neuroscience, where it models orientation dynamics in the primary visual cortex (V1). Depending on parameters such as decay and modulation, the control sets range from complete controllability to conic or fiber-like regions, capturing the limits of mutual reachability in the state space. The results have practical implications for mechanics, robotics, automation, and understanding cortical response properties. We invite the reader to consider the following references related to the topics discussed in the article [19,20,21,22,23]. The particular application problems are longstanding, but we approach and model these issues using the class of LCSs. In this sense, our work provides a new perspective and makes a valuable contribution.

Furthermore, the characterization of the control set structure and boundedness often stems from the underlying symmetries of the Lie groups involved. These symmetries reflect invariant properties of the systems under group actions, which in turn dictate the geometric and algebraic structure of control sets. The invariance of control sets under certain transformations highlights the role of symmetry in simplifying the analysis and classification of the control system’s behavior. By leveraging these symmetrical properties, the results demonstrate how fundamental features of the systems remain unchanged under specific transformations, thus providing deep insights into controllability and the geometric nature of the control sets within these Lie groups.

Extending the theoretical understanding of control sets, recent developments have focused on the interplay between geometric control theory and algebraic methods, particularly in the context of sub-Riemannian geometry and nonholonomic systems [1]. Advances in differential geometry have facilitated the classification and characterization of control sets using the structure of Lie algebras and Lie group actions [21,22,24,25]. Moreover, contemporary research investigates the robustness and stability properties of control sets under perturbations, as well as their applications to robotics and quantum control [26,27]. The integration of geometric approaches with control Lyapunov functions and barrier certificates has recently yielded new conditions for controllability and constructive design of control laws [28]. These innovations continue to expand the scope of control set analysis, enabling more comprehensive coverage of complex and high-dimensional systems. The ongoing research thus enhances the foundational theories on control sets on manifolds and Lie groups, fostering a deeper understanding of their role in modern control applications.

To facilitate understanding, we include geometric visualizations of control sets and orbit behaviors. These figures illustrate key theoretical results, providing visual insights into control sets and orbit behavior on Lie groups. They help clarify abstract concepts such as controllability regions, control set boundaries, and periodic orbits. By analyzing these visualizations, readers can gain a clearer understanding of how system parameters and control inputs shape the dynamics and structure of control sets within the geometric framework.

The paper is organized as follows. In Section 2, we introduce the concepts of linear and invariant vector fields on a connected Lie group G. Section 3 presents the definition of LCSs on Lie groups and discusses controllability, specifically when the control set is the entire state space. We begin with the classical LCS on Euclidean spaces, detailing the general solution’s form and the Kalman condition for controllability. Next, we explain how to generalize the drift and control vectors from Euclidean spaces to Lie groups. We then introduce the definition of an LCS on G and present the features of its general solution. Section 4 discusses the control sets for classical LCSs in the plane, which vary based on the nature of the eigenvalues of the drift matrix A. Here, the determinant $\det A$ and the trace $\mathrm{tr}A$ are instrumental [12,13]. In Section 5, we introduce in coordinates the general form of an LCS on the solvable Lie group of dimension two $G={\mathrm{Aff}}_{+}\left(2\right)$ and identify all control sets. Finally, in Section 6, we include several application models.

Notation: Throughout the paper, G denotes a connected Lie group with Lie algebra $\mathfrak{g}$, and e stands for the identity element of G. For any $x\in G$, $T_{x}G$ is the tangent space of G at x. The adjoint representation is $\mathrm{ad}\left(X\right)\left(Y\right)=[X,Y]$, and $L_g:G\to G$, $L_g\left(h\right)=gh$, with differential $d{L}_g$ the left-translation on G. Any derivation is denoted by D and a linear vector field by $\mathcal{X}$ with flow $\left\{{\phi }_t\right\}$. The exponential map is $exp:\mathfrak{g}\to G$. For control systems, $\mathsf{\Omega }\subset {\mathbb{R}}^m$ is the control range set, $\mathcal{U}$ is the class of admissible controls, ${\mathcal{O}}^{+}\left(x\right)$, ${\mathcal{O}}^{-}\left(x\right)$ are positive/negative orbits, and $\mathcal{C}\subset G$ denotes a control set.

2. Preliminaries

Control systems on Lie groups have garnered significant attention due to their rich mathematical and geometric structure, as well as their wide range of applications in robotics, physics, and neuroscience. Understanding the controllability properties and the geometry of control sets in these systems provides fundamental insights into their behavior and limitations.

Definition 2. 

A Lie group is a differentiable manifold G with a group structure, such that the analytical product and inverse maps, denoted by μ and I, read as follows

$$\begin{array}{ccccc}\mu :\hfill & G\times G⟶G\hfill & & I:\hfill & G⟶G\hfill \\ & (g,h)⟼\mu (g,h)=gh,\hfill & & & g⟼I\left(g\right)=g^{-1}\hfill \end{array}$$

The notion of Lie algebra is strongly related to the tangent space of G at the identity e. Specifically,

Definition 3. 

A Lie algebra is a finite-dimensional vector space $\mathfrak{g}$ endowed with a Lie bracket,

$$[·,·]:\mathfrak{g}\times \mathfrak{g}⟶\mathfrak{g},$$

a skew-symmetric bilinear map, i.e,

$$[X,Y]=-[Y,X],\forall X,Y\in \mathfrak{g},$$

which satisfies the Jacobi identity. That is, for any $X,Y,Z\in \mathfrak{g},$

$$[X,[Y,Z\left]\right]+[Z,[X,Y\left]\right]+[Y,[Z,X\left]\right]=0.$$

A subalgebra $\mathfrak{h}$ is a subspace of $\mathfrak{g}$ such that $[X,Y]\in \mathfrak{h}$ for $X,Y\in \mathfrak{h}$. For any X in $\mathfrak{g}$ the linear map $ad\left(X\right):\mathfrak{g}⟶\mathfrak{g}$ with $ad\left(X\right)\left(Y\right)=[X,Y]$. The map $ad$ is called the adjoint representation. The algebra $\mathfrak{g}$ is said to be

$1.$

Abelian, if $X,Y\in \mathfrak{g}$$\Rightarrow \left[X,Y\right]=0.$

$2.$

Solvable, if there exits $k\ge 1$: its derivative series stabilizes at 0:

$$0=a{d}^{\left(k\right)}\left(\mathfrak{g}\right)=\left[a{d}^{(k-1)}\left(\mathfrak{g}\right),a{d}^{(k-1)}\left(\mathfrak{g}\right)\right]\subset \dots \subset a{d}^{\left(1\right)}\left(\mathfrak{g}\right)=\left[\mathfrak{g},\mathfrak{g}\right].$$

If $\mathfrak{g}$ is Abelian or solvable, the associated Lie group G, i.e., a group whose tangent space at the identity element is isomorphic to $\mathfrak{g}$, will also be called Abelian or solvable, respectively.

Any $g\in G$ induces a left-translation diffeomorphism $L_g:G⟶G,h⟼L_g\left(h\right)=gh$, which allows us to introduce the notion of invariant vector field. In the following, G will denote a connected Lie group with Lie algebra $\mathfrak{g}$ identified with the set of left-invariant vector fields.

Definition 4. 

A vector field X on G is said to be left-invariant if for any $g\in G$,

$${\left(d{L}_g\right)}_h\left(X\left(h\right)\right)=X\left(gh\right),\forall h\in G.$$

By replacing h with e, any fixed vector $X\left(e\right)$ at the identity element determines, through the derivative of the left-translation, a tangent vector at the tangent space of G at $g\in G$. In other words, $X\left(e\right)$ induces a left-invariant vector field on the group. Thus, the tangent space of G inherits a Lie algebra structure isomorphic to $\mathfrak{g}$.

Recall that a derivation is a linear map $\mathcal{D}:\mathfrak{g}\to \mathfrak{g}$ respecting Leibniz’s rule concerning the Lie brackets specifically. For any $X,Y\in \mathfrak{g},$

$$\mathcal{D}[X,Y]=[\mathcal{D}X,Y]+[X,\mathcal{D}Y].$$

Definition 5. 

A vector field $\mathcal{X}$ on G is said to be linear if its flow ${\left\{{\phi }_t\right\}}_{t\in \mathbb{R}}$ is a one-parameter subgroup of $\mathrm{Aut}\left(G\right)$, the Lie group of automorphisms of G [14].

Associated with any linear vector field $\mathcal{X}$, there is a derivation $\mathcal{D}$ of $\mathfrak{g}$ that satisfies [7]

$${\left(d{\phi }_t\right)}_e={\mathrm{e}}^{t\mathcal{D}}for all t\in \mathbb{R}.$$

It is revealed that

$${\phi }_t(expY)=exp\left({\mathrm{e}}^{t\mathcal{D}}Y\right),for all t\in \mathbb{R},Y\in \mathfrak{g}.$$

Finally, $\mathcal{D}$ is a derivation if and only if for any $t\in \mathbb{R}$, $e^{t\mathcal{D}}$ is an automorphism of $\mathfrak{g}$ [29].

3. The Definition of LCSs on Lie Groups and Controllability

In this section, we give the general notion of an LCS on a connected lie group G, with Lie algebra $\mathfrak{g}$. We begin with classical linear systems on Euclidean spaces and explain how to generalize the dynamics from ${\mathbb{R}}^n$ to G. We examine the solutions of the system and discuss general results related to controllability in Euclidean spaces, focusing on two main outcomes. Additionally, we refer to controllability results for the LCS on classes of nilpotent, solvable, and semisimple Lie groups; however, we do not provide details as this topic is beyond the scope of this paper.

The intrinsic concept of symmetry in Lie groups plays a crucial role in describing control sets for linear systems defined on these groups. Lie groups are smooth manifolds equipped with group operations compatible with the differential structure, and their symmetries manifest through group actions and automorphisms that leave specific properties invariant. These symmetries enable a natural geometric and algebraic framework for analyzing control systems as they allow the classification and characterization of control sets based on invariant properties.

The analysis of control sets in two-dimensional groups will be addressed in the following sections.

3.1. The LCSs on Euclidean Spaces

The classical LCS on the Euclidean space ${\mathbb{R}}^n$ is determined by the family of ordinary differential equations (ODEs),

$${\mathsf{\Sigma }}_{{\mathbb{R}}^n}:\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),u\in \mathcal{U}.$$

where A belongs to $\mathfrak{gl}\left(n\right)$, the Lie algebra of real matrices of order n, and B is a real matrix of order $n\times m$. The admissible class of control $\mathcal{U}$ is as before.

This model applies to a significant number of applications. See, for instance, refs. [30,31,32,33].

Consider the initial condition $x_0\in {\mathbb{R}}^n$ and the control $u\in \mathcal{U}$. The solution of the system ${\mathsf{\Sigma }}_{{\mathbb{R}}^n}$

$$\varphi (t,x_0,u)=e^{tA}\left(x_0+{\int }_0^t{e}^{-\tau A}Bu\left(\tau \right)d\tau \right),$$

satisfies the Cauchy problem $\dot{x}=Ax+Bu$, $x\left(0\right)=x_0$.

In particular, $\varphi (t,x_0,u)$ with $t\in \mathbb{R}$ describes a curve in ${\mathbb{R}}^n$ starting from $x_0$. The states of the curve are reached from $x_0$ forward and backward through the dynamics determined by the control u.

Controllability

As was mentioned, the controllability property refers to a system’s ability to transfer any initial condition to a desired state in a positive time. For the unrestricted case, i.e., when $\mathsf{\Omega }={\mathbb{R}}^m$, the Kalman rank condition [34,35] provides a criterion for testing controllability.

Let us denote by $K=(BAB{A}^{2}B\dots A^{n-1}B)$ the $n\times nm$ matrix associated with A and B of ${\mathsf{\Sigma }}_{{\mathbb{R}}^n}$.

Theorem 1. 

The unrestricted system ${\mathsf{\Sigma }}_{{\mathbb{R}}^n}$ is controllable on ${\mathbb{R}}^n\iff rank\left(K\right)=n$.

The proof of this theorem can be found in reference [33], Theorem 1.3.

The controllability result for a restricted LCS requires a condition related to the Lyapunov spectrum $Spec{\left(A\right)}_{Ly}$ of the matrix A, i.e., the set of the real parts of the eigenvalues in $Spec\left(A\right)$.

Theorem 2. 

Let ${\mathsf{\Sigma }}_{{\mathbb{R}}^n}$ be a restricted linear control system that satisfies the Kalman condition. Therefore,

$${\mathsf{\Sigma }}_{{\mathbb{R}}^n}\mathrm{is}\mathrm{controllable}\mathrm{on}{\mathbb{R}}^n\iff Spec{\left(A\right)}_{Ly}=\left\{0\right\}.$$

3.2. The LCSs on Lie Groups

Here, we follow the first article presenting the notion of an LCS on Lie groups [7]. To extend the concept of classical LCS from Euclidean spaces to any connected Lie group G with Lie algebra $\mathfrak{g}$, we highlight the following facts:

$1.$

The flow of the linear differential equation induced by the matrix A of ${\mathsf{\Sigma }}_{{\mathbb{R}}^n}$ satisfies $e^{tA}\in \mathrm{Aut}\left({\mathbb{R}}^n\right)$, $t\in \mathbb{R}$. This is why we introduce the concept of a linear vector field on G, where its flow is defined by a one-parameter group of G-automorphisms.

$2.$

Any column vector $b^j$ of the matrix $B=(b^1{b}^2\dots b^m)$ induces by translation an invariant vector field on ${\mathbb{R}}^n$. Therefore, the control vectors of an LCS defined on a Lie group G are given by the elements in its Lie algebra $\mathfrak{g}$, i.e., left-invariant vector fields on the group.

$3.$

It is important to note here the relationship between the Kalman rank condition and the following sequence of Lie brackets between the linear vector field $Ax$ and the invariant vector field b. Specifically,

$$[Ax,b]=-Ab,[Ax,[Ax,b]]=A^{2}b,[Ax,[Ax,[Ax,b]]]=-A^{3}b,\dots$$

We observe that the matrix A leaves the Abelian Lie algebra ${\mathbb{R}}^n$ invariant.

Definition 6. 

In [7], the authors introduce the notion of an LCS ${\mathsf{\Sigma }}_G$ on G as the family of ordinary differential equations,

$${\mathsf{\Sigma }}_G:\stackrel{·}{g\left(t\right)}=\mathcal{X}\left(g\left(t\right)\right)+\sum _{j=1}^m{u}_j\left(t\right)Y^j\left(g\left(t\right)\right),g\left(t\right)\in G,t\in \mathbb{R},u\in \mathcal{U},$$

parameterized by the family of admissible class of control $\mathcal{U}$ as before. In this context, $\mathcal{X}$ represents a linear vector field, meaning for any real time t its flow ${\phi }_t$ is an element of $Aut\left(G\right)$, the Lie group of G-automorphisms. And for any index $j=1,\dots ,m$, $Y^j$ is a left-invariant vector field on G.

Let us denote by $\phi (g,u,t)$ the solution of ${\mathsf{\Sigma }}_G$ associated with the control u with initial condition g at the time $t.$ It follows that [7,14]

$$\phi (g,u,t)={\mathcal{X}}_t\left(g\right)\phi (e,u,t).$$

It is worth comparing this general solution with the classical LCS. Notice that ${\mathcal{X}}_t\left(g\right)$ corresponds to $e^{tA}{x}_0$. The remaining parts of both formulas represent the solutions of the system, starting from the identity elements.

Controllability of LCSs on Lie Groups

The controllability property of an LCS on arbitrary Lie groups presents a significant challenge. In this context, we refer to results related to various classes of Lie groups, such as nilpotent [15], solvable [17,18], and semi-simple groups [24]. It is important to note that the Levi Theorem [29] provides a decomposition of any arbitrary Lie group into solvable and semi-simple components. To illustrate key examples of these groups, we mention the Heisenberg group, which is nilpotent; the group of proper motions of the Euclidean space, which is solvable; the orthogonal group, which is compact and semi-simple; and the special linear group, which is a non-compact semi-simple Lie group.

Specifically, the symmetries of the Lie groups facilitate the understanding of how control sets behave under group transformations, revealing their structure and possible configurations. They often lead to the invariance of control sets with respect to specific group actions, simplify the analysis by reducing the problem to canonical forms, and help identify the essential features that determine controllability. In essence, the intrinsic symmetry inherent in Lie groups acts as a guiding principle that shapes the formation and description of control sets, making the analysis more elegant, systematic, and deeply connected to the geometric properties of the underlying group structure.

4. The Control Sets of LCSs on the Plane

In this section, we examine the control sets of classical control systems in the plane. We draw on several references, including [22]. Again, the Kalman rank condition plays a role. Furthermore, the control sets are defined based on whether the determinant $detA$ and the trace $trA$ of the matrix A are zero or not. In this section, we follow reference [12].

A classical LCS on the plane ${\mathbb{R}}^2$ is given by the family of ODEs

$$\begin{array}{ccccc}& & \hfill \dot{v}\left(t\right)=Av\left(t\right)+u\left(t\right)b,\hspace{1em}u\left(t\right)\in \mathsf{\Omega },\hspace{1em}t\in \mathbb{R},& & \hfill \left({\mathsf{\Sigma }}_{{\mathbb{R}}^2}\right)\end{array}$$

where $A\in \mathfrak{gl}\left(2\right)$, the control range $\mathsf{\Omega }:=[u^{-},u^{+}]$ with $u^{-}<u^{+}$, and $b\in {\mathbb{R}}^2$ is a nonzero vector.

By definition, the solution of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ with the initial condition $v\in {\mathbb{R}}^2$, and control $\mathbf{u}\in \mathcal{U}$ is the absolutely continue curve $t\in \mathbb{R}\mapsto \phi (t,v,\mathbf{u})$ such that

$${\displaystyle \frac{d}{dt}}\phi (t,v,\mathbf{u})=A\phi (t,v,\mathbf{u})+\mathbf{u}\left(t\right)b,$$

and is built by the concatenation of solutions associated with constant controls.

Assume the drift is invertible, meaning $detA\ne 0$. The solution for a constant control $u\in \mathsf{\Omega }$ is given by

$$\phi (t,v,u)={\mathrm{e}}^{tA}(v-v\left(u\right))+v\left(u\right),wherev\left(u\right):=-u{A}^{-1}b$$

are the equilibrium states of the system.

Furthermore, for $v\in {\mathbb{R}}^2$ the positive and negative orbits of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$, respectively, read as

$${\mathcal{O}}^{+}\left(v\right):=\{\phi (t,v,\mathbf{u}),t\ge 0,\mathbf{u}\in \mathcal{U}\}\mathrm{a}\mathrm{n}\mathrm{d}{\mathcal{O}}^{-}\left(v\right):=\{\phi (t,v,\mathbf{u}),t\le 0,\mathbf{u}\in \mathcal{U}\}.$$

It is straightforward to show that ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ satisfies the LARC if the inner product between $Ab$ and $\theta$b is non-zero. Here, $\theta$ denotes the counter-clockwise rotation of ${\displaystyle \frac{\pi }{2}}$-degrees. Equivalently, ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ satisfies the LARC if and only if b is not an eigenvector of A. As we mentioned, according to this hypothesis, the positive and negative orbits have a non-empty interior.

In the following, we will discuss the control sets of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$. We begin by assuming that the eigenvalues of the matrix A are real. The complex case will be addressed in the following section.

4.1. When the Eigenvalues of A Are Real

Under the assumption that the drift has two real eigenvalues, our analysis will be divided based on the possible values of the determinant and trace of A.

4.1.1. The Case $detA=0$ and $trA=0.$

Since LARC implies that $\{Ab,b\}$ is a basis, we consider the matrix A written in this basis to obtain

$$A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right).$$

The solutions of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ for a constant control is given as $\phi (t,v_0,u)=v_0+t(A{v}_0+ub)+u{\displaystyle \frac{t^2}{2}}Ab.$

On the new basis, the solution of the system with constant control and initial condition $v_0$ is given by

$$\phi (t,v_0,u)=\left(x_0+t{y}_0+u{\displaystyle \frac{t^2}{2}},y_0+ut\right),where{v}_0=(x_0,y_0).$$

We are willing to exhibit the control sets in this first case. We notice that the relative position of the real number 0 with respect to the control range is highly relevant.

Theorem 3. 

If the LCS ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ satisfies the LARC and $detA=trA=0$, it holds that

(a) 

$0\in int\mathsf{\Omega }$ implies that ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ is controllable;

(b) 

$0\in \partial \mathsf{\Omega }$ implies that $\mathbb{R}·Ab$ is a continuum of one-point control sets;

(c) 

$0\notin \mathsf{\Omega }$ reveals that ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ does not admit any control set.

The proof of this theorem can be found in reference [12], Theorem 3.1.

In Figure 1a, the behavior of two orbits under different control inputs is shown in the case when $0\in \mathrm{int}\left(\mathsf{\Omega }\right)$. In the subsequent figure, the behavior of the LCS is depicted for the case where $0\notin \mathsf{\Omega }$; when the control $u=1$, the orbits exhibit increasing behavior, whereas when $u=-1$, the orbits show decreasing behavior. This indicates that, if $0\notin \mathsf{\Omega }$, the LCS does not admit any control set.

4.1.2. The Case $detA=0$ and $trA\ne 0$

Under these conditions, there exists an orthonormal basis $\{e_1,e_2\}$ of ${\mathbb{R}}^2$ where $A=\left(\begin{array}{cc}\mu & 0\\ 0& 0\end{array}\right).$ The solutions of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ for constant controls are given by

$$\phi (t,v_0,u)=\left({\mathrm{e}}^{\mu t}\left(x_0+u{\displaystyle \frac{b_1}{\mu }}\right)-u{\displaystyle \frac{b_1}{\mu }},y_0+u{b}_{2}t\right),where{v}_0=(x_0,y_0),b=(b_1,b_2).$$

Concerning the new basis, the LARC is equivalent to $b_1{b}_2\ne 0$. Next, we present the control sets.

Theorem 4. 

Assume the LCS ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ satisfies the LARC, $detA=0$ and $trA=0$. Therefore,

(a) 

$0\in int\mathsf{\Omega }$ implies that there exists a unique control set ${\mathcal{C}}_{{\mathbb{R}}^2}$ for ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$, which is unbounded and given by

$${\mathcal{C}}_{{\mathbb{R}}^2}=-{\displaystyle \frac{b_1}{\mu }}\mathsf{\Omega }\times \mathbb{R}if\mu <0and{\mathcal{C}}_{{\mathbb{R}}^2}=-{\displaystyle \frac{b_1}{\mu }}int\mathsf{\Omega }\times \mathbb{R}if\mu >0.$$

(b) 

$0\in \partial \mathsf{\Omega }$ implies that $\mathbb{R}·{\mathbf{e}}_2$ is a continuum of one-point control sets.

(c) 

$0\notin \mathsf{\Omega }$ reveals that ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ does not admit any control set.

The proof of this theorem can be found in reference [12], Theorem 3.3.

Figure 2a,b illustrate the asymptotic behavior of certain orbits considering $\mu =1$ and $\mu =-1$, with controls $u=1$ and $u=-1$. As observed, when $0\in \mathrm{int}\left(\mathsf{\Omega }\right)$, there exist control sets with non-empty interiors. Furthermore, in the case $\mu >0$, the boundary of the control set is not considered.

4.1.3. The Case $detA\ne 0$

This section will analyze the case where the determinant of the matrix A is nonzero. In this scenario, we observe that the condition $0\in \mathsf{\Omega }$ does not affect the system’s behavior, and this behavior is also independent of the trace of A. The analysis will be structured according to the possible signs of $detA$.

The case $detA<0$.

By assumption, the eigenvalues of A are real. So, $detA<0$ implies that in some orthonormal basis

$$A=\left(\begin{array}{cc}\mu & 0\\ 0& \lambda \end{array}\right),\mu \lambda <0,$$

the drift A is diagonalizable. Without loss of generality, we can assume that $\lambda <0<\mu$.

Theorem 5. 

Assume the classical LCS ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ satisfies the LARC and $detA<0$. Therefore, ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ admits a unique control set ${\mathcal{C}}_{{\mathbb{R}}^2}$, which is bounded and given by

$${\mathcal{C}}_{{\mathbb{R}}^2}=-{\displaystyle \frac{b_1}{\mu }}int\mathsf{\Omega }\times -{\displaystyle \frac{b_2}{\lambda }}\mathsf{\Omega }.$$

The proof of this theorem can be found in reference [12], Theorem 3.6.

In [9], the authors introduce the notion of a control set and present the first example in the literature that we would like to highlight. This scenario fits perfectly as a particular case of Theorem 5.

Example 1. 

In [9] the authors consider the following system (see Figure 3),

$$\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}1\\ 1\end{array}\right)u\left(t\right),$$

(1)

where, $u\left(t\right)\in \mathsf{\Omega }=[-1,1]$. They prove that the control set is given by

$$\mathcal{C}=(-1,1)\times [-1,1].$$

The solution reads as

$$\phi (t,v_0,u)=\left({\mathrm{e}}^t{x}_0+{\int }_0^{t}u\left(\tau \right){\mathrm{e}}^{t-\tau }\mathrm{d}\tau ,{\mathrm{e}}^{-t}{y}_0+{\int }_0^{t}u\left(\tau \right){\mathrm{e}}^{\tau -t}\mathrm{d}\tau \right)$$

The case $detA>0$.

Assume the real eigenvalues of A are both positive or both negative. There exists a basis of ${\mathbb{R}}^2$ such that

$$A=\left(\begin{array}{cc}\lambda & 1\\ 0& \lambda \end{array}\right)\mathsf{o}\mathsf{r}A=\left(\begin{array}{cc}\lambda & 0\\ 0& \mu \end{array}\right),\lambda \mu >0.$$

Assume the eigenvalues of A are negative. The positive case is analogous. We get the following:

Theorem 6. 

If the LCS ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ satisfies the LARC and $detA>0$, ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ has only one control set ${\mathcal{C}}_{{\mathbb{R}}^2}:$

$$int{\mathcal{C}}_{{\mathbb{R}}^2}=\mathrm{Im}\left(f\right),$$

where f is the diffeomorphism

$$f:{(0,+\infty )}^2\to {\mathbb{R}}^2,f(s,t)=\phi (ϵs,\phi (ϵt,v\left(u^{-}\right),u^{+}),u^{-}),withϵ=\left\{\begin{array}{cc}1& iftrA<0\\ -1& iftrA>0\end{array}\right..$$

Since f is continuous and its domain is an open set in the plane, it follows that the control set has a non-empty interior. Moreover, ${\mathcal{C}}_{{\mathbb{R}}^2}$ is closed if $trA<0$ and open if $trA>0$. In the last case, the system admits two one-point control sets $\left\{v\left(u^{+}\right)\right\}$ and $\left\{v\left(u^{-}\right)\right\}$ at its boundary $\partial {\mathcal{C}}_{{\mathbb{R}}^2}$.

The proof of this theorem can be found in reference [12], Theorem 3.8.

In Figure 4, the construction of $\mathrm{Im}\left(f\right)$ is illustrated, which corresponds to the control set of the LCS. The equilibrium states $v\left(u^{+}\right)$ and $v\left(u^{-}\right)$ are also shown.

In Figure 5, the geometric behavior of some orbits of the LCS with

$$A=\left(\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right)\mathrm{and}b=\left(\begin{array}{c}-1\\ 2\end{array}\right)$$

for controls $u=1$ and $u=-1$ is shown, where both equilibrium points, besides being control sets, act as sinks.

4.2. When the Eigenvalues of A Are Complex

Here, we follow reference [13]. For the matrix $A\in \mathfrak{gl}(2,\mathbb{R})$, let us denote by ${\sigma }_A:={(trA)}^2-4detA,$ the discriminant. Of course, A has a pair of conjugate complex eigenvalues if and only if ${\sigma }_A<0$. In this section, we assume ${\sigma }_A<0$. Moreover, fix an orthonormal basis of ${\mathbb{R}}^2$ such that

$$A=\left(\begin{array}{cc}\lambda & -\mu \\ \mu & \lambda \end{array}\right).$$

It is revealed that ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ satisfies the Kalman rank condition if and only if $b\ne 0$.

Since $detA\ne 0$, as we mentioned before, the solution of the system reads as

$$\phi (t,v,u)={\mathrm{e}}^{tA}(v-v\left(u\right))+v\left(u\right),wherev\left(u\right):=-u{A}^{-1}b,$$

are the singularities of the system, when $u\in \mathsf{\Omega }$.

Furthermore, for any constant control the solution of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ is a spiral ${\phi }_A(t,v,v\left(u\right))$ if $trA\ne 0$ and lies on circumferences if $trA=0$. Please see the Appendix of [13].

Since b cannot be zero, we obtain a complete characterization of the control sets of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ by considering the possibilities for the trace of the matrix A. It is worth noting that $0\in int\mathsf{\Omega }$ does not play a role here.

4.2.1. The Case $detA\ne 0$ and $trA=0$

The solutions of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ for constant controls have the form

$$\phi (t,v,u)=R_{t\mu }(v-v\left(u\right))+v\left(u\right),$$

where $R_{t\mu }$ is the rotation by $t\mu$ degrees, clockwise if $t\mu <0$, and counterclockwise if $t\mu >0$. and they lie on the circumferences $C_{u,v}$ with center $v\left(u\right)$ and radius $|v-v(u\left)\right|$.

Theorem 7. 

If the drift A of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ satisfies $trA=0$ and $detA>0$, then ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ is controllable.

The proof of this theorem can be found in reference [13], Theorem 2.2.

4.2.2. The Case $detA\ne 0$ and $trA\ne 0$

In this section, we show how to construct a periodic orbit for ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$, which is the boundary of the unique control set with a non-empty interior.

Assume the eigenvalues of A are $\lambda \pm \mu i$ with $\lambda <0$ and $\mu >0$. Define recurrently

$$P_0=v\left(u^{+}\right),P_{2n+1}:=\phi \left({\displaystyle \frac{\pi }{\mu }},P_{2n},u^{-}\right)and{P}_{2n+2}:=\phi \left({\displaystyle \frac{\pi }{\mu }},P_{2n+1},u^{+}\right),n\ge 0.$$

It is possible to prove that the odd and even sequences are convergent. Specifically,

$$P_{2n}\to P^{+}:=\left({\displaystyle \frac{-u^{+}+{\mathrm{e}}^{\pi \frac{\lambda }{\mu }}{u}^{-}}{1-{\mathrm{e}}^{\pi \frac{\lambda }{\mu }}}}\right)A^{-1}band{P}_{2n+1}\to P^{-}:=\left({\displaystyle \frac{-u^{-}+{\mathrm{e}}^{\pi \frac{\lambda }{\mu }}{u}^{+}}{1-{\mathrm{e}}^{\pi \frac{\lambda }{\mu }}}}\right)A^{-1}b.$$

Moreover, it is revealed that the subset of ${\mathbb{R}}^2$ given by

$$\mathcal{O}:=\left\{\phi (t,P^{+},u^{-}),t\in \left[0,{\displaystyle \frac{\pi }{\mu }}\right]\right\}\cup \left\{\phi (t,P^{-},u^{+}),t\in \left[0,{\displaystyle \frac{\pi }{\mu }}\right]\right\},$$

(2)

is a periodic orbit of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$. Let us denote by $\mathcal{C}$ the closure of the region delimited by $\mathcal{O}$.

With all this information, we are willing to establish the control sets in this context.

In Figure 6, the behavior of the orbit described in (2) is shown in (2).

The proof of the following theorem can be found in reference [13], Theorem 2.4.

Theorem 8. 

For the LCS ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$ with ${\sigma }_A<0$ and $trA\ne 0$, it holds that

(a) 

$trA<0$ implies that $\mathcal{C}$ is a control set;

(b) 

$trA>0$ implies that $int\mathcal{C}$ and $\mathcal{O}$ are the only control sets of ${\mathsf{\Sigma }}_{{\mathbb{R}}^2}$.

In Figure 7 and Figure 8, a geometric description of the control set and some orbits considering controls $u=1$ and $u=-1$ is provided. In the case $\mathrm{tr}A<0$, these controls direct the orbits toward one of the equilibrium states $v\left(u^{+}\right)$ or $v\left(u^{-}\right)$, whereas in the case $\mathrm{tr}A>0$, the equilibrium states behave like sources.

5. The LCSs on the 2-Dimensional Solvable Lie Group

Here, we follow the reference [8]. We analyze the control sets of an LCS on the 2-dimensional solvable group:

$$G={\mathrm{Aff}}_{+}\left(2\right)=\left\{\left(\begin{array}{cc}x& y\\ 0& 1\end{array}\right);(x,y)\in {\mathbb{R}}^{+}\times C\right\}.$$

In order to simplify calculations, the authors in [8] introduce the G-automorphism: $\psi :G\to G$

$$\psi (x,y)=(x,c(x-1)+dy),d\in {\mathbb{R}}^{*}.$$

which preserves linear and left-invariant vector fields and hence conjugates the LCS.

The underlying manifold of G is the open half-plane structure ${\mathbb{R}}^2$ endowed with the product

$$(x_1,y_1)·(x_2,y_2)=(x_1{x}_2,y_2+x_2{y}_1).$$

Its Lie algebra $\mathfrak{g}=\mathrm{aff}\left(2\right)$ is generated by the left-invariant vector fields $X=x{\displaystyle \frac{\partial }{\partial x}}$ and $Y=x{\displaystyle \frac{\partial }{\partial y}}$. Since $[X,Y]=Y$, $\mathfrak{g}$ is solvable and also G.

Given the basis $X,Y$, any derivation D of $\mathrm{aff}\left(2\right)$ reads as $D=\left(\begin{array}{cc}0& 0\\ a& b\end{array}\right),$ where a and b are real numbers.

The corresponding linear vector field $\mathcal{X}$ on G associated with D has the form

$$\mathcal{X}(x,y)=(0,a(x-1)+by),with(a,b)\in {\mathbb{R}}^2.$$

Moreover, any left-invariant vector field of G depends on two parameters

$$Y(x,y)=(x\alpha ,x\beta ),forsome(\alpha ,\beta )\in {\mathbb{R}}^2.$$

Definition 7. 

An LCS on G is a system of the form,

$$(\dot{x},\dot{y})=\mathcal{X}(x,y)+uY(x,y),withu\in \mathsf{\Omega }.$$

Here, $\mathcal{X}$ is linear and Y is left-invariant, where, $\mathsf{\Omega }=[u_{-},u^{+}]$ with $u_{-}<0<u^{+}$.

In coordinates, the system reads as follows

$$\begin{array}{cccc}& & \hfill \left\{\begin{array}{c}\dot{x}=u\alpha x\hfill \\ \dot{y}=a(x-1)+by+ux\beta \hfill \end{array}\right.,whereu\in \mathsf{\Omega }and(a,b),(\alpha ,\beta )\in {\mathbb{R}}^2\setminus \left\{(0,0)\right\}.\left({\mathsf{\Sigma }}_G\right)& \end{array}$$

It is straightforward to show that

$$\mathcal{L}(x,y)=\mathrm{span}\left\{\right(u\alpha x,a(x-1)+by+ux\beta ),(0,ux(a\alpha +b\beta )),u\in \mathsf{\Omega }\}.$$

And the LARC holds for $\mathsf{\Sigma }$ if and only if $\alpha (a\alpha +b\beta )\ne 0$.

Remark 1. 

Since the identity element e of G is a singularity of the drift and by hypothesis 0 belongs to the interior of omega, it is revealed that there exists a control set $\mathcal{C}$ containing the identity. Moreover, e belongs to the interior of $\mathcal{C}$ if and only if ${\mathcal{O}}^{+}\left(e\right)$ is open. Again,

$$\mathcal{C}=cl\left({\mathcal{O}}^{+}\left(e\right)\right)\cap {\mathcal{O}}^{-}\left(e\right).$$

5.1. The Case $\alpha (a\alpha +b\beta )\ne 0$

In this section, we analyze the control sets of an LCS on $G={\mathrm{Aff}}_{+}\left(2\right)$ under the LARC. The existing control set is unique and has a non-empty interior. The analysis is according to the different possibilities of b.

It is worth mentioning that in [25], the authors prove that $b=0$ and the LARC are equivalent to the controllability of an LCS. Here, we explicitly show the curves connecting any two arbitrary states.

5.1.1. The Case $b=0$

According to the hypothesis, it follows that $a\ne 0$. Therefore, the diffeomorphism

$$\psi (x,y):=(x,a^{-1}y-\beta {\alpha }^{-1}(x-1)),$$

conjugates ${\mathsf{\Sigma }}_G$ and the new linear control system:

$$\left\{\begin{array}{c}\dot{x}=u\alpha x\hfill \\ \dot{y}=x-1\hfill \end{array}\right.,whereu\in \mathsf{\Omega }.$$

(3)

The solutions of (3) starting at $(x,y)\in G$ are given by the concatenations of the flows

$$\phi (t,(x,y),u)=\left({\mathrm{e}}^{u\alpha t}x,{\displaystyle \frac{({\mathrm{e}}^{u\alpha t}-1)x}{u\alpha }}-t+y\right),t\in \mathbb{R},u\ne 0$$

and

$$\phi (t,(x,y),0)=\left(x,(x-1)t+y\right),t\in \mathbb{R},u=0.$$

Theorem 9. 

If $b=0$ the system Σ is controllable in G.

The proof of this theorem can be found in reference [8], Theorem 3.2 (See Figure 9).

5.1.2. The Case $b\ne 0$

Assume $b<0$, the other case is analogous. Through the G-diffeomorphism defined by

$$\psi (x,y):=(x,{\gamma }^{-1}(a(x-1)+by)),$$

where $\gamma =a\alpha +b\beta \ne 0$, the system ${\mathsf{\Sigma }}_G$ is conjugated to the new LCS

$$\left\{\begin{array}{c}\dot{x}=u\alpha x\hfill \\ \dot{y}=by+ux\hfill \end{array}\right.,whereu\in \mathsf{\Omega }.$$

(4)

The integral curves starting at $(x,y)\in G$ of (4) are given by concatenations of the flows

$$\begin{array}{c}\phi (t,(x,y),u)=({\mathrm{e}}^{u\alpha t}x,m_u({\mathrm{e}}^{u\alpha t}-{\mathrm{e}}^{bt})x+{\mathrm{e}}^{bt}y),for u\alpha \ne band{m}_u={\displaystyle \frac{u}{u\alpha -b}}\\ \\ and\phi (t,(x,y),b{\alpha }^{-1})=\left({\mathrm{e}}^{bt}x,{\mathrm{e}}^{tb}(y+tb{\alpha }^{-1}x)\right),t\in \mathbb{R},whenu\alpha =b.\end{array}$$

Next, we mention the main result of this section.

Theorem 10. 

If $b<0$ the unique control set of (4) is $\mathcal{C}=cl\left({\mathcal{O}}^{+}(x,y)\right)$, for any $(x,y)\in \mathcal{C}$.

The proof of this theorem can be found in reference [8], Theorem 3.6.

The following results describe all control sets of an LCS on the two-dimensional solvable Lie group. It also considers the case when the system does not satisfy the LARC.

Theorem 11. 

For the LCS Σ it holds that

1. 

$\alpha =a\alpha +b\beta =0$ and any vertical line close to $(1,0)$ is a control set;

2. 

$\alpha =0$ and $a\alpha +b\beta \ne 0$, and the control sets are vertical segments intersecting

$$\{(x,y)\in G;y=-a{b}^{-1}(x-1)\};$$

3. 

$\alpha \ne 0$ and $a\alpha +b\beta =0$, and Σ admits only the control set

$$\{(x,y)\in G;y=\beta {\alpha }^{-1}(x-1)\};$$

4. 

$\alpha (a\alpha +b\beta )\ne 0$ with $b=0$ and the unique control set is the whole G;

5. 

$\alpha (a\alpha +b\beta )\ne 0$ with $b\ne 0$ and the unique control set is a cone in G with (open) edge on the point $(0,a{b}^{-1})$.

The proof of this theorem can be found in reference [8], Theorem 3.1.

In Figure 10, with parameters for the left cone $a=-1,b=-1,\alpha =-0.1,\beta =1$, and for the right cone $a=-1,b=-0.08,\alpha =-0.1,\beta =1$, and $\mathsf{\Omega }=[-1,1]$, we observe the geometric behavior of some orbits for controls $u=1$ and $u=-1$. Moreover, both controls either send the orbits toward the boundaries of the control set or into its interior, which is highlighted in yellow.

Let us notice that items 1 and 2 of Lemma 3.4 in [8] show that $\mathcal{C}$ is a cone in G with an (open) wedge on $(0,0)\in {\mathbb{R}}^2$ (see Figure 10 below).

6. Examples Based on Control Sets

In what follows, we perform the control set analysis for selected models on ${\mathbb{R}}^2$ and on the solvable group $G=Af{f}_{+}\left(2\right)$. In every case, we fix a bounded control range $\mathsf{\Omega }=[u^{-},u^{+}]$ with $u^{-}<0<u^{+}$ and emphasize (i) the LARC (when used) and (ii) the explicit description of the control set(s) $\mathcal{C}$ given by the results in Section 4 and Section 5.

6.1. Planar Drivetrain with One Neutral Mode ($detA=0$, $trA\ne 0$)

A planar drivetrain with one neutral mode is a 2D mechanical control system whose dynamics include one zero-eigenvalue direction (persistent motion without decay) and one decaying direction. Control theory interprets it as a system that is stable in one direction and marginally stable (neutral) in another, requiring input to regulate the neutral degree of freedom [36]. Let

$$A=\left(\begin{array}{cc}-1& 0\\ 0& 0\end{array}\right),b=\left(\begin{array}{c}1\\ 1\end{array}\right),\mathsf{\Omega }=[-1,1].$$

In an eigenbasis of A, this is the case $detA=0$, $trA=\mu =-1\ne 0$. LARC holds since $b_1{b}_2\ne 0$. By Theorem 4(a) with $\mu <0$,

$${\mathcal{C}}_{{\mathbb{R}}^2}=\left(-\frac{b_1}{\mu }\mathsf{\Omega }\right)\times \mathbb{R}=\mathsf{\Omega }\times \mathbb{R}.$$

Application meaning. The first coordinate (damped mode) is fully controllable within a bounded interval, while the neutral mode is transitively reachable along vertical fibers.

6.2. Planar Servo with Antagonist Damping (Real Eigenvalues, $detA<0$)

In a human elbow joint, the biceps flexes while the triceps extends. If both are slightly activated together, they produce antagonist damping that prevents the joint from trembling or overshooting. Or, in a robotic planar arm with two opposing motors, one motor pushes forward, and the other resists in the opposite direction. The control system introduces a velocity-proportional opposing force, effectively creating a virtual damper. This situation can be modeled through the classical LCS on ${\mathbb{R}}^2$ [37],

$$\dot{v}=Av+ub,u\in \mathsf{\Omega }=[-1,1],A=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),b=\left(\begin{array}{c}1\\ 1\end{array}\right).$$

Since b is not an eigenvector of A, LARC holds. This is the diagonal case with $\lambda <0<\mu$. By Theorem 5 the system has a unique control set with a non-empty interior,

$${\mathcal{C}}_{{\mathbb{R}}^2}=(-1,1)\times [-1,1].$$

Application meaning. The rectangle describes the maximal region in which any two admissible states (angle/velocity offsets) can be mutually reached using bounded torques.

6.3. Planar Oscillator with Complex Eigenvalues and Decay ($trA<0$)

Think of a mass on a spring with a dashpot (damper). Now forget the usual “position vs. time” plot and instead watch the system in a plane where the horizontal axis is the position and the vertical axis the velocity. Therefore, (x,v) in this state, the (phase) plane reveals exactly how the system is moving at a given moment. As time flows, the point traces a curve—its trajectory—showing how position and velocity co-evolve [19]. Let

$$A=\left(\begin{array}{cc}-1& -2\\ 2& -1\end{array}\right),b=\left(\begin{array}{c}1\\ 0\end{array}\right),\mathsf{\Omega }=[-1,1],$$

so that ${\sigma }_A={(trA)}^2-4detA<0$ and $trA<0$. As in Theorem 8, the system admits a unique control set $\mathcal{C}$ whose boundary is a periodic orbit built by alternating the constant controls $u^{\pm }=\pm 1$ over half-periods $\pi /\mu$ (here $\mu =2$):

$$\mathcal{O}=\left\{\phi (t,P^{+},u^{-}):0\le t\le \frac{\pi }{2}\right\}\cup \left\{\phi (t,P^{-},u^{+}):0\le t\le \frac{\pi }{2}\right\},$$

with

$$P^{\pm }=\left({\displaystyle \frac{-u^{\pm }+e^{\pi \frac{trA}{\mu }}{u}^{\mp }}{1-e^{\pi \frac{trA}{\mu }}}}\right)A^{-1}b=\pm {\displaystyle \frac{1+e^{-\pi /2}}{1-e^{-\pi /2}}}{A}^{-1}b,A^{-1}b={\displaystyle \frac{1}{5}}\left(\begin{array}{c}-1\\ -2\end{array}\right).$$

Then $\mathcal{C}$ is the closed region enclosed by $\mathcal{O}$ (non-empty interior).

$$int\mathcal{C}\ne \varnothing ,\partial \mathcal{C}=\mathcal{O}.$$

Application meaning. In a lightly damped planar mode with bounded actuation (e.g., piezo stage with saturation), the periodic boundary provides a constructive protocol to scan the boundary of the maximal controllable domain [38].

6.4. Linear Control on $Af{f}_{+}\left(2\right)$ with $b=0$ (Global Controllability)

On $G=Af{f}_{+}\left(2\right)$ in coordinates $(x,y)$, consider

$$\dot{x}=u\alpha x,\dot{y}=a(x-1)+by+ux\beta ,\mathsf{\Omega }=[-1,1],$$

with parameters $a=1,b=0,\alpha =1,\beta =0$. Then $a\alpha +b\beta =1\ne 0$ and $\alpha \ne 0$, hence LARC holds. By Theorem 3.2 of [8] (case $b=0$),

$${\mathcal{C}}_G=G(\mathrm{the}\mathrm{whole}\mathrm{group}).$$

By analyzing the system dynamics under specific parameter conditions, the entire group is reachable from any initial state, establishing complete controllability. The theoretical results can be illustrated through concrete examples, highlighting the practical implications for systems requiring precise maneuvering. These findings have broad relevance in robotics, automation, and control engineering, where ensuring system flexibility and maneuverability is critical for operational success.

6.5. $V1$ Neuroscience Application: Control Sets for Orientation Dynamics

The primary visual cortex (V1) is the first cortical brain area responsible for processing visual information received from the retina via the thalamus. It is characterized by a highly organized architecture that encodes fundamental visual features such as orientation, spatial frequency, and motion. V1 contains orientation-selective neurons that respond preferentially to edges and lines at specific angles, forming orientation maps essential for shape perception and visual interpretation. The neural responses in V1 are modeled using mathematical frameworks that involve cortical dynamics, often leveraging geometric and control theories to understand how orientation and other visual features are represented and processed within this cortical layer. The space states of models are in Lie groups of dimension two, three, and even four. See refs. [20,21,39].

A simplified model of cortical responses in the primary visual cortex $V1$ can be framed as an LCS on the solvable Lie group $G=Af{f}_{+}\left(2\right)$, where states $(x,y)\in {\mathbb{R}}^{+}\times \mathbb{R}$ encode (i) an orientation selectivity/gain-like variable ($x>0$) and (ii) a response bias/excitability coordinate (y), respectively. External visual drive and local circuitry are modeled as a bounded scalar control $u\in \mathsf{\Omega }=[-1,1]$ acting through a left-invariant field. With the linear field $\mathcal{X}$ associated with a derivation D and a left-invariant field Y, the system reads

$$(\dot{x},\dot{y})=\mathcal{X}(x,y)+uY(x,y),\mathcal{X}(x,y)=(0,a(x-1)+by),Y(x,y)=(\alpha x,\beta x).$$

Hence,

$$\dot{x}=u\alpha x,\dot{y}=a(x-1)+by+u\beta x,u\in [-1,1],$$

(5)

with parameters $a,b,\alpha ,\beta \in \mathbb{R}$.

Control set regimes and cortical interpretation. We summarize the geometry of control sets $\mathcal{C}$ for (5) and its direct meaning for $V1$ dynamics; all statements refer to the results in Section 5.

Homogeneous propagation (no decay): $b=0$. The homogeneous propagation refers to a process where the characteristics of propagation, such as speed or intensity, are uniform throughout the medium or space. This implies that the system’s properties are invariant under spatial or temporal translations, simplifying analysis and modeling.

If $b=0$ and $a\ne 0$ and $\alpha \ne 0$, the LARC holds, and by By Theorem 3.2 of [8] the system is globally controllable:

$$\mathcal{C}=G.$$

V1 meaning: Orientation preference/gain can be steered between any two states with bounded input; there are no excluded cortical response regions.

Constrained propagation (decay): $b<0$ involves signal or wave propagation subject to specific limitations or restrictions, such as physical barriers, boundary conditions, or control constraints. These limitations influence how and where the propagation can occur, often reflecting real-world conditions.

Assume $\alpha \ne 0$ and $a\alpha +b\beta \ne 0$ (LARC). By Theorem 10 and the classification of Lemma 3.4 of [8], there is a unique control set with a non-empty interior, which is a cone in G with (open) edge at $(0,a/b)$:

$$\mathcal{C}=cl\left({\mathcal{O}}^{+}(x_0,y_0)\right)(\mathrm{any}(x_0,y_0)\in \mathcal{C}).$$

V1 meaning: Only a conic domain of orientation–gain states is mutually reachable; “blind zones’’ appear outside the cone.

Degenerate modulation: $\alpha =0$, $a\alpha +b\beta \ne 0$ describes a scenario where the modulation process (such as changing amplitude, frequency, or phase) occurs in a degenerate or singular way. This can happen due to system parameters like zero modulation depth, resonance, or singularities, causing the modulation to lose its effectiveness or become ill-defined. By Lemma 3.4 of [8] the control sets collapse to vertical segments intersecting the line $\{y=-a{b}^{-1}(x-1)\}$:

$$\mathcal{C}=\mathrm{vertical}\mathrm{segments}\mathrm{in}x>0\mathrm{along}\mathrm{invariant}\mathrm{fibers}.$$

V1 meaning: Orientation selectivity cannot be changed by inputs; dynamics are confined to 1D fibers (response-bias modulation only).

Explicit envelope of the conic control set (case (C)). Choose concrete parameters

$$a=1,b=-1,\alpha =1,\beta =0,\mathsf{\Omega }=[-1,1].$$

Then (5) becomes

$$\dot{x}=ux,\dot{y}=(x-1)-y,u\in [-1,1],$$

which satisfies LARC and $b<0$. For any constant control $u\ne -1$,

$$x\left(t\right)=e^{ut}{x}_0,y\left(t\right)={\displaystyle \frac{u}{u+1}}\left(e^{ut}-e^{-t}\right)x_0+e^{-t}{y}_0,$$

and for $u=-1$,

$$x\left(t\right)=e^{-t}{x}_0,y\left(t\right)=e^{-t}\left(y_0-t{x}_0\right).$$

Launching from $(x_0,y_0)=(1,0)$ and eliminating t yields the envelope curves generated by extreme controls:

$$y_u\left(x\right)={\displaystyle \frac{u}{u+1}}\left(x-x^{-1/u}\right),u\in (-1,1],y_{-1}\left(x\right)=xlnx.$$

Therefore, the control set is precisely the conic domain:

$$\mathcal{C}=\left\{(x,y):x>0,y_{-1}\left(x\right)\le y\le y_1\left(x\right)=\frac{1}{2}(x-x^{-1})\right\}\cup \left\{\mathrm{orbits}\mathrm{connecting}\mathrm{the}\mathrm{envelopes}\right\}.$$

V1 meaning: Stimuli and intracortical inputs can steer responses only within the cone bounded by $y=xlnx$ and $y=\frac{1}{2}(x-x^{-1})$; outside this region, states are not mutually reachable by admissible controls—mathematically capturing the “restricted propagation’’ of orientation activity.

7. Conclusions and Future Work

The study of LCSs on Lie groups reveals that their inherent geometric and algebraic structures provide a natural and robust framework for understanding controllability properties. The presence of intrinsic symmetries, reflected through automorphisms and algebraic decompositions, facilitates the characterization and classification of control sets, as well as simplifying their analysis through invariances. In particular, the rich structure of Lie groups enables the tackling of complex problems in real-world applications, ranging from robotics and mechanical systems to neuroscience, where orientation dynamics in V1 are effectively modeled within this geometric setting.

Since 2017, our research team has focused on controllability, control sets, optimality, and stability on different classes of Lie groups: nilpotent, solvable, and semisimple [7,8,12,13,14], and many others. Significant progress has been made in identifying necessary and sufficient conditions for controllability, characterizing control sets in different contexts, and analyzing optimality and stability properties of solutions. We review recent results, focusing on how the geometric and algebraic features of low-dimensional Lie groups shape the structure, form, and properties of control sets—maximal regions of controllability. Control sets with non-empty interiors are essential as they identify where the system can be steered between states with bounded inputs. Key results regarding their existence, uniqueness, and boundedness are also highlighted. However, many questions remain open and motivate future investigations:

• Extending these properties to more general classes of groups, especially arbitrary connected Lie groups and their homogeneous spaces.
• Deepening the understanding of stability and robustness of control sets, including their topological and dynamical features.
• Developing control algorithms and planning methods considering constraints and high-dimensional settings, which pose inherent geometric challenges.

The continued integration of algebraic, geometric, and topological techniques will be crucial for advancing the theoretical understanding and practical applicability of control systems on Lie groups, with potential implications across engineering, sciences, and neuroscience.