Conjectures on the Stability of Linear Control Systems on Matrix Lie Groups

Abstract

Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner stability, asymptotic stability, and BIBO (Bounded Input, Bounded Output) stability, are well understood for classical linear control systems in Euclidean spaces. This paper aims to thoroughly address the stability problem for a class of linear control systems defined on matrix Lie groups. This approach generalizes classical models corresponding to the latter when the group is Abelian and non-compact. It is important to note that this generalization leads to a very difficult control system, due to the complexity of the state space and the special dynamics resulting from the drift and control vectors. Several mathematical concepts help us understand and characterize stability in the classical case. We first show how to extend these algebraic, topological, and dynamical concepts from Euclidean space to a connected Lie group of matrices. Building on classical results, we identify a pathway that enables us to formulate conjectures about stability in this broader context. This problem is closely linked to the controllability and observability properties of the system. Fortunately, these properties are well established for both classes of linear systems, whether in Euclidean spaces or on Lie groups. We are confident that these conjectures can be proved in future work, initially for the class of nilpotent and solvable groups, and later for semi-simple groups. This will provide valuable insights that will facilitate, through Jouan’s Equivalence Theorem, the analysis of an important class of nonlinear control systems on manifolds beyond Lie groups. We provide an example involving a three-dimensional solvable Lie group of rigid motions in a plane to illustrate these conjectures.

1. Introduction

A control system aims to perform actions that influence the development of a process by manipulating specific variables to achieve the desired behavior. From a mathematical point of view, this consists of a vector field known as the drift, which needs to be controlled through a set of vector fields referred to as control vectors. These vectors are activated by admissible control functions, which determine, at any given moment, which control vectors are active and to what extent within the selected strategy.

The theoretical and practical significance of the class of so-called linear control systems in Euclidean spaces is remarkable, with substantial contributions to society through various concrete applications across nearly all areas of human activity [1,2,3,4,5,6,7,8]. We mention Russia’s celebrated Pontryagin Maximum Principle [7,9].

In classical linear control systems, the drift is defined as a vector field, an ordinary differential equation, determined by a matrix A and controlled by a set of constant vectors in the state space. Thus, the associated vector field remains invariant under translations. This category of systems have been extensively studied since the early 1960s and generalized in various ambient spaces, both finite and infinite dimensional, while also considering different types of vector fields, including both drift and control vectors [1,2,3,5,6,7,8,10].

In 1999, our team introduced a generalization of classical linear control systems from Euclidean spaces over a Lie group G [11,12,13,14,15]. Here, the drift is called a linear vector field whose flux is a one-parameter group of G-automorphisms, as in the classical case. The control vectors are elements of the Lie algebra $\mathfrak{g}$ of the group, considered as left-invariant vector fields over G. The class of admissible controls is given by the space of piecewise-constant functions valued on a closed set of a Euclidean space whose dimension coincides with the number of control vectors.

As a motivational example, consider the dynamics of a cancer tumor, beginning from an initial state denoted as $x_0$, with dynamics intricately shaped by an ordinary differential equation within the state space M, representing the positive parameters volume–aggressiveness. This process is elegantly captured as the solution of $x\left(t\right)$,

$$\dot{x}\left(t\right)=f\left(x\left(t\right)\right),x\left(t\right)\in M,\mathrm{with}x\left(0\right)=x_0,$$

revealing the complex interplay of factors guiding this biological phenomenon. Several treatments are available for cancer, including surgery, chemotherapy, radiotherapy, hormonal therapy, immunotherapy, and bone marrow transplants. These treatments can be combined as control vectors to change the behavior of f using the different strategies, the controls. Determining the right combination of treatments for a patient is crucial. The introduction of specific treatments $g_1,\cdots ,g_m$, and the family of control function $u=(u_1,\dots ,u_m)\in \mathcal{U}$ define a control system that changes the behavior of the tumor, as solutions of the controlled family of differential equations

$${\Sigma }_M:\dot{x}\left(t\right)=f\left(x\left(t\right)\right)+\underset{j=1}{\stackrel{m}{\Sigma }}{u}_j\left(t\right)g_j\left(x\left(t\right)\right),x\left(t\right)\in M,u\in \mathcal{U},x\left(0\right)=x_0.$$

Here, $\mathcal{U}$ is the admissible class of control functions to be chosen. This process provides a way to combine a global tumor treatment in time.

A number of theoretical–practical problems involving algebra, topology, analysis, geometry, and differential equations appear.

Denote by $\mathcal{A}\left(x_0\right)\subset M$ the reachable set from $x_0$ through the controls $u\in \mathcal{U},$ in positive time. Some problems include the following:

-

To compute $\mathcal{A}\left(x_0\right)\subset M$. Furthermore, under what circumstances is the system ${\Sigma }_M$ controllable from $x_0$? i.e., when does $\mathcal{A}\left(x_0\right)=M?$

-

To characterize the stability properties of the system

-

To characterize the observability property of the system, meaning the ability to reconstruct its dynamics with limited information

-

Assume $x_1\in \mathcal{A}\left(x_0\right)$. Starting at $x_0,$ is it possible to reach $x_1$ with the minimum time, or with the minimum collateral damage?

As for tumor treatment problems, similar questions can be asked in the real world for many situations. From a practical point of view, given a manifold $M,$ the drift f to be controlled, and the control vectors $g_1,\cdots ,g_m,$ the admissible class of control $\mathcal{U}$ must be properly chosen according to the real situation.

The stability of control systems has been studied from various perspectives, including deterministic and stochastic approaches [5,16,17,18,19,20,21]. Deterministic systems on differential manifolds encompass systems when the manifold becomes a Lie group. There is the class of invariant control systems (ICSs) [3] and the class of linear control systems (LCSs) [22]. We also mention references [23,24] for models on Lie algebras. It is important to note that LCSs and ICSs differ significantly in several respects. The only similarity between the two is that both utilize elements from the Lie algebra of the group as control vectors to control the drift vector field. Nonetheless, the dynamics of the drift vary considerably in each case. We elaborate on these differences in detail in Section 4.1.

This manuscript aims to provide an overview of how to generalize the existing stability theory of linear control systems in classical Euclidean spaces to a Lie group. Our work is the first to analyze the stability of a LCS. Therefore, all the findings and conjectures presented in this manuscript are new.

There are several concepts of stability. Related to drift, at the end of the 19th century, H. Poincaré and I. Bendixon described the global topological properties of solutions to linear differential equations in the plane [25]. Shortly thereafter, A. Lyapunov introduced the concepts of stability and asymptotic stability for the solutions of systems of first-order differential equations [25]. He also proposed a method for analyzing these concepts and characterized them in a local context.

On the other hand, a control system is considered BIBO stable if a bounded input results in a bounded output. This property of a system means that small changes in input, initial conditions, and parameters do not lead to significant changes in the output. Moreover, a system is deemed unstable if responses grow indefinitely over time. An unstable system can lead to significant damage, both to itself and to nearby systems, and it may even pose a threat to human life; consider an elevator with an unstable control system as an example, or an unstable plane, etc.

The drift of a classical linear control system (CLCS) is a matrix. The control vectors are elements of the Euclidean space state. A CLCS is said to be internally stable if the solutions of the linear differential equations determined by the drift remain bounded as the time approaches infinity. If the output grows without limits, the system is called unstable.

The stability of a control system is closely linked to the concepts of controllability and observability, see Section 3. In this context, Theorem 4 establishes that BIBO stability is equivalent to inner asymptotic stability for a significant class of linear control systems.

The controllability property means that starting from any initial state, it is possible to transform this state into any desired state in the future using an appropriate control function. In other words, a system is controllable if the inputs can be designed to take the system from any initial state to any final state in a positive time. The system is observable if the initial state and the corresponding dynamics can be recovered from its outputs.

This manuscript is organized as follows: In Section 2, we start the article by defining a general control system on a differentiable manifold and explaining the concept of the Lie Algebra Rank Condition (LARC). According to Sussmann’s orbit theorem [26], we can assume that any control system satisfies the LARC.

In Section 3, we introduce the class of linear control systems in Euclidean spaces and discuss the concepts of controllability, observability, and stability. The existing results approached all these concepts algebraically. We use the Kalman rank condition for controllability, essentially the LARC for these systems when the controls are unbounded. Additionally, if we consider controls taking values in a compact set, we can further characterize controllability by adding to the LARC a dynamic condition related to the drift spectrum. To determine the observability, we introduce the classical observability matrix, which is determined by the drift and the output map. Due to the system’s solution structure, the control vector plays no role, even within Lie groups. We review the stability properties of classical systems, mainly focusing on the Single Input, Single Output (SISO) subclass, which has a wealth of literature detailing its stability properties. Examples suggest that internal stability is closely linked to the eigenvalue which corresponds to the drift of the linear system. In contrast, asymptotic stability relates to the overall behavior of the system.

In Section 4, we provide a comprehensive explanation of how to generalize the relevant mathematical concepts that determine the CLCS: from $G={\mathbb{R}}^n$ to an arbitrary Lie group G. We specifically address how to extend the definitions of constant control vectors and linear differential equations. To achieve this, we first introduce the concept of invariant vector fields. Subsequently, we define linear vector fields on G, with their associated derivations and a few formulas that illustrate their relationships. We conclude this section with an introduction to low-dimensional Lie groups and a discussion on the computation of linear and invariant vector fields for each case.

Section 5 introduces the concept of linear control systems on Lie groups as an extension of classical linear control systems. Following the same approach as with classical systems, we provide conditions for controllability and observability for different classes of groups, including Abelian, nilpotent, solvable, and semi-simple groups. We include examples of LCSs on nilpotent and solvable low-dimensional Lie groups.

Section 6 presents our main findings and conjectures. Building on classical results, we identify a pathway that allows us to formulate conjectures about stability in the broader context of Lie groups. This issue is closely related to the controllability and observability properties of the system, both of which are well-established concepts within group theory.

In Conjecture 1, we establish results about the inner stability of a linear control system. By utilizing the Jordan decomposition of the derivation associated with the drift, we derive Conjecture 2,which characterizes the BIBO (Bounded Input, Bounded Output) stability of an LCS. To illustrate these concepts, we provide Example 10 about the stability of a system within the group of rigid motions in the plane.

The class of Linear Control Systems on Lie groups deserves comprehensive analysis through the lens of control theory. For many years, our team and others have been working on controllability, observability, and control sets for this class of systems. Refer to the literature for further details, see references [22,27,28,29,30]. In this manuscript, we begin the exploration of stability properties for LCSs, and we hope to continue this study in depth. Given the importance of stability in any control system, the benefits of this study are evident. We are confident that these conjectures can be proved in future work, initially for nilpotent and solvable groups. However, we must acknowledge that limitations do exist for LCSs on semi-simple Lie groups, as we mention in our conjectures. This will provide valuable insights that will facilitate, through Jouan’s Equivalence Theorem [31], the analysis of an important class of nonlinear control systems on manifolds beyond Lie groups.

2. Preliminaries

For general purposes related to control theory, we refer to [2,3,5,8,18,32], and for general mathematic issues to [11,13,14].

A control system with observation $\Sigma =(M,D,h,N)$ is determined by

1. M represents the differential manifold of the state space.

2. D is the set of vector fields on M determined by the controlled family of ordinary differential equations,

$$D:\dot{x}\left(t\right)=f\left(x\left(t\right)\right)+\sum _{j=1}^m{u}_j\left(t\right)g_j\left(x\left(t\right)\right),u\in \mathcal{U}.$$

In this context, f denotes a vector field called drift, and the vector fields $g_j,j=1,\cdots ,m,$ are called control vectors. On the other hand, $\mathcal{U}$ is the set of admissible piecewise constant control functions $u:[0,T_u]⟶\Omega$ valued in a closed set $\Omega \subset {\mathbb{R}}^m,$ with $0\in int(\Omega ).$ The system is said to be unrestricted if $\Omega$ coincides with ${\mathbb{R}}^m$. Otherwise, the control system is called restricted.

3. There exists a differentiable map $h:M\to N$, where N is a manifold such that $dim\left(N\right)\le dim\left(M\right).$ A control system (without observation) is determined by $\Sigma =(M,D)$.

The Lie Algebra Rank Condition (LARC) states that the Lie algebra generated by the drift vector and the control vectors of the system coincide, at any state x of the manifold, with the tangent space of M at the point x. In simpler terms, any direction starting on x can be expressed as a linear combination of these vector fields and their Lie brackets. The orbit of the system on x is the collection of all terminal points that can be reached by the system from x through the admissible controls over positive and negative time. The Sussmann Orbit Theorem [26] demonstrates that the orbits of any family of smooth vector fields are immersed submanifolds. Specifically, if a system does not satisfy the LARC over the entire manifold, it is still possible to define the same system on the orbit $\mathcal{O}\left(x\right)$ of any x. Moreover, on this orbit, the system satisfies the Lie Algebra Rank Condition.

Our goal is to start with an analysis of the stability behavior of a special class of control systems: Linear Control Systems (LCSs) when the manifold M is a Lie group G. The dynamic of the entire system is induced by two classes of dynamics: the drift called a linear vector field on G and the control vectors, which are left-invariant vector fields on the group, see Section 4 for details. We aim to find topological, algebraic, differentiable, and geometric properties to impose on $\Sigma$ stability properties such as classical linear control systems on Euclidean spaces.

As we can see, the stability notion depends on the system’s controllability and observability properties. We will discuss these two notions for both classes of linear systems: classical systems and LCSs on Lie groups.

A control system is said to be controllable if, for any $x,y$ in M, there exists an admissible control u and a time $t>0$ such that the associated $\Sigma$-solution carries out $\phi (x,u,t)=y$.

For the next notion, we follow [27]. For a system with an observation, the indistinguishable equivalence relation associated with the data permits decomposing the manifold into subsets where two points in the same set cannot be distinguishable for D and h in $N.$ More precisely, given two distinguishable states $x,y\in M$, there exists positive times $t_1,\cdots ,t_l>0$ with $t=t_1+\cdots +$$t_l$ and a concatenation ${\phi }_t={\phi }_{t_1}\circ \cdots \circ {\phi }_{t_l}$ of flows associated with vector fields $X_1,\cdots ,X_l\in D$ such that

$$h\left({\phi }_t\left(x\right)\right)\ne h\left({\phi }_t\left(y\right)\right).$$

The system is observable if any indistinguishable class is a singleton set. In other words, any two points of M are distinguishable by $\Sigma$. Essentially, the observability property allows recovering the dynamics on M from the partial information given by $h.$ For a LCS on Euclidean spaces and groups, the previously mentioned partition of time is not necessary. This is because the observability property depends solely on one vector field, the drift, indicating that the control vectors do not play any role.

The goal of this article is two-fold: First, to compare the stability properties of the class of Linear Control Systems in two different manifolds: on the Euclidean Space ${\mathbb{R}}^n,$ and in a more general setup on a connected Lie Group G. Second, to find the symmetries involved in this theory, which allow us to conjecture the stability properties of the generalized LCS.

3. Linear Control Systems on Euclidean Spaces

To clarify concepts related to states, dynamics, and controls, we start with an example that highlights key elements of the classical theory. In the book by Pontryagin et al. [7], the authors established the following optimal problem:

How can a train stop at the station in the shortest time?

Here, as well as at the comment ending Section 3.1, we reference our paper [33]. For any $t\ge 0$, we denote by $x\left(t\right)$ the distance from the train to the station that we consider the origin. Then, $\dot{x}\left(t\right)=y,\dot{y}\left(t\right)=u\left(t\right),$ give the velocity and acceleration, respectively. The train is guided by the control u, which takes values in $[-1,1]$. Therefore, we obtain a restricted linear system:

$${\Sigma }_{\left(Lin\right)}\left({\mathbb{R}}^2\right):\left(\begin{array}{c}\dot{x}\left(t\right)\\ \dot{y}\left(t\right)\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\left(\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right)+\left(\begin{array}{c}0\\ 1\end{array}\right)u\left(t\right),u\in \mathcal{U}.$$

Geometrically, if $(x_0,y_0)\in {\mathbb{R}}^2$, we need to find a control such that the corresponding solution of the system transfers the initial condition to the origin $(0,0)$ in the minimum time. The Pontryagin Maximum Principle, which received the Lenin Prize in Russia, addresses this problem effectively [7]. Here, the controllability property of the system is fundamental. We use this example to show that the distance information recovers the optimal solution through the observability property of the system but not the velocity. In addition, the stability of these systems will be analyzed.

Let us start with the canonical coordinates of the Euclidean n-dimensional vector space ${\mathbb{R}}^n$. Any $x\in {\mathbb{R}}^n$ reads as $x=(x_1,x_2,\cdots ,x_n)$. Denote by $e_i={\left({\displaystyle \frac{\partial }{\partial x_i}}\right)}_0$ the ith element of the canonical basis of ${\mathbb{R}}^n$, $i=1,2,\cdots ,n$. The tangent space of ${\mathbb{R}}^n$ based on x, denoted by $T_x{\mathbb{R}}^n$, is the space of all vectors starting at x; that is,

$$T_x{\mathbb{R}}^n=Span\left\{{\left({\displaystyle \frac{\partial }{\partial x_i}}\right)}_x|i=1,2,\cdots ,n\right\}.$$

$T_x{\mathbb{R}}^n$ is a vectorial space of dimensión n obtained by translation of the ${\mathbb{R}}^n=T_0{\mathbb{R}}^n$ from the origin to the state x. In other words, for each $i=1,2,\cdots ,n$, ${\left({\displaystyle \frac{\partial }{\partial x_i}}\right)}_x$ denotes the vector ${\left({\displaystyle \frac{\partial }{\partial x_i}}\right)}_0$ transferred to x.

Next, we introduce the notion of the fiber bundle, which is the natural space of classical mechanics: position and momentum. In our particular case ${\mathbb{R}}^n$, the fiber bundle is denoted by $T{\mathbb{R}}^n$ and defined as the disjoint union of all tangent spaces of ${\mathbb{R}}^n$; that is, $T{\mathbb{R}}^n={\cup }_{x\in {\mathbb{R}}^n}{T}_x{\mathbb{R}}^n$.

This concept allows us to describe geometrically an ordinary differential equation. By definition, a vector field X on ${\mathbb{R}}^n$ is a differential map $X:{\mathbb{R}}^n⟶T{\mathbb{R}}^n$ determined by the election of a tangent vector $X\left(x\right)\in T_x{\mathbb{R}}^n$, for each $x\in {\mathbb{R}}^n$. We notice that there exists an isomorphism between the vector space $X\left({\mathbb{R}}^n\right)$ of all vector fields on ${\mathbb{R}}^n$ and the space $F\left({\mathbb{R}}^n\right)$ of the differential application from ${\mathbb{R}}^n$ to itself. In fact, any function $f:{\mathbb{R}}^n⟶{\mathbb{R}}^n$ induces a vector field $X^f\in X\left({\mathbb{R}}^n\right)$ defined on $x\in {\mathbb{R}}^n$ by

$$X^f\left(x\right)=\sum _{i=1}^n{f}_i\left(x\right){\left({\displaystyle \frac{\partial }{\partial x_i}}\right)}_x\in T_x{\mathbb{R}}^n,$$

where $f=(f_1,f_2,\cdots ,f_n)$. Geometrically, the correspondence $f⟶X^f$ is established by translation of the vector $f\left(x\right)$ at the state x for any $x\in {\mathbb{R}}^n$.

In the following, the notion of the constant vector field will be clear, as commented in the introduction section. Denote by $b:{\mathbb{R}}^n⟶{\mathbb{R}}^n$ the constant function equal to b. Therefore, the vector field $X^b$ applied to $x\in {\mathbb{R}}^n$ is the vector b determined by translation to the tangent space $T_x{\mathbb{R}}^n$; that is, $X^b\left(x\right)={\displaystyle \sum _{i=1}^n}{b}_i{\left({\displaystyle \frac{\partial }{\partial x_i}}\right)}_x$.

In a vector space, translations are possible, but this is no longer true for manifolds. However, in a Lie group, it is possible to define left and right translations at the group level. Its derivatives act on vector fields.

On the other hand, the solution of a linear differential equation

$$\dot{x}\left(t\right)=Ax\left(t\right),$$

with initial condition $x_0\in {\mathbb{R}}^n$ is given by the exponential map, as follows: Its flow

$$\left\{e^{tA}:t\in \mathbb{R}\right\},$$

(1)

acts on the initial condition. This flow will be crucial for the extension to Lie groups.

Next, we follow the very well-known book in [3]. A classical linear control system ${\Sigma }_{Lin}({\mathbb{R}}^n,\mathcal{D})$ on the Euclidean space ${\mathbb{R}}^n$ is determined by the dynamics of $\mathcal{D}$ coming from the family of ODE,

$$\dot{x}\left(t\right)=Ax\left(t\right)+\sum _{j=1}^m{u}_j\left(t\right)b^j,b^j\in {\mathbb{R}}^n\mathrm{and}u\in \mathcal{U}.$$

Here, $A\in \mathfrak{gl}(n,\mathbb{R})$ is the Lie algebra of the real matrices of order n. The matrix B constructed using the column vectors $b^j$, $j=1,\cdots ,m$ is called the cost matrix.

The linear differential equation $\dot{x}\left(t\right)=Ax\left(t\right)$ on ${\mathbb{R}}^n$ is controlled by m invariant constant vector fields $b^j$, $j=1,\cdots ,m$.

Given an initial condition $x_0\in {\mathbb{R}}^n$ and $u\in \mathcal{U}$, it is possible to describe the solution of the system completely, as follows:

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

which satisfies the Cauchy problem with an initial value $\dot{x}=Ax+Bu$, and $x\left(0\right)=x_0$.

Thus, ${\varphi }_t^u\left(x_0\right)$ with $t\in \mathbb{R},$ describes a curve in ${\mathbb{R}}^n$ such that, starting from $x_0$, the elements on the curve are reached from $x_0$ forward and backward through the dynamics determined by the controls.

In the following, we present the main results regarding the controllability and observability properties of classical linear control systems. The proofs can be found in [33]. For the stability properties of these systems, we refer to [5].

3.1. Controllability

The controllability property refers to a system’s ability to transfer any initial condition to a desired state in a positive time. The following result from [4] provides a criterion for testing controllability in the unrestricted case, i.e., when $\Omega ={\mathbb{R}}^m$. Let us denote by

$$K=(BAB{A}^{2}B\cdots A^{n-1}B),$$

the $n\times nm$ matrix associated with the dynamics A and B of ${\Sigma }_{Lin}\left({\mathbb{R}}^n\right)$.

Theorem 1 

([4] rank condition). The unrestricted system ${\Sigma }_{Lin}\left({\mathbb{R}}^n\right)$ is controllable ⟺ $rank\left(K\right)=n$.

According to the Cayley–Hamilton theorem, it is unnecessary to compute $A^k{b}^j$ for $j=1,2,\cdots ,m$ when $m\ge n$.

The controllability result for a restricted linear control system 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 ${\Sigma }_{Lin}\left({\mathbb{R}}^n\right)$ be a restricted linear control system that satisfies the Kalman condition. Therefore,

$${\Sigma }_{Lin}\left({\mathbb{R}}^n\right)\mathit{is}\mathit{controllable}\mathit{on}{\mathbb{R}}^n⟺Spec{\left(A\right)}_{Ly}=\left\{0\right\}.$$

In the train example, we have $rank\left(BAB\right)=2$ and $Spec\left(A\right)=\left\{0\right\}$. According to the previous Theorem, the system is controllable. Hence, given an initial condition $(x_0,y_0)$, there exists a control connecting $(x_0,y_0)$ to $(0,0)$ in positive time. Therefore, using the Pontryagin Maximum Principal, it turns out that the optimal control $u^{*}$ exists and takes the optimal values in the boundary $\partial \Omega =\{-1,1\}$. The solutions of $u=-1$ and $u=1$, are parallel parabolas.

3.2. Linear Control Systems with Observation

Next, we will explain the concept of unobservability, which allows us to break down the state space into equivalence classes. These classes group elements that cannot be distinguished from one another by the observation function h based on the dynamics of the system ${\Sigma }_{Lin}({\mathbb{R}}^n,\mathcal{D},h,{\mathbb{R}}^s)$. h is a linear transformation between vector spaces.

Definition 1. 

Two states $x_0,x_1$ in ${\mathbb{R}}^n$ are said to be indistinguishable by ${\Sigma }_{Lin}$, denoted by $x_{0}I{x}_1$, if

$$x_1-x_0\in Ker\left(C{e}^{tA}\right),t\ge 0.$$

If the equivalence class of 0 is trivial, the system is said to be observable.

We notice that the observation map h does not differentiate between states $x_0$ and $x_1$ across the control of the system, resulting in identical outcomes for each control u in $\mathcal{U}$ and for any positive time, as follows:

$$C\left(e^{tA}(x_0+{\int }_0^t{e}^{-\tau A}Bu\left(\tau \right)d\tau )\right)=C\left(e^{tA}(x_1+{\int }_0^t{e}^{-\tau A}Bu\left(\tau \right)d\tau )\right).$$

In the following, we present a fundamental result that helps classify the indistinguishable equivalence classes of ${\Sigma }_{Lin}.$

Proposition 1. 

For the system ${\Sigma }_{Lin}$, we have

1. 

I is an equivalence relation.

2. 

If $I\left(x\right)$ denotes the equivalence class of x by the relation I, then

$\left(a\right)$

$$I\left(0\right)=\bigcap _{j=1}^{n-1}Ker\left(C{A}^j\right).$$

$\left(b\right)$

Moreover, $I\left(x\right)=x+I\left(0\right)$.

Proof. 

The demonstration is straightforward.

• Since, for any $t\ge 0,$$Ker\left(C{e}^{tA}\right)$ is a vector space, it follows immediately that I is an equivalence relation.

• (a)

  By definition, $x\in I\left(0\right)⟺x\in Ker\left(C{e}^{tA}\right),\forall t\ge 0$. Since all the objects are analytical, it turns out that the previous equivalence is also true for any real time t. In particular, the analytical curve $\gamma \left(t\right)=C{e}^{tA}x=0$. Taking the derivative, we obtain $\dot{\gamma }\left(t\right)=CA{e}^{tA}x=0$. Thus, if $t=0$, we obtain $\dot{\gamma }\left(0\right)=CAx=0$, and $x\in KerCA$. The second derivative $\ddot{\gamma }\left(t\right)=C{A}^2{e}^{tA}x=0$ at $t=0$ gives $\ddot{\gamma }\left(0\right)=C{A}^{2}x=0$. Then, $x\in KerC{A}^2$. We can continue with this process. However, the Cayley–Hamilton theorem implies that $x\in {\bigcap }_{j=0}^{n-1}Ker\left(C{A}^j\right)$. The other inequality is analogous.

  (b)

  Let $x\in {\mathbb{R}}^n$ and $y\in I\left(x\right)$. So, $(y-x)\in KerC{e}^{tA}$, $\forall t\ge 0$. Therefore,

  $$C{e}^{tA}(y-x)=0,\forall t\ge 0.$$

  In particular,

  $$y-x\in \bigcap _{j=1}^{n-1}Ker\left(C{A}^j\right),$$

  and,

  $$y-x\in I\left(0\right).$$

  Thus, we obtain

  $$y=x+I\left(0\right).$$

From the previous Proposition, we obtain that the observability matrix

$$\mathcal{O}={\left(\begin{array}{ccccc}\hfill C& \hfill CA& \hfill C{A}^2& \hfill \cdots & \hfill C{A}^{n-1}\end{array}\right)}^{\top },$$

has a maximum rank if, and only if, ${\Sigma }_{Lin}$ is observable. See also [27].

In the context of the train example, let us explore the concept of observability. Observability refers to the ability to fully reconstruct the dynamics of a control system using only partial information. We can pose two key questions: Is it possible to reconstruct any initial state and its optimal trajectory by observing

• The distances traveled?
• The velocities involved in the movement?

Let us consider the train system with an observation map: $\Sigma =({\mathbb{R}}^2,\mathcal{D},\pi ,\mathbb{R}).$ Here, the optimal dynamic comes from the Pontryagin Maximum Principle with two possible controls: $u=1$ and $u=-1$, and $\pi :{\mathbb{R}}^2\to \mathbb{R}$ a linear map from the plane to the line.

The projection of the first variable enables observability, while the projection on the y-axis does not. In fact, consider ${\pi }_1(x,y)=x$. Its matrix reads $C=\left(\begin{array}{cc}1& 0\end{array}\right)$, and $CA=\left(\begin{array}{cc}0& 1\end{array}\right).$ Thus, $rank\left(\mathcal{O}\right)=2.$ So, the system is observable. For the second case, $rank\left(\mathcal{O}\right)=1.$

Therefore, from the curve of distances along the x-axis determined by the output map, it is possible to recover the initial condition and the optimal trajectory on the plane. This analysis can be understood in a straightforward geometric manner through the associated parabolas and their projections. □

3.3. Inner and BIBO Stability for SISO Systems

A control system is considered stable if it produces a bounded output. Conversely, if the output grows without limits, the system is referred to as unstable.

An unstable system can lead to significant damage, both to itself and to nearby systems, and it may even pose a threat to human life. Consider an elevator with an unstable control system as an example.

There are various definitions of stability, for our purposes, we will focus on two main ones: the inner stability and the BIBO stability.

Definition 2. 

Let us consider a single input, single output (SISO) system

$$\begin{array}{ccc}\hfill \dot{x}\left(t\right)& =& Ax\left(t\right)+bu\left(t\right)\hfill \\ \hfill y\left(t\right)& =& c^{\top }x\left(t\right)\hfill \end{array}$$

The system $\sum =(A,b,c^{\top })$ is said to be

1. 

Inner stable if

$$\underset{t\to +\infty }{\lim \sup }∥x\left(t\right)∥<+\infty ,$$

for any solution $x\left(t\right)$ of $\dot{x}\left(t\right)=Ax\left(t\right).$

2. 

Asymptotically inner stable if

$$\underset{t\to +\infty }{\lim \sup }∥x\left(t\right)∥=0,$$

for any solution $x\left(t\right)$ de $\dot{x}\left(t\right)=Ax\left(t\right).$

3. 

Inner unstable if the system is not inner stable.

We observe that the inner stability notion only depends on the matrix A.

Example 1. 

Let us consider the SISO system $\sum =(A,b,c^{\top })$ with

$$A=\left[\begin{array}{cc}0& 1\\ -b& -2a\end{array}\right]$$

such that $a^2-b<0$. The characteristic polynomial of A reads

$$\begin{array}{ccc}\hfill p_A\left(s\right)& =& s^2-2as+b,\hfill \\ \hfill s& =& -a\pm \sqrt{a^2-b}.\hfill \end{array}$$

The eigenvalues of A are $-a\pm \mu i$ with $\mu =\sqrt{\left|a^2-b\right|}.$ Therefore, modulo conjugation, the solution of $\dot{x}\left(t\right)=Ax\left(t\right)$ with initial condition $x\left(0\right)=x_0$ is given by $x\left(t\right)=e^{tA}{x}_0,$ where

$$e^{tA}=e^{-at}{R}_{\mu t}\mathit{with}{R}_{\mu t}=\left(\begin{array}{cc}cos\mu t& -sin\mu t\\ sin\mu t& cos\mu t\end{array}\right)$$

In particular,

$$∥x\left(t\right)∥=e^{-at}∥R_{\mu t}{x}_0∥=e^{-at}∥x_0∥.$$

Thus, ∑ is inner stable if $a=0,$ asymptotically inner stable if $a>0$, and unstable if $a<0$.

The following theorem summarizes the key issues related to inner stability:

Theorem 3. 

Let $\sum =(A,b,c^{\top })$ be a SISO system. Then,

1. 

∑ is inner unstable if $spec\left(A\right)\cap {\mathbb{C}}_{+}\ne ⌀$

2. 

∑ is asymptotically inner stable if $spec\left(A\right)\subset {\mathbb{C}}_{-}$

3. 

∑ is inner stable if $spec\left(A\right)\cap {\mathbb{C}}_{+}=⌀$ and $m_g\left(\lambda \right)=m_a\left(\lambda \right),$$\lambda \in spec\left(A\right)\cap \left(i\mathbb{R}\right).$

4. 

∑ is inner unstable if $m_g\left(\lambda \right)<m_a\left(\lambda \right),$ and ∀$\lambda \in spec\left(A\right)\cap \left(i\mathbb{R}\right).$

where ${\mathbb{C}}_{-}=\left\{z\in \mathbb{C};\Re \left(z\right)<0\right\},{\mathbb{C}}_{+}=\left\{z\in \mathbb{C};\Re \left(z\right)>0\right\},i\mathbb{R}=\left\{z\in \mathbb{C};\Re \left(z\right)=0\right\}.$

The second concept of stability is directly linked to the entire system.

Definition 3. 

A SISO system $\sum =(A,b,c^{\top })$ is bounded input, bounded output (BIBO) if there exists $k>0$ such that

1. 

$x\left(0\right)=0$ and

2. 

If $\left|u\left(t\right)\right|\le 1$, $\forall t\ge 0,$ then $y\left(t\right)\le k,$ where $x\left(t\right)$ is the solution of

$$\dot{x}\left(t\right)=Ax\left(t\right)+bu\left(t\right).$$

The following example illustrates that BIBO stability is weaker than inner stability:

Example 2. 

Let $\sum =(A,b,c^{\top })$ be a SISO system with

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

A direct computation gives

$$e^{tA}=\left(\begin{array}{cc}{e}^{2t}& 0\\ 0& e^{-t}\end{array}\right)andx\left(t\right)={\int }_0^t\left(\begin{array}{c}{e}^{2(t-s)}\\ e^{-(t-s)}\end{array}\right)u\left(s\right)ds.$$

Thus,

$$y\left(t\right)=\left(\begin{array}{cc}0& 1\end{array}\right){\int }_0^t\left(\begin{array}{c}{e}^{2(t-s)}\\ e^{-(t-s)}\end{array}\right)u\left(s\right)ds={\int }_0^t{e}^{-(t-s)}u\left(s\right)ds.$$

Therefore,

$$\left|y\left(t\right)\right|\le {\int }_0^t\left|e^{-(t-s)}u\left(s\right)\right|ds\stackrel{\left|u\left(s\right)\right|\le 1}{\le }{\int }_0^t{e}^{-t+s}ds=1-e^{-t}.$$

It turns out that $\left|y\left(t\right)\right|\le 1,$ and the system is BIBO stable.

We conclude this section with a result that connects inner and BIBO stability when the SISO system is both controllable and observable.

Theorem 4. 

Assume the SISO system $\sum =$$(A,b,c^{\top })$ is controllable and observable. The following are equivalent:

1. 

∑ is asymptotically inner stable.

2. 

∑ is BIBO stable.

Despite the fact that the train system is controllable, and with the projection to the first variable also being observable, we find that the exponential curve related to the matrix A, starting from any non-zero initial condition, is unbounded. Specifically, this indicates that the system is neither inner stable nor asymptotically inner stable.

4. Lie Groups and Linear Control Systems

4.1. From Euclidean Spaces to Lie Groups

For general facts about the standard Lie theory in Section 4, we refer to [11,12,13,14,15]. Moreover, for specific information about linear control systems in Section 4.1, we rely on our sources [22,33] for reference.

Let G be a connected Lie group with Lie algebra $\mathfrak{g}$ considered as the set of left-invariant vector fields on G. To simplify, when we refer to a Lie group, we are specifically talking about a matrix Lie group.

We first explain how to extend the notion of a linear differential equation from $G={\mathbb{R}}^n$ to an arbitrary Lie group G. We follow our references [22,33].

The exponential map is defined as follows:

$${exp}_G:\mathfrak{g}\to G,Y\in \mathfrak{g}\to Y_1\left(e\right).$$

(2)

where $\left\{Y_t:t\in \mathbb{R}\right\}$ is the flow associated with the left-invariant vector field induced by the vector $Y\in T_{e}G,$ and e denotes the identity element of G.

The first example of this extension was presented by [6]. In [22], the authors proposed a general definition that involves the concept of a normalizer, which is beyond the scope of this review. Therefore, we choose to provide a straightforward generalization instead. Let A be a real matrix of order n and $b\in {\mathbb{R}}^n$.

From a dynamic point of view, the flow induced by A is satisfied:

$$\left\{e^{tA}:t\in \mathbb{R}\right\}\subset Au{t}^{+}(n,\mathbb{R}).$$

(3)

Here, $Au{t}^{+}(n,\mathbb{R})$ denotes the connected component containing the identity element of $Aut\left({\mathbb{R}}^n\right),$ which is the set of real invertible matrices of order $n.$

On the other hand, from an algebraic point of view, the Lie bracket

$$\left[A,b\right]=-Ab\in {\mathbb{R}}^n,$$

It shows that $\left[A,\right]:{\mathfrak{r}}^n\to {\mathfrak{r}}^n$ leaves invariant the Lie algebra ${\mathfrak{r}}^n\cong T_0{\mathbb{R}}^n$ of ${\mathbb{R}}^n.$

For the following definition and results, we follow references [15,22].

Definition 4. 

Let G be a Lie group. Any $g\in G$ induces the diffeomorphism $L_g,R_g$ called the left and right g-translations, respectively, and defined by

$$\begin{array}{ccccc}{L}_g:\hfill & G⟶G\hfill & & R_g:\hfill & G⟶G\hfill \\ & h⟼L_g\left(h\right)=gh,\hfill & & & h⟼R_g\left(h\right)=hg.\hfill \end{array}$$

Here, d denotes the derivative of a differential function.

Definition 5. 

Let G be a Lie group. A vector field X on G is said to be

left-invariant if for any $g\in G$,

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

It turns out that left-invariant vector fields are entirely determined by their values at the identity element e of G. In other words, the Lie algebra of the group is diffeomorphic to the tangent space at e of G. The same holds for right-invariant vector fields. However, we focus solely on left-invariant vector fields since they are the control vectors in our definition of a linear control system on G.

Definition 6. 

A linear vector field $\mathcal{X}$ on G is determined by its flow $\left\{{\mathcal{X}}_t:t\in \mathbb{R}\right\}$, which is a 1-parameter subgroup of $Aut\left(G\right)$, the Lie group of G-automorphism.

Even though, in general, $\mathcal{X}$ is a nonlinear vector field, we keep the linear name based on the picture coming from the following equivalence:

$$\mathcal{X}\left(gh\right)={\left(d{L}_g\right)}_h\mathcal{X}\left(h\right)+{\left(d{R}_h\right)}_g\mathcal{X}\left(g\right),\mathrm{for}\mathrm{all}g,h\in G.$$

(4)

Recall that a derivation on a Lie algebra $(\mathfrak{g},\left[,\right])$ is a linear map $\mathcal{D}:\mathfrak{g}\to \mathfrak{g}$ which satisfies the Leibnitz rule,

$$\mathcal{D}\left[X,Y\right]=\left[\mathcal{D}X,Y\right]+\left[X,\mathcal{D}Y\right],X,Y\in \mathfrak{g}.$$

(5)

We denote by $\partial \mathfrak{g}$ the Lie algebra of $\mathfrak{g}$-derivations.

From the Jacobi identity of the Lie algebra, we can associate to each $\mathcal{X}$ an element $\mathcal{D}\in \partial \mathfrak{g}$ determined by the formula

$$\mathcal{D}Y=-[\mathcal{X},Y]\left(e\right),\mathrm{for}\mathrm{all}Y\in \mathfrak{g}.$$

For a real-time t, the relationship between ${\mathcal{X}}_t$ and $\mathcal{D}$ is given through

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

(6)

In addition, from the commutative diagram

$$\begin{array}{ccc}\mathfrak{g}& \stackrel{{\left(d{\mathcal{X}}_t\right)}_e}{⟶}& \mathfrak{g}\\ exp\downarrow & & \downarrow exp\\ G& \underset{{\mathcal{X}}_t}{⟶}& G\end{array}$$

we obtain

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

which allows computing ${\mathcal{X}}_t(expY)$. Since the group is connected, any $g\in G$ can be expressed as a finite product of exponentials, and ${\mathcal{X}}_t$ respects the algebraic structure of $G.$

Particular linear dynamics come from inner automorphisms. Any element $Z\in \mathfrak{g}$ is an invariant vector field whose solution starting on $g\in G$ is obtained by left translation of the solution through the identity element. In other words,

$$Z_t\left(g\right)={exp}_G\left(tZ\right)g,g\in G,$$

defines by conjugation a 1-parameter group of inner automorphisms on G by

$${\mathcal{X}}_t\left(x\right)=Z_{-t}\left(e\right)g{Z}_t\left(e\right),g\in G.$$

Thus, ${\mathcal{X}}_t\in Aut\left(G\right)$ for any $t\in \mathbb{R}$. In this case, the associated derivation $\mathcal{D}:\mathfrak{g}\to \mathfrak{g}$ reads $\mathcal{D}\left(Y\right)=-[Z,Y]$, Y in $\mathfrak{g}$.

The following notion is standard:

Definition 7. 

A Lie algebra $\mathfrak{g}$ is said to be

$1.$

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

$2.$

Nilpotent, if there exits $k\ge 1$: its descendant central series stabilizes at $0,$

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

$3.$

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

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

$4.$

Simple, if $\mathfrak{g}$ does not contain proper ideals and it is not Abelian.

$5.$

Semi-simple, if the largest solvable subalgebra $\mathfrak{r}\left(\mathfrak{g}\right)$ of $\mathfrak{g}$ is null.

If the Lie algebra $\mathfrak{g}$ is Abelian, nilpotent, solvable, simple, or semi-simple, the associated Lie group G is Abelian, nilpotent, solvable, simple, or semi-simple, respectively.

4.2. Examples of LCSs on Low-Dimension Lie Groups

In this section, we present examples of LCSs on low-dimensional Lie groups. We start by detailing several specific Lie groups, discussing their algebraic structures and the associated linear and invariant vector fields. All results related to these systems are derived from our papers.

4.2.1. Solvable Lie Group with Two Dimensions

Details of the following example can be found in [28].

Example 3. 

A two-dimensional affine group can be seen as a semi-direct product $G=\mathbb{R}{\times }_{\rho }\mathbb{R}$, with Lie algebra the semi-direct product $\mathfrak{g}=\mathbb{R}{\times }_{\theta }\mathbb{R}.$ Here, the actions on G and $\mathfrak{g}$ are given by ${\rho }_x=e^x$ and $\theta =I{d}_{\mathbb{R}},$ respectively. The product in G reads

$$(x_1,y_1)\ast (x_2,y_2)=(x_1+x_2,y_1+e^{x_1}{y}_2).$$

Any $(\alpha ,\beta )\in {\mathbb{R}}^2$ determines a left-invariant vector field as follows:

$$Y(x,y)=(\alpha ,e^x\beta ).$$

Furthermore, the bracket between two elements of $\mathbb{R}{\times }_{\theta }\mathbb{R}$ is

$$[({\alpha }_1,{\beta }_1),({\alpha }_2,{\beta }_2)]=(0,{\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1).$$

Let $Y^1$ and $Y^2$ be the canonical basis of $\mathfrak{g}.$ From the previous formula, we obtain the rule $\left[Y^1,Y^2\right]=Y^2.$ It follows that the Lie algebra is solvable [13]. The exponential map is explicitly given by

$${exp}_G(a,b)=\left\{\begin{array}{cc}(0,b)& ifa=0\hfill \\ \left(a,{\displaystyle \frac{1}{a}}(e^a-1)b\right)& ifa\ne 0\hfill \end{array}\right..$$

The Lie algebra of $\mathfrak{g}$-derivations is 2-dimensional. Precisely,

$$\partial \mathfrak{g}=\left\{\mathcal{D}=\left(\begin{array}{cc}0& 0\\ a& b\end{array}\right):a,b\in \mathbb{R}\right\}.$$

(7)

Since G is connected and simply connected, any pair $(a,b)\in {\mathbb{R}}^2$ induces a well-defined linear vector field on G, which reads as

$$\mathcal{X}(x,y)=(0,by+(e^x-1)a).$$

(8)

The associated 1-parameter group of automorphisms defining its flow is given by the formula

$${\mathcal{X}}_t(x,y)=\left\{\begin{array}{cc}\left(x,y+t(e^x-1)a\right)& ifb=0\\ \left(x,e^{tb}y+{\displaystyle \frac{1}{b}}(e^{tb}-1)(e^x-1)a\right)& ifb\ne 0\end{array}\right..$$

4.2.2. Heisenberg Lie Group with Three Dimensions

This example can be found in [34] see also [33].

Example 4. 

Let us consider a 3-dimensional connected and simply connected Heisenberg Lie group G homeomorphic to ${\mathbb{R}}^3,$

$$G=\left\{\left(\begin{array}{ccc}1& x& z\\ 0& 1& y\\ 0& 0& 1\end{array}\right);x,y,z\in \mathbb{R}\right\},$$

(9)

with Lie algebra

$$\mathfrak{g}=Span\left\{Y^1,Y^2,Y^3\right\},$$

(10)

where the only non-null brackets are $\left[Y^1,Y^2\right]=Y^3$. The Lie algebra $\partial \mathfrak{g}$ has six dimensions and is given by

$$\partial \mathfrak{g}=\left\{\mathcal{D}=\left(\begin{array}{ccc}a& b& 0\\ c& d& 0\\ e& f& a+d\end{array}\right):a,b,c,d,e,f\in \mathbb{R}\right\}.$$

Any invariant vector field is a linear combination of the basis of $\mathfrak{g}.$ Furthermore, according to [34], the linear vector field associated with a derivation $\mathcal{D}\in \partial \mathfrak{g}$ reads as follows:

$$\mathcal{X}(x,y,z)=(ax+dy){\displaystyle \frac{\partial }{\partial x}}+(bx+ey){\displaystyle \frac{\partial }{\partial y}}+({\displaystyle \frac{b}{2}}{x}^2+{\displaystyle \frac{d}{2}}{y}^2+cx+fy+(a+e)z){\displaystyle \frac{\partial }{\partial z}}.$$

(11)

4.2.3. A Non-Nilpotent Solvable Group with Three Dimensions

Details of the following example can be found in [29].

Example 5. 

Consider the solvable Lie algebra $\mathfrak{g}$ as a semi-direct Lie group product $\mathbb{R}{\otimes }_{\theta }{\mathbb{R}}^2$ with bracket rule

$$[(z_1,v_1),(z_2,v_2)]=(0,z_1\theta v_2-z_2\theta v_1)\in \mathfrak{g}.$$

Here, $\theta =\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right),$ see [29] for details.By considering the canonical basis of $\mathfrak{g}$, we obtain

$$\mathfrak{g}=Span\left\{Y^1=(1,0,0),Y^2=(0,1,0),Y^3=(0,0,1)\right\}.$$

Therefore, $\left[Y^1,Y^2\right]=0,$ and $[Y^1,Y^3]=Y^3.$ The connected and simply connected Lie group G with Lie algebra $\mathfrak{g}$ is given by the semi-direct product$G=\mathbb{R}{\otimes }_{\rho }{\mathbb{R}}^2$ via $\rho \left(t\right)=e^{t\theta }.\mathit{Recall}\mathit{that}$

$$e^{t\theta }=\underset{n=0}{\stackrel{\infty }{\Sigma }}{\displaystyle \frac{t^n}{n!}}{\theta }^n,{\theta }^0=Id,t\in \mathbb{R}.$$

A left-invariant vector field is written as $Y=(a,w)\in \mathfrak{g}$ with flow

$$Y_t=(a,e^{t\theta }w),t\in \mathbb{R}$$

(12)

On the other hand, the general shape of a linear vector field on G reads as

$$\mathcal{X}(t,v)=(0,{\mathcal{D}}^{*}v+{\Lambda }_t\xi ).$$

(13)

In this context, ${\mathcal{D}}^{*}$ is defined through the formula $\mathcal{D}(0,v)=(0,{\mathcal{D}}^{*}v),$ where

$$(0,\xi )=\mathcal{D}(1,0),{\Lambda }_t=\left(\begin{array}{cc}t& 0\\ 0& e^t-1\end{array}\right).$$

5. Linear Control System on Lie Groups

Our team introduced the following general notion in [22]; see also [6], which gives the first example of this class of control systems.

A linear control system ${\Sigma }_G$ on G is determined by the family

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

of differential equations parametrized by the class of admissible control $\mathcal{U}$.

The drift $\mathcal{X}$ is a linear vector field. In addition, for any $j,$ the control vector $Y_j\in \mathfrak{g}$ is considered a left-invariant vector field [22]. In Section 5.2, we also consider a homomorphism h between the Lie groups G and H as the output map, which means this application respects the product on G.

We assume ${\Sigma }_G$ satisfies the Lie algebra rank condition $\left(LARC\right)$, which means that for any $g\in G$,

$$Spa{n}_{\mathcal{L}A}\left\{\mathcal{X},Y_1,\cdots ,Y_m\right\}\left(g\right)=T_{g}G.$$

(14)

Denote by $\phi (g,u,t)$ the solution of ${\Sigma }_G$ associated with the control u with initial condition g at time t. It turns out [35] that

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

(15)

Thus, to compute the system’s solution through an initial condition g, we need to translate the solution through the identity element, using the flow of the linear vector field acting on g.

The positive and negative orbit of ${\Sigma }_G$ at $g\in G$ is defined as

$${\mathcal{O}}^{+}\left(g\right)=\{\phi (g,u,t),t\ge 0,u\in \mathcal{U}\}\mathrm{and}{\mathcal{O}}^{-}\left(g\right)=\{\phi (g,u,-t),t\ge 0,u\in \mathcal{U}\}.$$

It is interesting to note the symmetry in the solution of ${\Sigma }_G$ on G compared to the classical linear system in Euclidean space:

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

(16)

As we will see, the observability property of this class of systems depends solely on the drift and an observation map, similarly to classical linear control systems in vector spaces, see Section 5.2.

5.1. Controllability

A control system enables us to influence the behavior of a target to achieve a specific goal. The system acts as a mathematical model, providing a framework for adjusting its states to reach a designated target through various strategies known as controls.

A linear control system ${\Sigma }_G$ is said to be controllable if any two points in G can be connected through a solution of the system in positive time, that is if $G={\mathcal{O}}^{+}\left(g\right)$ for all $g\in G.$

References for studying the controllability property of linear control systems on Abelian, nilpotent, solvable, and semi-simple Lie groups are as follows: [6,22,28,29,30,34,35,36]. Several of these references suggested a geometric approach focusing on the eigenvalues of a derivation associated with the system’s drift. It has been demonstrated that a linear system is controllable if its reachable set from the identity is open and if the associated derivation only has eigenvalues with zero real parts. In certain cases, the LARC is equivalent to the ad-rank condition, which implies the openness of the reachable set, see [35].

Here, we present a characterization of the controllability property of a generic example from one of the five classes of solvable groups with three dimensions [29].

Example 6. 

Consider the linear system ${\Sigma }_G$ on the solvable three dimensional Lie group $G=\mathbb{R}{\otimes }_{\rho }{\mathbb{R}}^2$, with Lie algebra $\mathfrak{g}=\mathbb{R}{\otimes }_{\theta }{\mathbb{R}}^2$ determined by

$${\Sigma }_G:\dot{g}\left(t\right)=\mathcal{X}\left(g\left(t\right)\right)+u_1\left(t\right)Y_j\left(g\left(t\right)\right)+u_2\left(t\right)Y_2\left(g\left(t\right)\right),g\left(t\right)\in G,t\in \mathbb{R},u\in \mathcal{U},$$

which satisfy the Lie algebra rank condition.

Here, ${\rho }_t=e^{t\theta },\Delta =Span\left\{Y_1,Y_2\right\}$ has dimensions $2,$ and θ is the real matrix of order 2 with all the coefficients 0 except 1 at position $2,2.$

Theorem 5. 

The system ${\Sigma }_G$ is controllable if, and only if, dim$\left({\mathfrak{g}}^0\right)>1,$ or

$$dim\left({\mathfrak{g}}^0\right)=1\mathit{and}\left[Y_1,Y_2\right]=Y_2.$$

As for restricted classical linear systems in Euclidean spaces, the next result characterizes the controllability property on nilpotent groups.

Theorem 6. 

A restricted linear control system ${\Sigma }_G$ on a nilpotent Lie group G is controllable if, and only if, $O^{+}\left(e\right)$ is open and $G=G^0.$

We observe that Theorem 6 is a perfect generalization of Theorem 2, from an Euclidean space to a nilpotent Lie group.

5.2. Observability

In this section, we follow reference [27]. We examine a linear control system ${\Sigma }_G$, which features a homomorphism output map $h:G⟶H$ with G and H Lie groups. Similarly to the Euclidean case, the form of the solution

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

(17)

demonstrates that the observability of the system is independent of the control inputs. This conclusion arises directly from the homomorphism property of the output map.

Definition 8. 

A pair $(\mathcal{X},h)$ is said to be

1. 

Observable at $x_1$ if $\forall x_2\in G\setminus \left\{x_1\right\}$, $\exists t\ge 0$ such that,

$$h\left({\phi }_t\left(x_1\right)\right)\ne h\left({\phi }_t\left(x_2\right)\right).$$

2. 

Locally observable at $x_1$ if there exists a neighborhood of $x_1$ such that the condition 1 is satisfied for each x in the neighborhood.

3. 

Observable (locally observable) if it is observable (locally observable) for every element $x\in G$.

In other words, two states $g_1,g_2\in G$ are said to be indistinguishable if

$${\mathcal{X}}_t\left(g_1{g}_2^{-1}\right)\in K,t\ge 0.$$

Here, the kernel of h is K. Furthermore, the indistinguishable class of the identity element e reads as

$$I=\{g\in G:{\mathcal{X}}_t\left(g\right)\in K,\forall t\in \mathbb{R}\},$$

is a closed normal subgroup of G, and the equivalence class of g is $Ig.$

Next, we establish the main observability results for this class of systems (see [27], [Theorem 2.5]).

Theorem 7. 

The pair $(\mathcal{X},h)$ is observable in G if I is discrete and,

$$\mathit{Fix}\left(\phi \right)\cap K=\left\{e\right\}.$$

where $Fix\left(\phi \right)$ denotes the set of points that remain fixed under the entire flow associated with the linear vector field $\mathcal{X}$.

It is important to understand, in the previous theorem, how the first condition contributes to the concept of local observability. Specifically, a pair is locally observable if and only if I is discrete, which is equivalent to saying that its algebra vanishes.

6. Towards the Stability for Linear Control Systems on Lie Groups: The Conjectures

6.1. Inner Stability

When exploring the stability of a linear control system, similarly to the Euclidean case, our focus lies first on the flow associated with the drift. We seek to broaden the understanding of this concept.

Definition 9. 

Let ${\Sigma }_G$ be a linear control system with drift $\mathcal{X}$.

1. 

${\Sigma }_G$ is inner stable if $\forall g\in G$, the orbit ${\left\{{\phi }_t\left(g\right)\right\}}_{t\in {\mathbb{R}}^{+}}$ is bounded.

2. 

${\Sigma }_G$ is asymptotically inner stable if $\forall g\in G$, ${\phi }_t\left(g\right)\to e.$

3. 

${\Sigma }_G$ is inner unstable if it is not inner stable.

As in the Euclidean case, the eigenvalues of the derivation are closely linked to the previous concept, as illustrated in the following example:

Example 7. 

Any $A\in \mathfrak{gl}(n,\mathbb{R})$ induces a linear vector field in $Gl(n,\mathbb{R})$ defined by the Lie bracket,

$${\mathcal{X}}_A\left(g\right)=Ag-gA=[A,g]=ad\left(A\right)g.$$

Assume $A=diag({\lambda }_1,\cdots ,{\lambda }_n)$ is diagonal. The derivation associated with $\mathcal{X}$

$$D=ad\left(A\right):\mathfrak{gl}(n,\mathbb{R})⟶\mathfrak{gl}(n,\mathbb{R}),$$

is also diagonal. If $E_{ij}$ is the matrix with 1 in position $i,j$ and 0 in other places,

$$ad\left(A\right)E_{ij}=({\lambda }_i-{\lambda }_j)E_{ij}.$$

The flow of $\mathcal{X}$ is determined by conjugation,

$${\phi }_t\left(g\right)=e^{tA}g{e}^{-tA}.$$

In particular, ${\phi }_t$ is the restriction to G of the linear map in $\mathfrak{gl}(n,\mathbb{R})$ given by the same expression. If

$$g=\sum _{ij}{a}_{ij}{E}_{ij},$$

it follows that

$${\phi }_t\left(g\right)=\sum _{ij}{a}_{ij}{e}^{tA}{E}_{ij}{e}^{-tA}=\sum _{i,j}{e}^{t{\lambda }_j}{E}_{ij}{e}^{-tA}=\sum _{i,j}{e}^{t{\lambda }_j}{e}^{-t{\lambda }_i}{E}_{ij}.$$

$${\phi }_t\left(g\right)=\sum _{i,j}{e}^{t({\lambda }_j-{\lambda }_i)}{a}_{ij}{E}_{ij}.$$

(18)

$|{\phi }_t\left(g\right)|$ is bounded, $t>0$⟺$g={\sum }_{i,j}{a}_{ij}{E}_{ij}$, where $a_{ij}=0$ if ${\lambda }_j-{\lambda }_i>0$.

Moreover, if ${\Sigma }_G$ is a linear control system with drift $\mathcal{X}$, we have

1. 

${\Sigma }_G$ is inner stable if, and only if, $\forall g\in G$; $g={\sum }_{i,j}{a}_{ij}{E}_{ij}$ with $a_{ij}=0$ if ${\lambda }_i-{\lambda }_j>0$.

2. 

${\Sigma }_G$ is internally asymptotically stable if, and only if, $\forall g\in G$; $g={\sum }_{i,j}{a}_{ij}{E}_{ij}$ with $a_{ij}=0$ if ${\lambda }_i-{\lambda }_j\ge 0$.

3. 

${\Sigma }_G$ is inner unstable if, and only if, there exits $g\in G$; $g={\sum }_{i,j}{a}_{ij}{E}_{ij}$ with $a_{ij}\ne 0$ if ${\lambda }_i-{\lambda }_j>0.$

It is important to note that the internal asymptotic stability of $\mathcal{X}$ guarantees that every solution of a linear system with drift $\mathcal{X}$ and bounded control is also bounded based on the nature of the solutions.

According to the main result of the stability properties of the standard linear control systems on Euclidean spaces, we are willing to establish the first conjecture based on the spectrum of the derivation associated with the drift.

Conjecture 1. 

Let ${\Sigma }_G$ be a linear control system. We conjecture that

1. 

${\Sigma }_G$ is inner unstable if $\mathrm{Spec}\left(D\right)\cap {\mathbb{C}}^{+}\ne \varnothing .$

2. 

${\Sigma }_G$ is internal asymptotic stable if $\mathrm{Spec}\left(D\right)\subset {\mathbb{C}}^{-}.$

3. 

${\Sigma }_G$ is inner stable if $\mathrm{Spec}\left(D\right)\cap {\mathbb{C}}^{+}=\varnothing$, $m_g\left(\lambda \right)=m_a\left(\lambda \right),$$\forall \lambda \in \mathrm{Spec}\left(D\right)\cap i\mathbb{R}.$

4. 

${\Sigma }_G$ is inner unstable if $\exists \lambda \in \mathrm{Spec}\left(D\right)\cap i\mathbb{R}$ with $m_g\left(\lambda \right)<m_a\left(\lambda \right).$

Here, $m_a\left(\lambda \right)$ and $m_g\left(\lambda \right)$ denote the algebraic and geometric multiplicity of the eigenvalue $\lambda ,$ respectively.

6.2. BIBO Stability

The second concept of interest is BIBO stability, which means that through the system a bounded input gives a bounded output.

Let ${\Sigma }_G$ be a linear control system on the Lie group G, and $h:G⟶H$ a homomorphism between Lie groups. Next, we introduce the notion of BIBO stability.

Definition 10. 

${\Sigma }_G$ is BIBO stable if $\forall u\in \mathcal{U},\left|u\right(t\left)\right|\le M,\forall t>0$ the set

$${\left\{h\left(\phi (e,u,t)\right)\right\}}_{t\in {\mathbb{R}}^{+}}\mathit{is} \mathit{bounded}.$$

Similarly to the Euclidean case, BIBO stability is expected to depend on the eigenvalues of A and also on the homomorphism $h.$

Example 8. 

In the example above, let us consider a normal subgroup L invariant for ${\phi }_t$, and pick an element

$$g=\sum _{i,j}{a}_{ij}{E}_{ij},$$

such that $\exists a_{ij}\ne 0$, with ${\lambda }_i-{\lambda }_j>0.$ It follows that $g\in L.$

Since $H=G/L$ is a homogeneous space and also a Lie group, the canonical projection $\pi :G\to H,$ induces a linear control system on H. It turns out that all eigenvalues of the spectrum corresponding to the derivative of the projected drift have negative real parts. Therefore, the set

$${\left\{\pi \left(\varphi (e,u,t)\right)\right\}}_{t>0},$$

is bounded if the range of the admissible class of control $\mathcal{U}$ is bounded.

6.3. D-Decomposition and the Dynamic Subgroups

We recall the Jordan decomposition of a real matrix D through its blocks $D_i$.

If D has only real eigenvalues, the possibilities are

Case 1. $D_i={\lambda }_{i}I$ or

Case 2. $D_i=\left(\begin{array}{cccc}{\lambda }_i& 1& \cdots & 0\\ ⋮& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & 1\\ 0& \cdots & \cdots & {\lambda }_i\end{array}\right).$

If the eigenvalues are complex, we have

Case 3.

$$D_i=\left(\begin{array}{ccc}{R}_i& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & R_i\end{array}\right)\mathit{where}{R}_i=\left(\begin{array}{cc}{\lambda }_i& -{\mu }_i\\ {\mu }_i& {\lambda }_i\end{array}\right),$$

or lastly,

Case 4. $D_i=\left(\begin{array}{cccc}{R}_i& I& \cdots & 0\\ ⋮& R_i& \ddots & ⋮\\ ⋮& \ddots & \ddots & I\\ 0& \cdots & \cdots & R_i\end{array}\right),$ $I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right).$

According to these possibilities, we have

(1) $D_i={\lambda }_{i}I.$

(2) $D_i={\lambda }_{i}I+N_i,$ with $N_i=\left(\begin{array}{cccc}0& 1& \cdots & 0\\ ⋮& 0& \ddots & ⋮\\ ⋮& \ddots & \ddots & 1\\ 0& \cdots & \cdots & 0\end{array}\right).$

(3) $D_i={\lambda }_i\left(\begin{array}{cccc}1& 0& \cdots & 0\\ ⋮& 1& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \cdots & \cdots & 1\end{array}\right)+{\mu }_i\left(\begin{array}{cccc}\begin{array}{cc}0& -1\\ 1& 0\end{array}& 0& \cdots & 0\\ ⋮& \begin{array}{cc}0& -1\\ 1& 0\end{array}& \ddots & ⋮\\ ⋮& \ddots & \ddots & ⋮\\ 0& \cdots & \cdots & \begin{array}{cc}0& -1\\ 1& 0\end{array}\end{array}\right).$

(4) $D_i={\lambda }_i\left(\begin{array}{cccc}1& 0& \cdots & 0\\ ⋮& 1& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \cdots & \cdots & 1\end{array}\right)+{\mu }_i\left(\begin{array}{cccc}\theta & 0& \cdots & 0\\ ⋮& \theta & \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \cdots & \cdots & \theta \end{array}\right)+N_i,$

$$\mathit{with}\theta =\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),N_i=\left(\begin{array}{cccc}0& I& \cdots & 0\\ ⋮& 0& \ddots & ⋮\\ ⋮& \ddots & \ddots & 1\\ 0& \cdots & \cdots & 0\end{array}\right).$$

And

$$E_i=\left(\begin{array}{cccc}\begin{array}{cc}0& -1\\ 1& 0\end{array}& 0& \cdots & 0\\ ⋮& \begin{array}{cc}0& -1\\ 1& 0\end{array}& \ddots & ⋮\\ ⋮& \ddots & \ddots & ⋮\\ 0& \cdots & \cdots & \begin{array}{cc}0& -1\\ 1& 0\end{array}\end{array}\right).$$

Therefore, a basis exists that D decomposes into $D=H+N+E$. H is diagonalizable with real eigenvalues, N is nilpotent, and E is diagonalizable with pure complex eigenvalues. Furthermore,

$$[H,N]=[H,E]=[N,E]=0.$$

This is the Jordan decomposition of D. Furthermore, if D is a derivation, then $H,N$ and E are also derivations.

Let $\mathcal{X}$ be a linear vector field on G, and $D\in Der\left(\mathfrak{g}\right)$ be the corresponding derivation associated with $\mathcal{X}$ through the equation

$${\left(d{\phi }_t\right)}_I=e^{tD},\mathit{with flow} {\phi }_t.$$

The Jordan decomposition of D induces a decomposition of the associated linear vector field $\mathcal{X}$$={\mathcal{X}}_H+$${\mathcal{X}}_N$$+{\mathcal{X}}_E$, such that its flow is given by

$${\phi }_t={\phi }_t^H+{\phi }_t^N+{\phi }_t^E.$$

Since the lie brackets between H, N, and E are null, their flows commute.

As we have seen, the dynamics of the drift $\mathcal{X}$ of any linear control system has a great influence on its behavior. That is why a study of $\mathcal{X}$ becomes fundamental.

Denote by ${\mathfrak{g}}_{\lambda }$ the eigenspace of D in $\mathfrak{g}$ associated with $\lambda \in \mathbb{R}:$

$${\mathfrak{g}}_{\lambda }=\{X\in \mathfrak{g}:DX=\lambda X\}.$$

If $\lambda ,\mu \in \mathbb{R}$ satisfy $DX=\lambda X$ and $DY=\mu Y$, then

$$\begin{array}{cc}\hfill D[X,Y]& =[DX,Y]+[X,DY]\hfill \\ \hfill & =\lambda [X,Y]+\mu [X,Y]\hfill \\ \hfill & =(\lambda +\mu )[X,Y].\hfill \end{array}$$

Thus, $[{\mathfrak{g}}_{\lambda },{\mathfrak{g}}_{\mu }]\subset {\mathfrak{g}}_{\lambda +\mu }$. In particular, we obtain

$${\mathfrak{g}}^{+}=\underset{\stackrel{\lambda \in Spec\left(D\right)}{\lambda >0}}{⨁}{\mathfrak{g}}_{\lambda },{\mathfrak{g}}^{-}=\underset{\stackrel{\lambda \in Spec\left(D\right)}{\lambda <0}}{⨁}{\mathfrak{g}}_{\lambda },{\mathfrak{g}}^0=⨁KerD.$$

And if D is diagonalizable, it follows that $\mathfrak{g}={\mathfrak{g}}^{+}\oplus {\mathfrak{g}}^{-}\oplus {\mathfrak{g}}^0$, where

• ${\mathfrak{g}}^{+},{\mathfrak{g}}^{-},{\mathfrak{g}}^0$ are Lie subalgebras.
• ${\mathfrak{g}}^{+}$ and ${\mathfrak{g}}^{-}$ are nilpotent.
• $[{\mathfrak{g}}^0,{\mathfrak{g}}^{+}]\subset {\mathfrak{g}}^{+}$ and $[{\mathfrak{g}}^0,{\mathfrak{g}}^{-}]\subset {\mathfrak{g}}^{-}.$

According to the general theory of matrix dynamics,

$$e^{tD}X⟶0\mathrm{if}t⟶+\infty ,\forall X\in {\mathfrak{g}}^{-},$$

$$e^{-tD}X⟶0\mathrm{if}t⟶+\infty ,\forall X\in {\mathfrak{g}}^{+}.$$

This behavior is represented in G through ${\phi }_t$, via the conmutation formula

$${\phi }_t\left(expX\right)=exp\left(e^{tD}X\right).$$

The subalgebras ${\mathfrak{g}}^{+},{\mathfrak{g}}^0,{\mathfrak{g}}^{-}$, determine the corresponding dynamic subgroups

$$G^{+}=exp{\mathfrak{g}}^{+},G^{-}=exp{\mathfrak{g}}^{-},G^0=Fix\left(\phi \right).$$

where

$$Fix\left(\phi \right)=\{g\in G;\mathcal{X}\left(g\right)=0\}=\{g\in G;{\phi }_t\left(g\right)=g,\forall t\}.$$

In what follows, we identify several issues regarding these subgroups [35].

Theorem 8. 

With the previous notation, we have

1. 

$G^{+}\cap G^{-}=G^{+}\cap G^0=G^{-}\cap G^0=\left\{e\right\}$.

2. 

$G^{+}$, $G^{-}$, and $G^0$ are closed; $G^{+}$, $G^{-}$ are simply connected.

3. 

If $g\in G^{+}$, ${\phi }_t\left(g\right)⟶e$, $t⟶-\infty$

If $g\in G^{-}$, ${\phi }_t\left(g\right)⟶e$, $t⟶+\infty$

4. 

If $G$ is solvable,

$$G=G^{+}{G}^0{G}^{-}=G^{-}{G}^0{G}^{+}.$$

Each element of G its broken down into its components $G^{+},G^0$ and $G^{-}$.

5. 

$H\subset G$ es φ-invariant and compact, then $H\subset G^0$.

The proofs of all these facts can be found in [35].

In the following, we provide a specific example that illustrates the decomposition of Lie subalgebras and their corresponding subgroups. This example involves rotations and translations, which form the group of rigid motions in the Euclidean plane. It turns out that this group is solvable; therefore, the decomposition is guaranteed by item 4 of Theorem 8.

Example 9. 

Let us consider a solvable three-dimensional Lie group,

$$G=\left\{\left(\begin{array}{ccc}cost& -sint& a\\ sint& cost& b\\ 0& 0& 1\end{array}\right)|a,b,t\in \mathbb{R}\right\}.$$

Its structure is characterized by the semi-direct product, $S^1{\times }_{\rho }{\mathbb{R}}^2$, where

$$(t_1,v_1)·(t_2,v_2)=(t_1+t_2,v_1+\rho \left(t_1\right)v_2)with\rho \left(t\right)=\left(\begin{array}{cc}cost& -sint\\ sint& cost\end{array}\right).$$

Its Lie algebra is the semi-direct product $\mathbb{R}{\times }_{\theta }{\mathbb{R}}^2$, where

$$[(a_1,{\eta }_1);(a_2,{\eta }_2)]=(0,\theta (a_1{\eta }_2-a_2{\eta }_1))con\theta =\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right).$$

With these notations, a linear vector field on the group reads as

$$\mathcal{X}(t,v)=(0,Av+{\Delta }_t^{\theta }\xi )withA\in \mathfrak{gl}(2,\mathbb{R})and{\Delta }_t^B\eta ={\int }_0^t{e}^{sB}\eta ds.$$

The flow of $\mathcal{X}$ is given by

$${\phi }_s(t,v)=(t,e^{sA}v+{\Delta }_t^{\theta }{\Delta }_s^A\xi ),$$

and its corresponding derivation reads

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

Since $[A,\theta ]=0$, we obtain

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

The Jordan decomposition of D is

$$D=\left(\begin{array}{cc}0& 0\\ \lambda A^{-1}\xi & \lambda I\end{array}\right)+\left(\begin{array}{cc}0& 0\\ \mu A^{-1}\theta \xi & \mu \theta \end{array}\right)$$

Here,

$$P=\left(\begin{array}{cc}1& 0\\ -A^{-1}\xi & I\end{array}\right)and{P}^{-1}DP=\left(\begin{array}{cc}0& 0\\ 0& A\end{array}\right).$$

Hence, if $\lambda \ne 0$ it follows that

$$G^0=\{(t,v);\lambda v+{\Delta }_t^{\theta }\left(\lambda A^{-1}\xi \right)=0\}=\{(t,v):v=-{\Delta }_t^{\theta }{A}^{-1}\xi \}=Fix\left(\phi \right).$$

$G^0=exp{\mathfrak{g}}^0$$=\left\{(t,-{▵}_t^{\theta }{A}^{-1}\xi ),t\in S^1\right\}$, $G^{-}=\left\{0\right\}\times {\mathbb{R}}^2$.

We notice that

$${\phi }_s(t,v)=(t,e^{sA}v+{▵}_t^{\theta }{▵}_s^A\xi ),$$

so, we obtain

$${\phi }_s(t,-{▵}_t^{\theta }{A}^{-1}\xi )=(t,-{▵}_t^{\theta }{A}^{-1}\xi ),$$

since

$$\mathcal{X}(t,-{▵}_t^{\theta }{A}^{-1}\xi )=(0,\underset{=0}{\underbrace{A(-{▵}_t^{\theta }{A}^{-1}\xi )+{▵}_t^{\theta }\xi }}).$$

Moreover, $(t,v)\in G=S^1{\times }_{\rho }{\mathbb{R}}^2,$ so we obtain

$$(t,v)=(t,-{▵}_t^{\theta }{A}^{-1}\xi ).(0,{\rho }_t(v+{▵}_t^{\theta }{A}^{-1}\xi )).$$

Therefore, the decomposition $S^1{\times }_{\rho }{\mathbb{R}}^2=G^0{G}^{-}$ is unique.

Let ${\Sigma }_G$ be a linear control system defined on the group G with a drift $\mathcal{X}$ that induces the dynamic subgroups through the Jordan decomposition. In this context, we highlight several relationships between these subgroups and the accessible sets of ${\Sigma }_G$ from the identity element e of G, both positive and negative. Recall that the positive orbit from e is the set of states that can be reached by the system over positive time. In other words, for any x in $O^{+}\left(e\right)$ there exists a control u such that the solution of the system starting from the identity and following the control u reaches x in a positive amount of time. Similarly, for the negative orbit, we consider the dynamics over negative time.

Theorem 9. 

Consider the subgroups $G^{+}$, $G^{-}$, $G^0$ induced by $\mathcal{X}$ of ${\Sigma }_G$ on G. If the identity element belongs to the interior of its positive orbit, i.e., $e\in int{O}^{+}\left(e\right),$ then

1. 

$G_e^{+,0}\subset int{O}^{+}\left(e\right)$.

2. 

$G_e^{-,0}\subset int{O}^{-}\left(e\right)$.

3. 

If $G=G^0⟹{\Sigma }_G$ is controllable.

4. 

If $G^0$ is compact $⟺O^{+}\left(e\right)\cap O^{-}\left(e\right)$ is bounded.

The proofs of all these statements can be found in [35].

6.4. BIBO Conjecture for a Linear Control System ${\Sigma }_G$

Let us consider a linear control system

$${\Sigma }_G:\dot{g}=\mathcal{X}\left(g\right)+\sum u_i{Y}_i\left(g\right)\mathrm{on}G,$$

with a homomorphism $h:G⟶H$ as an output map.

The system is observable if, and only if, the indistinguishable class of the identity

$$I=\{g\in G:{\mathcal{X}}_t\left(g\right)\in K,\forall t\in \mathbb{R}\}$$

is discrete and if K denotes the kernel of h,

$$Fix\left(\phi \right)\cap K=\left\{e\right\}.$$

Assume ${\Sigma }_G$ is observable and satisfies the ad-rank condition, we believe that it is possible to demonstrate the following issues:

1. 

$G^{+}\subset I$.

2. 

${\Sigma }_G$ BIBO stable $⟹h\left(G^0\right)$ is compact.

Furthermore, if the Lie group G is not semi-simple, $G^{+,0}\subset int{\mathcal{O}}^{+}\left(e\right)$.

Finally, we establish our main conjecture which characterize the BIBO stability of a linear control system on a Lie group G.

Let ${\Sigma }_G$ be a controllable and observable linear control system. We conjecture that

$${\Sigma }_G\mathit{is}\mathit{BIBO}\mathit{stable}⟺G=G^{-,0}\mathit{and}h\left(G^0\right)\mathit{is}\mathit{compact}.$$

Example 10. 

Let us consider the linear vector field $\mathcal{X}$ as in Example 9, whose derivation D decomposes the group G of the rigid motions of the plane as $G=G^{0,-}$. Next, consider any element Y of the Lie algebra of G, i.e., a left-invariant vector field, such that the system

$${\Sigma }_G:\dot{g}=\mathcal{X}\left(g\right)+uY\left(g\right)\mathrm{on}G,$$

is controllable; for such systems, there exists [29] Take a Lie group H and a homomorphism $h:G⟶H$ as an output map such that the system is observable, for such systems, there exists [37]. Since h is a continuous function and $S^1$ is compact, then $h\left(G^0\right)$ is compact. According to Conjecture 2, ${\Sigma }_G$ is $BIBO$ stable.

The same ideas can be used for a linear control system with two outputs.

Finally, according to Section 3, the train system is controllable and observable with projection to the first variable. However, it is not BIBO stable. Simply observe that the projection of the eigenspace of 0 is not compact.

7. Conclusions

The primary aim of this work was to investigate the stability properties of a linear control system ${\Sigma }_G$ on a Lie group G. Our work is the first to analyze the stability of a LCS. Therefore, all the findings and conjectures presented in this manuscript are new.

In classical linear control systems defined in Euclidean spaces, the concepts of stability are closely related to two essential properties: controllability and observability.

To begin, we presented the key results associated with these properties in both Euclidean spaces and Lie groups. We then explored the different classes of stability concepts relevant to classical Euclidean systems. Building on this foundation and incorporating additional insights, we conjectured the inner and internal asymptotic stability properties for ${\Sigma }_G$ in Section 6.1. Furthermore, in Section 6.4, we introduced the concept of BIBO (Bounded Input, Bounded Output) stability for a linear control system that included an observation map. We also conjectured a characterization of this stability using Jordan decomposition and the dynamic subgroups that emerged from the drift of ${\Sigma }_G$. We are confident that these conjectures can be proved in future work, initially for the class of nilpotent and solvable groups, and later for semi-simple groups. Practical examples were incorporated to illustrate our conjectures.

In response to the Reviewer’s suggestions, we have decided to incorporate study of the stability properties of LCSs on Lie groups using the Lyapunov function technique in our future research and compare this theory with our approach.