Control Sets of Linear Control Systems on Matrix Groups and Applications
We review recent results on control sets of linear control systems on matrix groups. We also mention some applications of systems with algebraic structures.
This review is about control system on matrix groups and its applications. Given a matrix group with matrix algebra we introduce the notion of normalizer of , and we mention some classes of control systems inside of the normalizer which have relevant applications in many areas. We concentrate the review in the class of linear control systems on Euclidean spaces and matrix groups. A fundamental notion in this theory is the controllability property of a control system answering the following hard question. Given a specific state of the system, is it possible to reach any arbitrary state through admissible trajectories in positive time? Or better, are there some regions of the space of state where controllability holds? For that, we introduce the notion of a control set in Euclidean spaces and after on matrix groups, which are the main topics of this short review.
Why do we need to consider dynamics or even control systems on matrix groups? Well, many relevant applications are coming from physical problems where the state space is a matrix group. The Noether Theorem  states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. And, it is possible to associate symmetry with dynamic through the notion of invariant vector fields on matrix groups.
For instance, the splendid challenge problem called The Brachistochrone introduced by Bernoulli in Acta Eruditorum in 1696:
“Find the shape of the curve down which a ball sliding from rest and accelerated by gravity will slip (without friction) from one point to another”.
This problem was solved by I. Newton, J. Bernoulli, and others and open a new era in mathematics . Recently, the authors in  show that this problem like many others can be modeled as a control system on some specific matrix group as follows. Let us consider the setHere, is the gravitational constant, is the set of admissible control functions with values on a compact set containing , and is a control. It turns out that this model is equivalent to the control systemThe dynamic defined by the transposed column vectorsgenerates the dimension diamond Lie algebrawith the bracket relationships , , and and belongs to the center of These two vectors are invariant vector fields on the space state which is a matrix group with matrix Lie algebra
The model of controlling the attitude of a satellite in orbit is given by the matrix group which is the semidirect product between the rotation group of dimension with the Euclidian space [4, 5]. In this case, the dynamic of the system is determined by two invariant vector fields on , , elements of the matrix Lie algebra defined by the semidirect product between of skew-symmetric matrices of order three and
We also mention the problem of optimal controls for a two-compartment model for cancer chemotherapy with a quadratic objective . Here, the system is determined by two elements of , the algebra of trace zero matrices of order two. Controllability of this kind of control systems means the possibility of transforming any initial state; let us say sick in another one healthy. Mathematical controllability conditions for this kind of systems can be found in  and the references therein. It is worth mentioning that in this reference the authors use fundamental notions of the Lie theory as the Cartan-Killing form for semisimple algebras. This is the beauty of mathematics: no matter how abstract a mathematical concept can be, there is always the possibility of using it in a specific application.
For further references on real applications of the Geometric Control Theory on Matrix Groups we mention [8–15]. For more theoretical point of view we recommend [16–20]. For specific theoretical results on linear control system on groups we refer the reader to  and the references therein.
2. Control Sets of Linear Control System on Euclidean Spaces
The Euclidian space of dimension is given by the Cartesian product of copies of the real numbers byconsidered with it canonical topology and differential structure.
2.1. A Classical Example
We start this section with a very well-known example from the Pontryagin, Book, . Stop a train in a station in minimum time. Consider the ideal case of the train moving on a straight line without friction and denote by , the distance from the train to the station at time , which is considered as the origin of the line. From the Newton law, the force is given by , the mass by the acceleration. Of course, we can assume . We get a couple of ordinary differential equations controlled by the possible combinations between acceleration and brake as follows:Here, is a family of functions that we call admissible, depending on the possible strategies you consider to control the train,The numbers and represent the maximum and minimum standard normalized speeds. From the mathematical points of view we could consider , the set of locally integrable measurable functions defined on intervals of the real line But, also the space of continuous functions or even piecewise constant functions are possible and appropriate. These three kinds of controls guarantee the existence and uniqueness of the solution associated with each initial state and for each strategy . Now, is there a solution to this problem? In the affirmative case, how to prove its existence, or even better, how to compute the optimal solution?
As we can see, this model can be represented in a matrix form. In fact,
Each choice of control generates an ordinary differential equation. If the solution with control and a specific initial condition reaches the origin at some positive time, then is a successful control. There may be no such control or many. If there are several, one of them can be preferred depending on the considered criteria, in our case, minimum time. The control problem here is as follows: given find such that the integral curve of the system starting in and control reaches the origin of the plane at minimum time.
2.2. Linear Control System
In a more general setting, we introduce the classical linear control system as follows . A classical linear control system on the space state is determined by the family of ordinary differential equations,guided by real functions In other words, the drift is controlled by engines through the different component of the integrable function , where is topologically closed with The system is called unrestricted if and restricted in the other case.
Classically, this system is written asThe columns of the cost matrix are called control vectors.
Given the initial condition and the control , the solution of at any time denoted by is described through the variation of parameters formulaActually, from the Caratheodory Theorem, there exists an unique solution of such that . In fact, from the Fundamental Theorem of Calculus, it follows thatHere, the exponential map of a matrix is defined by where is the identity map. In particular, the set describes a curve in starting from at the instant , reaching every element of determined by the control , in positive and negative time.
The positive orbit of is defined as the union of the positive orbit up to the time , of which elements are the reachable points from the origin at the time . Precisely,On the other hand, the negative orbit of the system which is given byis the set of the states which can reach the origin through a solution of in positive time and the same for . Finally, for each ,is the reachable set of from
For a real matrix of order we denote by the spectrum of , i.e., the set of -eigenvalues and by the Lyapunov spectrum of , which means the sets of the real parts of the eigenvalues in Let us denote by the smallest -invariant subspace of which contains the image of That is, it is possible to restrict the linear map and , where Denote by the Kalman matrix, determined by the products of the power of with We say that satisfy the Kalman rank condition if , which is the topological dimension of .
2.3. Control Sets and Controllability
The controllability property of a control system is the possibility of connecting any two points of the state space through concatenation of controls in positive time. However, this property is too strong and it is true just for few systems. Thus, it is important to know some regions where controllability holds. This is the main idea to introduce the notion of control set.
Definition 1. is controllable in if for each
A more realistic approach is the notion of control set .
Definition 2. A subset is called a control set of if for each (i)there exists such that , for any ;(ii) , where denote the topological closure;(iii) is maximal with respect to conditions (i) and (ii).
This concept is applied on arbitrary restricted control systems on surfaces or even manifolds. For Euclidian linear control systems, there exists a fundamental result due to Colonius-Kliemann , as follows.
Theorem 3. Let be a restricted linear control system which satisfies the Kalman rank condition. Hence,(1)There exists a unique control set with nonempty interior with shapeFurthermore, is controllable in the interior of the control set.(2)A global controllability results can be obtained through ,
Example 4. The next example appears in the book  and we include it because it is the first one in this theory. Consider the two-dimensional restricted linear control system,The solutions of are explicitly given byIt turns out that(1)the singularity coming from and from are both saddle points;(2), and .
These facts are easily viewed by drawing a picture of both saddle points. The only control set of reads as
Example 5. In the train example the system is restricted, satisfying the Kalman rank condition, rank, and Hence, the associated system is controllable. Therefore, is the only control set.
Furthermore, in this case, it is possible to show this fact directly; i.e., from any initial condition there exists an explicitly curve transferring this point to the origin through a positive solution of . We inform that the Pontryagin Maximum Principle (Lenin Price in Russia) solves the optimal problem showing that the optimal control lives in the boundary of and it is of the type bang-bang, i.e., a piecewise constant control with values in the corner of the interval . In this case, and and the minimum time curve is building with at most one change of the control. So, inside the family of parabolas generated by the solutions ofyou find two specific parabolas with control and with control , reaching the origin. Hence, starting from any arbitrary initial condition outside this curves, you choose the unique parabola where starting from and moving in positive time hit one of the curves or and then you change the control to reach the target.
Next, we describe without proof the main facts about the controllability of a unrestricted linear control system on . These properties strongly depend on the unboundedness of the set Obviously, the unrestricted case gives more possibilities to characterize controllability.
If the system is unrestricted, the positive orbit is a vectorial subspace of In fact, the algebraic structure of depends on the structure of a vector space of under the hypotheses of . Let and be a real number. It turns out that and are also elements of . Furthermore,Thus, is a vector subspace for any . On the other hand, if it follows that For that, you just need to consider first the control and rest at the origin units of time. Since the union of a family of crescent subspaces is also a subspace we conclude that is also a subspace. In this setting, the Kalman rank condition is equivalent to controllability.
The next results are fundamental for the theory of linear control systems on Euclidean spaces .
Theorem 6. Let be a unrestricted linear control system on . It holds the following:(1)For each , .(2) is controllable .(3), for each
Example 7. Consider the unrestricted linear control system in the plane defined by the dynamicWe have . So, and according to the previous result the system is not controllable. Furthermore, for each , - Hence, is a control set but with empty interior. In particular, any solution starting from the origin cannot leave the -axis.
Example 8. Consider a linearization at some point of a nonlinear control system coming from a satellite in Earth orbit, whereand is a closed subset of with
The system satisfies the Kalman rank condition. So, if we assume , the linearized system is controllable. Now, from a practical point of view, is a restricted control system which also satisfies Hence, the linearized system is controllable. Thus, the original nonlinear system is locally controllable in a neighborhood . If for any reason the satellite comes out of the orbit you can travel through a selected optimal path in as you wish, approaching the original orbit up to some point; let us say . If it is possible, you can repeat the some idea on and so on, to reach effectively the orbit.
3. From Euclidean Spaces to Matrix Groups
In this section, we first introduce few ingredients about matrix groups and matrix Lie algebras that we need to extend the notion of a linear control system, from Euclidean spaces to matrix groups; see [22–26].
3.1. About Matrix Groups
Let us denote by the set of all real invertible matrices of order ,We denote by the connected component of which contains the identity map In other words, for any
Every matrix group considered in this review will be a subset of which is an open subset of the vector space of all real matrices of order Since is isomorphic to the Euclidian space , it follows that the topology and the differentiable structure of the groups we consider come from this Euclidian space of dimension .
The analytical map defined by called the left translations on is a diffeomorphism, which means that and its inverse are differentiable maps.
The Lie algebra of comes from the notion of invariant vector fields. Denote by the set of -vector fields on By definition, an element of satisfies the following: for any the value of in denoted by or some times is a vector of the tangent space of at , whereWe observe that in our caseHere, denotes the tangent space of at the identity element, which is nothing more than the vector space of all real matrices of order In fact, for any matrix the differentiable curve satisfies and
Definition 9. We say that is a left-invariant vector field on ifHere, or denotes the differential of at the identity
In other words, to define a left-invariant vector field on we just need to determine a tangent vector at the identity element. In fact, at any point the value is given by the derivative of left translations. Precisely,
Since any left-invariant vector field is determined by its value at the identity, it turns out that the set of all left-invariant vector fields on denoted by is isomorphic to the tangent space
The vector space with the application called bracket and defined byturns into a matrix algebra, which is called the Lie algebra of .
Of course, the bracket is bilinear and skew-symmetric. This last property means that . Furthermore, its satisfies the Jacobi identity,
A subspace is a subalgebra if and it is an ideal if
Example 10. In the sequel we show the Lie algebra of the corresponding matrix group:(1).(2) , the set of real matrices of order n.(3) , where for and is the -dimensional sphere.(4) , the skew-symmetric matrices. Here,is the matrix group of orthogonal matrices.(5) , where and for any , (6)The trace zero matrices are the matrix algebra of the matrix group , the order matrices with determinant 1.(7)The Heisenberg Lie algebra has the basis such that is the only nonvanishing bracket. In fact, the Heisenberg group has the matrix representationThe derivative of , at , determines ,
Definition 11. A Lie algebra is as follows:(1)Abelian if for any we have (2)Nilpotent if the central series stabilizes at (3)Solvable if the derivative series stabilizes at (4)Semisimple if the largest solvable subalgebra of is null
A Lie group is said to be Abelian, nilpotent, solvable and semisimple, if its Lie algebra satisfy the same property.
Remark 12. It is well known that the exponential map is a local diffeomorphism. In fact, is invertible. Thus, from the Inverse Map Theorem, it follows that there exists a neighborhood of the identity such that is a diffeomorphism. Furthermore, for nilpotent and simply connected Lie groups such as the Heisenberg group is a global diffeomorphism, which means
A homomorphism between two matrix groups and is called a matrix group homomorphism. A bijective matrix group homomorphism of with itself is called a matrix group automorphism. If is connected, the set of -automorphisms is a matrix group with Lie algebra , .
Remark 13. An intrinsically relationship between a matrix group homomorphism and its derivative is given by , which comes from the commutative diagram
Since is the trace map, it follows thatIn particular, if its exponential has determinant 1.
3.2. The Normalizer and Linear Vector Fields
As we saw, a linear control system on is written asEssentially, depends on two classes of dynamics,(1)the linear differential equation , to be controlled;(2)the control vectors , which are invariant vector fields on .
If we would like to extend from to a matrix group we first need to understand its dynamic. The solution of the linear differential equation determined by the matrix with the initial condition reads asWe observe that it associated flow ,So, for any , belongs to the automorphisms group of .
On the other hand, for any , the vector field on defined by , , is invariant by translation, depending just by its value at the origin. In fact, the solution of the associated differential equation with initial condition is given by
A simple computation shows that . Hence, the linear application transforms the invariant vector field in the invariant vector field The previous discussion allows to us to reach the concept of normalizer, which plays a role in Geometric Control Systems Theory.
Let be a matrix Lie group with Lie algebra as the set of left-invariant vector fields on . Let us denote by the Lie algebra of all smooth vector fields on . The normalizer of in is the setOf course, for any constant control , the vector field is an element of the normalizer In fact, since is an Abelian Lie algebra, then for any invariant vector filed So, we obtain , which of course is also invariant.
In the sequel, denotes the Lie algebra of the matrix group and the Lie algebra of all derivations of By definition, an element is a linear transformation which satisfies the Leibnitz rule with respect to the Lie brackets, i.e.,
Definition 14. A linear vector field on a matrix group is determined by the following requirement: its flows is an infinitesimal automorphism of , which means that
It turns out that (see )
The relationship between derivations and linear vector fields is given as follows.
Remark 15. For any In particular, from the commutative diagram
If the group is a connected and simply connected nilpotent matrix group the exponential map is a diffeomorphism. Thus, given a derivation, it is possible to explicitly compute the drift through the formula above via the logarithm map, as follows. Let , then
Example 16. Consider the solvable Lie group of dimension two
with Lie algebra and Since is simply connected, the Lie algebra of derivations of coincides with the normalizer,Take . The induced linear vector field on given by
Example 17. Let be the simply connected Heisenberg matrix group of dimension three with Lie algebra and A simple computation shows that the Lie algebra of the -derivations is six dimensional and given byAccording to , the face of the linear vector field associated with a derivation is given by
A special class of linear vector field is given by inner automorphisms of the group. Let be a non-Abelian Lie algebra and . Define the vector field as follows:As can be seen, is an automorphism with inverse . The evaluation of the vector field at any point is by definition
Example 18. Let us consider the group of Euclidean motions of the plane and In this case,For the generator , the associated linear vector field defined by inner automorphisms reads as
For the generator , we get
4. Control Sets of Linear Control Systems on Matrix Groups
In  the author studies a particular class of linear systems on matrix groups. After that, Ayala and Tirao give a formal definition of a linear control system on Lie groups as follows; see  and the references therein.
4.1. The Definition and the Solution
Definition 19. A linear control system on a connected matrix group is determined by the family of differential equationsHere, denotes the drift which is a linear vector field on The control vectors , , belong to and we shall think as the set of left-invariant vector fields. The input functions belong to, the class admissible controls. More precisely, the elements of are locally integrable functions , with
Linear control systems are important for at least two reasons. First, as we showed they are a natural generalization of the classical linear control system on the Euclidean space Besides that, Jouan  proved that is relevant from theoretical and practical point of view; see also. Actually, he shows that any general control system,on a differential manifold is equivalent to a an invariant control system on or equivalent to a linear control systems on a homogeneous space of
For the sake of completeness we show a formula which allow to compute the solutions when you know the flow of the drift .
Theorem 20. Let us consider a constant admissible control Therefore, the vector field has the solution given by
where and for each , is a homogeneous polynomial map of degree . The first terms of are obtained by a recursive formula as follows:
Example 21. Let us consider the Lie algebra with the following generators:where the only nonnull bracket is . The group is ; its elements are of the form , with the group operation “” defined by
On the other hand, the exponential and Logarithm maps are given by
Let us consider the system given bywhere the linear vector field is determined by its flows as follows:Hence, the system reads asThe -solutions are given bywhereThe derivation associated with has the matrix Since is nilpotent with the nilpotency degree 2, is zero for The nonnull terms of the series are listed below:In such a case, the series is in fact a finite sumSo, using the exponential rule, the -solution is explicitly given by
4.2. The -Decomposition Induced by a Derivation
Before to state the main results about control sets of linear control systems on a matrix group , we need to explicitly a decomposition of the Lie algebra induced by any given derivation . Recall that the linear transformation satisfies the Leibnitz rule with respect to the Lie bracket.
For an eigenvalue of , the real generalized -eigenspaces of is defined byHere, is the complexification of and , the generalized eigenspace of , and the extension of from to .
It turns out that if is an eigenvalue of and zero otherwise . Now, consider the -subspaces , where if is not the real part of any eigenvalue of . We obtain
Therefore, it is possible to decompose as , whereIt turns out that are -invariant Lie algebras and , are nilpotent.
This decomposition allows understanding the topological-dynamical behavior of For general properties of vector fields we refer the readers to .
4.3. Control Sets and Controllability
The corresponding connected matrix groups of the Lie algebras , , , , and are denoted by , , , , and , respectively. These groups play a fundamental role in the understanding of the dynamics of the system as showed in  and the references therein. All these groups are closed and invariant by the flow . Moreover, if is a solvable group, then is decomposable, which means that .
Next, we establish the main properties about control sets of a linear control system on matrix group . As before, and have the same meaning but now with respect to the system
4.3.1. The Existence of Control Sets
Our control system is said to be locally accessible at if for all the sets and have nonempty interior and locally accessible if it is locally accessible at any state In this review we always assume that is an open set. In particular, the system satisfies the Lie algebra rank condition (), which means thatTherefore, is locally accessible and under this condition, there exists a control set with nonempty interior for , . In fact, the only control set containing the identity element reads asFurthermore, the system restricted to the interior of is controllable.
To compute effectively a control set is a hard task. See, for instance, the book , where the authors use techniques from numerical analysis to approach the shape of a control set.
4.3.2. Topological Properties of Control Sets
Let us denote that and If is decomposable, it is possible to prove thatThe sets , and are contained in Hence, if the control set is bounded we get that and are also bounded. Reciprocally, the boundedness of these three sets could imply in some cases the boundedness of . In  and the references therein the following two results are proved.
Theorem 22. Assume that is semisimple or nilpotent. If , , and are compact subsets of then is bounded.
Recall that a linear transformation is said to be hyperbolic if has just eigenvalues with nonzero real parts, i.e., .
Theorem 23. Let be a nilpotent simply connected Lie group. Then,
We start by establishing an appropriate notion to understand the topology about the control set containing the identity element of
Definition 24. A connected matrix group has the finite semisimple center property if any semisimple Lie subgroups of has finite center.
For example, any solvable group and have the semisimple center property. The next results appears in 
Theorem 25. Assume that has the finite semisimple center property and the reachable set of is open. For the existent control set it holds that(1) is closed if and only if ;(2) is open if and only if ;(3)furthermore, if is nilpotent, we have(i) is closed if and only if has only eigenvalues with nonpositive real part;(ii) is open if and only if has only eigenvalues with nonnegative real part;(iii) if and only if has only eigenvalues with zero real part.
4.3.3. The Uniqueness of Control Sets
In the classical linear systems on Euclidean spaces, the Kalman rank condition implies the existence and uniqueness of one control set with nonempty interior. In , the authors prove the same result for any decomposable group.
Theorem 26. The set is the only control set of the linear control system whose interior intersects and .
From that, we obtain the following consequence 
Corollary 27. Assume that is decomposable. Then, is the only control set with nonempty interior.
Just observe that, for any linear control system which satisfies the on a solvable matrix group , we obtain that is the only control set with nonempty interior.
For general matrix groups we do not know if the control set around the identity is the only one. So, to understand how many control sets with nonempty interior could have on an arbitrary connected matrix group , we should study the situation when is semisimple. The Levi decomposition theorem together with the information on the solvable case we have in our hand will solve the question.
We finish this section with a conjecture: the uniqueness of control sets of a linear control system on a connected matrix group is no longer true. More precisely,
Claim. Assume the derivation associated with the drift vector field of is semisimple on . Then, the control sets of are exact translations of , which gives, in particular, an upper bound for the number of control sets.
For the two-dimensional solvable group , an important information comes from , where the authors proved that a linear control system whose associated derivation admits a nonzero eigenvalue cannot be controllable. Therefore, each control set is a proper subset of .
Next we give a short abstract about a number of examples that will be published elsewhere in detail.
Example 28. Let us consider a linear control system on the solvable matrix group , in three cases:(1)First, we consider the dynamicThe solution with initial condition and control is given by concatenations of the curvesDenote by the line in determined byIt turns out that the only control set containing the identity element in is given by