Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups

1 Introduction

The linear control system (LCS) notion was first introduced on matrix groups in [1]. Then, a general definition on arbitrary Lie group was given in [2]. In this article, we are concerned with minimal-time optimal problems for the class of controllable LCS, on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups.

Our specific contributions are twofold. First, we show how it is possible to solve concrete problems by using abstract mathematics: in our case, how to connect two states on a bi-dimensional cylinder through a given dynamics. The analysis involves Lie theory, dynamical systems, and control theory. On the other hand, we challenge the readers to work out similar problems in three-dimensional.

Minimal-time problems are a particular branch of mathematical optimization. It has applications in many different situations. For instance, connecting two states at the minimum time is one of the fundamental issues in any kind of geometry, i.e., the calculus of geodesics. Therefore, the controllability property is essential to face this kind of problem. Precisely, a control system is said to be controllable if any two arbitrary states can be connected through a solution of the systems in nonnegative time. A more realistic point of view comes from the notion of control sets, which are the subsets of the manifold where controllability holds at its interiors. The class of systems in our study is controllable, so the optimal problem to work out makes sense. On the other hand, homogeneous spaces represent the concrete real manifolds. In fact, they form a family of differentiable manifolds of particular importance in mathematics and physics. In fact, planes, spheres, cylinders, torus, hyperbolic planes, and others are classical examples of two-dimensional manifolds that can be seen as homogeneous spaces.^[1] More generally, the sphere S n ≃ S O ( n + 1 ) ⁄ S O ( n ) is a homogeneous space of the special rotational group. The flat Euclidean, projective, affine, Grassmannian, and hyperbolic spaces, are also examples of homogeneous spaces. On the other hand, the LCS class on a Lie group G generalizes the well-known class of LCS on Euclidean spaces [3,4]. And it is worth mentioning that due to the equivalence theorem of Jouan [5], any affine control system on a differentiable manifold, which dynamics generate a finite dimensional Lie algebra, it is equivalent to a LCS on homogeneous spaces of the corresponding Lie group. Equivalent systems share the main aspect of their dynamics. Therefore, it is relevant to classify LCSs on homogeneous spaces from all the possible aspects of the control theory, namely, controllability, control sets (maximal regions where the controllability property holds), its topological algebraic properties, and optimality.

The structure of this article is as follows: first, we explain in detail the notion of LCS on any arbitrarily connected Lie group and their homogeneous spaces. We establish the Pontryagin maximum principle (PMP) for this class of control systems, as appears in [6]. Next, we review the recent results of minimal-time problems on a horizontal cylinder realized as a homogeneous space of the solvable Lie group of two-dimensional, [7]. By applying the PMP, we show how to build on the cylinder an optimal solution, transferring among all the admissible curves of the system, each initial state to any arbitrarily desired final state, at the minimum time. In this case, we explicitly show the system in coordinates, the Hamiltonian functions, the associated Hamiltonian equations, and how to use the PMP, to build an appropriate optimal control in order to effectively compute the optimal solution. Furthermore, for the five categories of solvable Lie algebras of three-dimensional, we describe in detail the corresponding groups, the LCSs that satisfy the Lie algebra rank condition (LARC), and the controllability property in any case, as described in [8].

By following the ideas developed for two-dimensional groups, we propose the readers a very interesting challenge: To study minimal-time problems for linear control systems on the homogeneous spaces of nonnilpotent, solvable Lie groups of three-dimensional. Furthermore, the scenario is open to research the same kind of problems on nilpotent, solvable, and semi-simple Lie groups of arbitrary finite dimension.

Finally, for general facts about Lie theory, we suggest [9–12]. For control systems, we refer to [3,13–15]. And for applications, we mention [3,14,16].

2 Two special dynamics on a Lie group G

In this article, we deal with LCS, defined on a connected Lie group and their homogeneous spaces. Its dynamics depends on two kinds of differential equations on G , left-invariant and linear vector fields on the group. In this section, we explain with some details the basic fact about these dynamics.

Let G be a Lie group with identity element e and g be denoted by its Lie algebra. Remind that a vector field X on G is defined by the selection of a tangent vector X g inside the tangent space T g G of G at g , for any g ∈ G A left-invariant vector field X on G is determined first by a specific vector X e ∈ T e G and then by the group translations. Precisely, let g ∈ G and consider the diffeomorphism L g : G → G , defined by L g ( h ) = g h , and its derivative ( d L g ) e : T e G → T g G . By definition, the value of the left-invariant vector field X on g is given by:

The vector space g of all left-invariant vector fields of G is a Lie algebra isomorphic to T e G [10]. In fact,

and the bilinear map [ , ] : g × g → g has two properties. It is skew symmetric, which means that for any X , Y ∈ g , [ X , Y ] = − [ Y , X ] . And satisfy the Jacobi identity, i.e., for any X , Y , Z ∈ g ,

As usual, to compute the vector field X at any point X g , we consider the derivative of its flows { X t : t ∈ R } . On a matrix Lie group, the situation is favorable, since X t ( g ) = g exp ( t X ) . In other words,

Also, the bracket between two matrices [ X , Y ] = X Y − Y X is the commutator, and the exponential of X ∈ g is given by the series:

is the identity matrix.

On the other hand, a vector field X is called a linear vector field if its flow { X t : t ∈ R } is a one-parameter group of A u t ( G ) , the group of G -automorphisms [2]. Precisely,

One of the nicest properties of linear vector fields is the possibility to associate with X a g -derivation, i.e., a linear transformation on D : g → g , which respects the Leibnitz rule, i.e.,

The relationship between X and D is given by the following identities (see [11]):

Since we consider connected groups, any element g ∈ G is a finite product of exponential members of g . Thus, the computation of X follows applying the homomorphism property of its flow. A special situation happens when the associated derivation is inner. In this case, there exists an invariant vector field Y such that D ( ⋅ ) = [ ⋅ , Y ] . Therefore, D is easily computed using the following formula:

where

We finish this section introducing the definition of some special Lie algebras [12]. A Lie algebra g is said to be Abelian, if a d 1 = [ g , g ] = 0 , i.e.,

g is called nilpotent, if

And, g its say to be solvable, if

Here, for any natural number j ≥ 2 , a d ( j ) = [ a d ( j − 1 ) , a d ( j − 1 ) ] . A Lie group is called Abelian, nilpotent, or solvable, if its Lie algebra has the corresponding property, respectively.

3 LCSs on Lie groups and their homogeneous spaces

Let G be a connected Lie group with Lie algebra g . A LCS Σ G on G is defined by the family of differential equations:

Here, the set U is the class of admissible control functions. It reads as

with Ω ⊂ R m a closed and convex subset of R m with 0 ∈ int Ω . The vector field X is linear, and for any j = 1 , … , m , the vector field Y j is left-invariant. The system Σ G is called unbounded if Ω = R m and bounded if Ω is compact.

Given any initial condition g ∈ G and a control u ∈ U , there exists a solution φ ( g , u , t ) of Σ G . It is easy to prove that

where φ ( e , u , t ) is the solution of the system with the same control u through the identity element e , [18].

The reachable set from g ∈ G is by definition the set

i.e., A ( g ) is the set of states of G that can be reached from the initial state g , in nonnegative time, through all the strategies induced by the admissible controls.

The system Σ G is called controllable if for any two arbitrary states g and h in G , there exists an admissible control function u ∈ U and a positive time T such that φ ( g , u , T ) = h . In other words, if the system is controllable, it turns out that

Recall that the system satisfies the LARC, if the Lie algebra generated by the vector fields of the system coincides with g , i.e.,

Moreover, the system satisfies the ad-rank condition (ADRC), if

Of course, ADRC implies LARC. And due to the orbit theorem [19], we can assume, without loss of generality, that our system satisfies LARC.

We denote by Δ the Lie algebra generated by the control vectors, i.e.,

The control system Σ G is a natural extension of the classical LCS Σ R n from the Abelian group R n to a generally connected Lie group G . In the R n case, both the LARC and ADRC are the same and correspond to the very well-known Kalman rank condition [4]. We remark that in both cases, the solution starting on an arbitrary state of G just depends on the solution starting on the identity element of the group.

Next, we show how to extend this class of control systems into a more general setup, the homogeneous spaces of the Lie group G . This is relevant because of the Jouan equivalence theorem [5]. And because homogeneous spaces are strongly related to concrete manifolds. Just in two-dimensional, it includes the plane, sphere, cylinder, torus, hyperbolic plane, and others.

Let H ⊂ G be a closed subgroup of G and H \ G the associated well-defined differential manifold called a homogeneous space of G . Denote by π : G → H \ G the canonical projection and by π ∗ its derivative.

It is well-known that any left-invariant vector field can be projected to every homogeneous space of G . The same is not true, in general, for a linear vector field X . However, it is possible to prove that π ∗ X is a well-defined vector field on H \ G if and only if its flow { X t : t ∈ R } let H invariant. Precisely,

In order to define a LCS on a homogeneous space, it is necessary to be sure that the derivation associated with the linear vector field leaves invariant the Lie subalgebra h determined by the closed Lie subgroup H . Precisely,

Just observe that the conditions in (13) and (14) are equivalent if the group H is connected.

Let us assume that the drift vector field of a LCS Σ G is projectable on H \ G . Therefore, the projection of the LCS Σ G to the quotient H \ G is an affine control system Σ H \ G on the homogeneous space determined by the drift vector field X ˜ , the left-invariant vector fields Y j ˜ , j = 1 , … , m , and the class of admissible control U , defined as follows.

First, X ˜ and Y j ˜ are the vector fields on H \ G satisfying

and the LCS Σ H \ G on H \ G reads as:

We note that the number of control vector fields of the new systems can be reduced. In fact, Y j ∈ h ⇒ Y j ˜ = 0 .

4 PMP

In order to solve minimal-time problems for LCSs on homogeneous spaces, we introduce the PMP, [20], which gives necessary condition for the existence of an optimal control function transferring an initial state to a desired final state at minimum time ([3,16] for applications).

Let M be a differential manifold with co-tangent bundle T M = ∪ x ∈ M T x ∗ M . Here, T x ∗ M is the dual of the tangent vector space T x M of M at x .

Consider the affine control system Σ M on M as follows:

where, f 0 , f 1 , … , f m are the smooth vector fields on M , and u ∈ U , as previously.

The Σ M -Hamiltonian functions are given by:

The PMP for Σ M , [20], reads as follows.

In the sequel, we show how the authors in [6,21] adapt the general PMP, for a LCS Σ G on a connected Lie group G .

In this context, the principle can be written in a more favorable way. In fact, the co-tangent bundle T ∗ G of G is trivial and isomorphic to the bundle g ∗ × G . Here, g ∗ denotes the dual of the vector space g .

For u ∈ U , the associated Hamiltonian function H u : g ∗ × G ⇒ R reads as:

where

We note that any vector field was translated from g ∈ G to the identity e .

Then, they compute the lifting of any vector field of Σ G . It turns out that, the derivations − D and a d ( Y j ) = [ Y j , ⋅ ] are the lifting vector fields from G to T ∗ G , of the corresponding dynamic X and Y j , j = 1 , … , m , respectively.

Therefore, the Hamiltonian equations in Theorem 1 reads in our case as:

The second relationship is a linear differential equation in g , which gives additional information in order to compute the optimal control.

In order to apply the PMP for LCS on homogeneous spaces, it will be convenient to identify the co-tangent bundle T ∗ ( H \ G ) inside the co-tangent bundle T ∗ ( G ) . To do that, we follow reference [22].

Let G be a Lie group and H ⊂ G a closed subgroup. It is well-known that H \ G = { H g : g ∈ G } is an analytic differentiable manifold [11]. The co-tangent bundle reads,

The canonical projection π : G → H \ G is defined by π ( z ) = H z . For any z ∈ G , the differential of π at z and its transposes is given by:

The linear map d π z is onto, therefore its transpose ( d π z ) ∗ is an injective linear map. Thus, for each z ∈ G , the co-tangent space T H z ∗ ( H \ G ) is isomorphic to its image under ( d π z ) ∗ . In other words, elements in T H z ∗ ( H \ G ) can be viewed as elements in T z ∗ G . Precisely,

5 Minimal time on a two-dimensional cylinder

Let G = a f f ( R ) = R × ρ R be the connected solvable Lie group of two-dimensional. Precisely, the representation ρ is defined by the usual exponential map ρ x = e x , acting as the semi-direct product through the rule:

The Lie algebra g of G is the semi-direct product g = R × θ R , with Lie brackets between the vectors ( α 1 , β 1 ) , ( α 2 , β 2 ) ∈ g , defined as follows:

The exponential map exp : g → G is explicitly given by the formula:

It turns out that any left-invariant vector field Y ∈ g , and any linear vector field X on G read, respectively as,

where ( α , β ) , ( a , b ) ∈ R 2 .

As we mention, the flow of a linear vector field is a one-parameter group of automorphisms (see [2, Theorem 1]), which, in this case, is given by:

It turns out that a LCS Σ G on G is defined by the family of ordinary differential equations:

As shown in [23, Section 2.2] Σ G satisfies L A R C if and only if α ( a α + b β ) ≠ 0 .

In order to build an LCS on a homogeneous space, we consider the closed subgroup H = { 0 } × Z , which is invariant by the drift vector field if and only if b = 0 .

For this case, the projected LCS on the horizontal cylinder C H ≔ H \ G is controllable and given by:

In addition, through the map f : C H → C H defined by:

it is possible to obtain an equivalent projected system. In fact, the diffeomorphism f conjugates the systems

The solutions of Σ C H starting at P = ( x , [ y ] ) for u ∈ Ω are given by:

and

Since

we obtain that Σ C H is controllable.

As a consequence, one can use the PMP to analyze minimal-time trajectories between any two given points of the horizontal cylinder.

It turns out that

Therefore, any element λ ∈ T P ∗ C H is identified with a vector of R 2 .

Writing

we obtain that the Hamiltonian of Σ C H , for the minimal-time problem, is given by:

Moreover, the Hamiltonian equations of Σ C H are

Therefore, a control u ∈ U associated with a minimal-time trajectory t ↦ ϕ ( t , P 0 , u ) connecting P 0 and P 1 gives rise to a curve

From the fact that λ ( t ) satisfies the Hamiltonian equations, we obtain that

showing that

Therefore, the function p ( t ) changes sign at most one time. On the other hand, the fact that

implies that u ( t ) ∈ { − ρ , ρ } a.e. and that ω changes from − ρ to ρ (or vice-versa) at most one time.

6 LCSs on solvable Lie groups of three-dimensional

In this section, we show the five classes of three-dimensional solvable nonnilpotent Lie algebras g and their associated Lie groups G , as they appear in [24]. At any case, we show explicitly the shape of each left-invariant vector field, the linear vector fields, and all possible LCSs on G , which satisfies the LARC. We finish the section with a characterization of the controllability property of each class of system.

7 Minimal-time problem challenges

In this section, we invite to the readers search for time optimal linear problems for the class of LCSs on Lie groups with three-dimensional and their homogeneous spaces.

Furthermore, Chapters 2, 3, and 4 give information for any LCS on a connected Lie group of arbitrary dimension, and their homogeneous spaces. Thus, the challenges also include the search for time optimal problem of linear control system on any of Lie group and their homogeneous spaces for nilpotent, solvable, and semi-simple groups of arbitrary finite dimension.

Given a Lie group G with Lie algebra g , to face this kind of research, it is necessary to consider the following algorithm:

Compute:

1. The left-invariant vector fields Y of G ,

2. The linear vector fields X of G and its associated derivations D ,

3. The LCSs Σ G on G ,

4. The systems Σ G with u ∈ U , which satisfy L A R C ,

5. The closed subgroups H of G ,

6. The homogeneous spaces H \ G ,

7. The closed subgroups of G that are invariant by the flow { X t : t ∈ R } of X ,

8. The projected homogeneous system Σ H \ G ,

9. The controllability property of Σ H \ G .