Lie geometric methods in the study of driftless control affine systems [620639]

Lie geometric methods in the study of driftless control affine systems
with holonomic distribution
Aurelia Florea, Liviu Popescu
Abstract
In the present paper we study the optimal control problems using Lie geometric methods. We
prove that the framework of Lie algebroids is more suitable than the cotangent bundle in order to
find the optimal solutions of driftless control affine systems with holonomic distribution and positive
homogeneous cost.
MSC2010: 49J15, 93B05, 93B27, 70H05, 17B66
Keywords: control affine systems, controllability, optimal control, Hamilton-Jacobi-Bellman equa-
tions, Lie geometric methods.
1 Introduction
In the last decades, Lie geometric methods have been applied successfully in different domains of research
as dynamical systems or optimal control theory. One of the motivations for this work is the study of La-
grangian systems with some external holonomic or nonholonomic constraints. These systems have a wide
application in many different areas as optimal control theory, econometrics, cybernetics or mathematical
economics [2, 3, 5, 7, 8, 19, 20, 22, 23]. The Lie geometric methods in the control theory have been
applied by many authors (see for instance [1, 4, 6, 10, 11, 12, 15, 16, 17, 18, 21] and references therein).
One of the most important method in the geometric approach is the analysis of the solution for the
optimal control problem as provided by Pontryagin’s Maximum Principle. A curve c(t) = (x(t); u(t)) is
an optimal trajectory if there exists a lifting of x(t) to the dual space ( x(t); p(t)) satisfying the Hamilton-
Jacobi-Bellman equations. However, finding a complete solution to an optimal control problem remains
extremely difficult for several reasons. First of all, we are dealing with the problem of integrating a
Hamiltonian system, which is generally difficult to integrate, except for particular costs. Secondly, some
special solutions so-called abnormal, should be studied. Finally, even if all solutions are found, there
remains the problem of selecting optimal solutions from them. For these reasons, it is important to find
new methods that simplify the study.
In this paper we prove that the framework of Lie algebroids is more suitable than the cotangent bundle
in the study of driftless control affine systems with holonomic distribution and positive homogeneous costs.
The paper is organized as follows. In the second section we present the known results about Lie geometric
methods in optimal control theory for control affine systems, including the controllability problems in
1

the case of holonomic and nonholonomic distributions. Also, we present only the necessary notions
about Lie algebroids and the geometric viewpoint of the optimal control. Also, we find the relation
between the Hamiltonian Hon dual Lie algebroid E¤and the Hamiltonian Hon the cotangent bundle
T¤M, that is very useful in the study of control affine systems. The strategy is to apply the Pontryagin
Maximum Principle at the level of Lie algebroids. Finally, we give an application of driftless control
affine system with positive homogeneous cost, which is more general than the quadratic cost and show
that the Hamilton-Jacobi-Bellman equations, provided by Pontryagin Maximum Principle on cotangent
bundles, lead to a very complicated system of differential equations. Moreover, it is very difficult to
find the Hamiltonian function without dependence on control variables. For these reasons we will use a
different approach considering the framework of Lie algebroids. However, we prove that the distribution
generated by vector fields is holonomic and it determines a foliation in three dimensional space. In the
last part of the paper we find the complete solution of the problem using the framework of Lie algebroids.
2 Lie Geometric Methods in Optimal Control
LetMbe a smooth n-dimensional manifold. We consider the control system given by differential
equations, depending on some parameters
dxi
dt=fi(x; u);
where x2Mrepresents the state of the system and u2U½Rmrepresents the controls . Let x0andx1
be two points of M. An optimal control problem consists of finding the trajectories of our control system
which connects x0andx1and minimizing the cost
minZT
0L(x(t); u(t))dt; x (0) = x0; x(T) =x1;
where Lis the Lagrangian orrunning cost (energy, cost, time, distance, etc.). Control theory deals with
systems whose evolution can be influenced by some external agents. The most important and powerful
tool for studying the optimal solutions in control theory is Pontryagin’s Maximum Principle. It generates
the differential equations of first order, necessary for the optimal solutions. For each optimal trajectory,
c(t) = ( x(t); u(t)), it offers a lift on the cotangent space ( x(t); p(t)) satisfying Hamilton-Jacobi-Bellman
equations. The Hamiltonian is given by
H(x; p; u ) =hp; f(x; u)i ¡L(x; u); p2T¤M;
while the maximization condition with respect to the control variables u, namely
H(x(t); p(t); u(t)) = maxvH(x(t); p(t); v);
leads to@H
@u= 0 and the extreme trajectories satisfy the equations
˙x=@H
@p;˙p=¡@H
@x: (1)
2

2.1 Control affine systems
A control affine system has the form [12]
˙x=X0(x) +mX
i=1uiXi(x); (2)
where x= (x1; :::; x n) are local coordinates on a smooth ndimensional manifold M,u(t) = (u1(t); :::; u m(t))
2U½Rm,m·nandX0; X1:::Xmare smooth vector fields on M. Usually, X0is called the drift vector
field describing the dynamics of the system in the absence of controls, and the vector fields Xi; i=1; m
are called the input vector fields. The function u(t) is called the control or the input function, which
may be specified freely in order to steer the system in a desired direction.
Definition 1 The system is controllable if for any two points x0andx1onMthere exists a finite T
and an admissible control u: [0; T]!Usuch that for xsatisfying x(0) = x0we have that x(T) =x1.
In the other words, the system is controllable if for any two states x0,x1, there exists a solution
curve of (2) connecting x0tox1. Controllability is the ability to steer a system from a given initial state
to any final state, in finite time, using the available controls. The reachable set Rof a point x02M
characterizes the states x2Mthat can be reached from a given initial state x0in positive time, by
choosing various controls and switching from one to another from time to time. A system is controllable
ifR(x) =M;8x2M. Controllability doesn’t care about the quality of the trajectory between two
states, neither the amount of control effort.
Definition 2 A distribution ∆on the manifold Mis a map which assigns to each point in Ma subspace
of the tangent space at this point
M3x!∆(x)½TxM:
The distribution ∆ is called locally finitely generated if there is a family of vector fields fXigi=1;m
(called local generators of ∆) which spans ∆, i.e. ∆( x) =spanfX1(x); :::; X m(x)g ½TxM. The distri-
bution ∆ has dimension kif dim ∆( x) =k, for all points xinM. We recall that the Lie bracket of two
vector fields is given by
[f; g](x) =@g
@x(x)f(x)¡@f
@x(x)g(x);
(@g
@xis the Jacobian matrix of g). A distribution ∆ on Mis said to be involutive if, 8x2Mthen
f(x); g(x)2∆(x))[f; g] (x)2∆(x):
If the involutive distribution is generated by vector fields fXigi=1;mthen we have
[Xi; Xj] (x) =mX
k=1Lk
ij(x)Xk(x):
3

In other words, every Lie bracket can be expressed as a linear combination of the system vector fields, and
therefore it already belongs to ∆. The Lie brackets are unable to escape ∆ and generate new directions
of motion. We recall that a foliation fS®g®2AofMis a partition of M=[
®2AS®ofMinto disjoint
connected (immersed) submanifolds S®, called leaves.
Definition 3 A distribution ∆of constant dimension on Mis called integrable (holonomic) if there
exists a foliation fS®g®2AonMwhose tangent bundle is ∆, that is TxS= ∆( x), where Sis the leaf
passing through x.
Theorem 4 (Frobenius) Suppose that a distribution ∆has constant dimension. Then, ∆is integrable if
and only if ∆is involutive.
Definition 5 The distribution ∆ =spanfX1; :::; X mgonMis said to be bracket generating if the iterated
Lie brackets
Xi;[Xi; Xj];[Xi;[Xj; Xk]];¢ ¢ ¢;1·i; j; k·m;
span the tangent space TM ofMat every point.
Using the Lie brackets of vector fields, we construct the flag of subsheaves
∆½∆2½ ¢ ¢ ¢ ½ ∆r½ ¢ ¢ ¢ ½ TM
with
∆2= ∆ + [∆ ;∆]; :::;∆r+1= ∆r+ [∆;∆r]
where
[∆;∆r] =spanf[X; Y] :X2∆; Y2∆rg:
If there exists an r¸2 such that ∆r=TM, we say that ∆ is a bracket generating distribution and r
is called the step of the distribution ∆. In this case the distribution ∆ is not integrable and is called
nonholonomic. This condition is also known as strong H¨ ormander condition , orLie algebra rank condition .
Ifr= 2 the distribution is called strong bracket generating .
The presence of the drift X0in the study of control affine systems, significantly complicates the
question of controllability. In the following we consider the driftless control affine system (X0= 0) in
the form
˙x=mX
i=1uiXi(x): (3)
The vector fields Xi; i=1; m, generate a distribution ∆ on M(assumed to be connected) such that the
rank of ∆ is constant. Let x0andx1be two points of M. An optimal control problem consists of finding
those trajectories of the distributional system which connect x0andx1, while minimizing the cost
min
u(¢)ZT
0F(u(t))dt; (4)
where Fis a positive homogeneous function on ∆. We will characterize the controllability using the
properties of vector fields which generate the distribution ∆.
4

Theorem 6 (Chow-Rashevsky) If the distribution ∆ =spanfX1; :::; X mgis bracket generating (nonholo-
nomic), then the driftless control affine system is controllable.
If ∆ is not bracket generating and is integrable (holonomic) then the system is not controllable and ∆
determines a foliation on Mwith the property that any curve is contained in a single leaf of the foliation,
and the restriction of ∆ to each leaf of the foliation is bracket generating. We will study in this paper
the case of holonomic distributions.
We assume that the distribution ∆ = spanfX1; X2; :::; X mgis holonomic with constant rank, which
means that [ Xi; Xj]2Dfor every i; j=1; m,i6=j:From the Frobenius theorem, the distribution ∆ is
integrable, it determines a foliation on Mand two points can be joined if and only if they are situated
on the same leaf. Next, we will present some notions about Lie algebroids, which are useful in the study
of driftless control affine systems.
2.2 Lie algebroids
LetMbe a real, C1-differentiable, n-dimensional manifold and TxMits tangent space at x2M. The
tangent bundle of Mis denoted ( TM; ¼ M; M);where TM=[
x2MTxMand¼Mis the canonical projection
map ¼M:TM!Mtaking a tangent vector X(x)2TxM½TM to the base point x2M:A vector
bundle is a triple ( E; ¼; M ) where EandMare manifolds, called the total space and the base space, and
the map ¼:E!Mis a surjective submersion. Using [14] we have:
Definition 7 A Lie algebroid over a manifold Mis a triple (E;[¢;¢]E; ¾), where (E; ¼; M )is a vector
bundle of rank mover M;which satisfies the conditions:
a)C1(M)-module of sections Γ(E)is equipped with a Lie algebra structure [¢;¢]E.
b)¾:E!TM is a bundle map, called the anchor, which induces a Lie algebra homomorphism (also
denoted ¾) from the Lie algebra of sections (Γ(E);[¢;¢]E)to the Lie algebra of vector fields (Â(M);[¢;¢])
satisfying the Leibniz rule
[s1; fs2]E=f[s1; s2]E+ (¾(s1)f)s2;8s1; s22Γ(E); f2C1(M):
Also, it results:
1±[¢;¢]Eis aR-bilinear operation,
2±[¢;¢]Eis skew-symmetric, i.e.
[s1; s2]E=¡[s2; s1]E;8s1; s22Γ(E);
3±[¢;¢]Everifies the Jacobi identity
[s1;[s2; s3]E]E+ [s2;[s3; s1]E]E+ [s3;[s1; s2]E]E= 0;
and¾being a Lie algebra homomorphism, we have
¾[s1; s2]E= [¾(s1); ¾(s2)]:
5

Iffis a function on M, then d f(x)2E¤
xis given by hd f(x); ai=¾(a)f, for8a2Ex. For !2Vk(E¤)
theexterior derivative dE!2Vk+1(E¤) is given by the formula
dE!(s1; :::; s k+1) =k+1X
i=1(¡1)i+1¾(si)!(s1; :::;ˆsi; :::; s k+1) +
+X
1·i<j·k+1(¡1)i+j!([si;sj]E; s1; :::;ˆsi; :::;ˆsj; :::s k+1):
where si2Γ(E),i=1; k+ 1, and the hat over an argument means the absence of the argument. It
results that ( dE)2= 0. If we take the local coordinates ( xi) on an open U½M, a local basis fs®gof the
sections of the bundle ¼¡1(U)!Ugenerates local coordinates ( xi; y®) onE. The local functions ¾i
®(x),
L°
®¯(x) onMgiven by
¾(s®) =¾i
®@
@xi;[s®; s¯]E=L°
®¯s°; i=1; n; ®; ¯; ° =1; m;
are called the structure functions of Lie algebroids. Some examples of Lie algebroids are:
Example 8 The tangent bundle E=TM itself, with identity mapping as anchor. With respect to the
usual coordinates (x;¢x), the structure functions are Li
jk= 0,¾i
j=±i
j, but if we were to change to another
basis for the vector fields, the structure functions would become nonzero.
Example 9 Any integrable subbundle of TM is a Lie algebroid with the inclusion as anchor and the
induced Lie bracket.
By a control system on the Lie algebroid ( E;[¢;¢]E; ¾) (see [13]) with the control space ¿:A!Mwe
mean a section ½ofEalong ¿. A trajectory of the system ½is an integral curve of the vector field ¾(½).
Given the cost function L 2C1(A), we have to minimize the integral of Lover the set of those system
trajectories which satisfy certain boundary conditions. The Hamiltonian function H 2C1(E¤£MA) is
defined by
H(¹; u) =h¹; ½(u)i ¡ L (u);
whereas the associated Hamiltonian control system ½His given by the symplectic equation on Lie algebroid
i½H!E=dEH:
where !Eis the canonical symplectic form. The critical trajectories are given by [13]
@H
@uA= 0;dxi
dt=¾i
®@H
@¹®;d¹®
dt=¡¾i
®@H
@xi¡¹°L°
®¯@H
@¹¯: (5)
We can associate to any positive homogeneous cost L:E!Ron Lie algebroids E;a cost Lon
Im¾½TMdefined by
L(v) =fL(u)ju2Ex; ¾(u) =vg;
where v2(Im¾)x½TxM,x2M.
6

Theorem 10 The relation between the Hamiltonian function Hon the cotangent bundle T¤Mand the
Hamiltonian function Hon the dual Lie algebroid E¤is given by
H(p) =H(¾?(p)); ¹ =¾?(p); p2T¤
xM; ¹ 2E¤
x: (6)
Proof. The Fenchel-Legendre dual of Lagrangian Lis the Hamiltonian Hgiven by
H(p) = sup
vfhp; vi ¡L(v)g= sup
vfhp; vi ¡ L (u);¾(u) =vg
= sup
ufhp; ¾(u)i ¡ L (u)g= sup
ufh¾?(p); ui ¡ L (u)g=H(¾?(p));
and we get
H(p) =H(¹); ¹ =¾?(p); p2T¤
xM; ¹ 2E¤
x;
or locally
¹®=¾¤i
®pi; (7)
where the Hamiltonian H(p) is degenerate on Ker¾?½T¤M: u t
2.3 Application
Let us consider the following driftless control affine system with positive homogeneous cost:
8
><
>:˙x1=u2
˙x2=u1+u2x2
˙x3=u1+u2x3(8)
min
u(¢)ZT
0µq
u2
1+u2
2+"u1¶
dt; 0·" <1;
We are looking for the optimal trajectories starting from the point (0 ;1;0)tand parameterized by
arclength (minimum time problem) and free endpoint. The system can be written in the form
˙x=u1X1+u2X2; x =0
B@x1
x2
x31
CA2R3; X1=0
B@0
1
11
CA; X2=0
B@1
x2
x31
CA
min
u(¢)RT
0F(u(t))dt;F(u) =p
(u1)2+ (u2)2+"u1;0·" <1(9)
The vector fields are given by
X1=@
@x2+@
@x3; X 2=@
@x1+x2@
@x2+x3@
@x3;
7

The Lie bracket is
[X1; X2] =·@
@x2+@
@x3;@
@x1+x2@
@x2+x3@
@x3¸
=X1:
and it results that the associated distribution ∆ = spanfX1; X2gis holonomic and has the constant rank
2. Moreover, from the system (8) we obtain
˙x2¡˙x3= ˙x1(x2¡x3);
which yields
ln¯¯¯x2¡x3¯¯¯=x1+c: (10)
(cis a constant) and it results that ∆ determines a foliation on R3given by the surfaces (10). In order to
solve this optimal control problem we can use the Pontryagin Maximum Principle on the cotangent bundle.
The Lagrangian has the form L=1
2F2(for minimum time problem, every minimizer parametrized by
arclength, or constant speed F= 1 is also a minimizer of the so called energy cost L=1
2F2) and we get
the Hamiltonian
H(u; x; p ) =pi˙xi¡ L=p1u1+p2(u1+u2x2) +p3(u1+u2x3)¡1
2µq
(u1)2+ (u2)2+"u1¶2
;
The Hamilton-Jacobi-Bellman equations@H
@ui= 0;dxi
dt=@H
@pi;dpi
dt=¡@H
@xilead to the following system
8
>><
>>:p1+p3x1¡µq
(u1)2+ (u2)2+"u1¶µ
"+u1p
(u1)2+(u2)2¶
= 0
p2¡µq
(u1)2+ (u2)2+"u1¶µ
u2p
(u1)2+(u2)2¶
= 0(11)
and to a very complicated system of implicit differential equations. From (11) is difficult to find the
Hamiltonian Hwithout dependence on the control variables. For this reason we will use a different
approach, involving the framework of Lie algebroids.
In order to use the framework of Lie algebroids, we consider E= ∆ (holonomic distribution with
constant rank), the anchor ¾:E!TMis the inclusion and [ ;]Ethe induced Lie bracket. In the case of
previous example, the anchor ¾has the components
¾i
®=0
B@0 1
1×1
1×21
CA;
and we consider the Lagrangian function given by
L=1
2µq
(u1)2+ (u2)2+"u1¶2
:
8

Using [9] we can find the Hamiltonian on E¤given by
H(¹) =1
2Ãs
(¹1)2
(1¡"2)2+(¹2)2
1¡"2¡"¹1
1¡"2!2
: (12)
Using (4) we can calculate the Hamiltonian HonT¤Min the form H(x; p) =H(¹); ¹=¾?(p), where
Ã
¹1
¹2!
=Ã
0 1 1
1x1x2!0
B@p1
p2
p31
CA:
We get that ¹1=p2+p3; ¹2=p1+p2x1+p3x2and it results the Hamiltonian on the cotangent bundle
H(x; p) =1
2Ãs
(p2+p3)2
(1¡"2)2+(p1+p2x1+p3x2)2
1¡"2¡"(p2+p3)
1¡"2!2
: (13)
Unfortunately, with H(x; p) from (13) the Hamilton’s equations (1) on T¤Mlead to a complicated system
of differential equations. For this reason, we will use the geometric model of a Lie algebroid. From the
relation [ X®; X¯] =L°
®¯X°we obtain the non-zero components L1
12= 1; L1
21=¡1 while from (3) we
deduce that
˙x1=@H
@¹2;˙x2=@H
@¹1+x2@H
@¹2;˙x3=@H
@¹1+x3@H
@¹2;
˙¹1=¡¹1@H
@¹2;˙¹2=¹1@H
@¹1;
where
@H
@¹1=¡1 +"2¢¹1
(1¡"2)2¡"r
(¹1)2
(1¡"2)2+(¹2)2
1¡"2
1¡"2¡"¹2
1
(1¡"2)3r
(¹1)2
(1¡"2)2+(¹2)2
1¡"2;
@H
@¹2=¹2
1¡"2¡"¹1¹2
(1¡"2)2r
(¹1)2
(1¡"2)2+(¹2)2
1¡"2:
The form of the last relations leads to the following change of variables
¹1(t) = (1 ¡"2)r(t) sech µ(t); ¹ 2(t) =p
1¡"2r(t) tanh µ(t): (14)
where
sinhµ=eµ¡e¡µ
2;coshµ=eµ+e¡µ
2;tanhµ=sinhµ
coshµ;sechµ=1
coshµ:
9

In these conditions we have s
(¹1)2
(1¡"2)2+(¹2)2
1¡"2=jrj;
and the differential equations
˙¹1=¡¹1@H
@¹2;
with the relations (14) yields
p
1¡"2µ˙r
r¡˙µtanhµ¶
=r(¡tanhµ+"sechµtanhµ): (15)
Also, from the equation
˙¹2=¹1@H
@¹1;
and (14) we get
p
1¡"2µ˙r
rtanhµ+˙µsech2µ¶
=r((1 + ")2sech2µ¡"sechµ¡"sech3µ): (16)
Now, reducing ˙µand˙r
rfrom the equations (15) and (16), we obtain
p
1¡"2˙r=r2"sechµtanhµ("sechµ¡1);
andp
1¡"2˙µ=r("sechµ¡1)2:
The last two equations lead to
˙r
˙µ=r"sechµtanhµ
"sechµ¡1;
and respectively to
dr
r="sechµtanhµ
"sechµ¡1dµ;
with the solution
lnjrj=¡ln("sechµ¡1)¡lnc:
Therefore
jrj=1
c("sechµ¡1):
10

Since the optimal trajectories are parameterized by arclength, the conclusion corresponds exactly to the
1=2 level of the Hamiltonian and we have
H=r2
2(1¡"sechµ)2=1
2c2:
Now, c=§1 and
r=§1
"sechµ¡1:
The equation
˙¹1=¡¹1˙x1;
implies that
x1(µ) = lnc1(1¡"sechµ)
(1¡"2)sech µ; c 12R:
Since we are looking for the trajectories starting from the point (0 ;1;0)t, we have x1(0) = 0 and
lnc1
1 +"= 0)c1= 1 + ";
which leads to
x1(µ) = ln1¡"sechµ
(1¡")sech µ= lncoshµ¡"
1¡":
We obtain also that
˙¹2=¹1µ
˙x2¡x2@H
@¹2¶
=¹1˙x2+x2˙¹1;
and, consequently, ¹2=¹1×2+c2. Further,
x2(µ) =sinhµp
1¡"2§c2(1¡"sechµ)
(1¡"2) sech µ:
From x2(0) = 1 we obtain that c2= 1 + "and this yields
x2(µ) =sinhµp
1¡"2+coshµ¡"
1¡":
In the same way we get
x3(µ) =sinhµp
1¡"2§c3(1¡"sechµ)
(1¡"2) sech µ:
11

From x3(0) = 0 we obtain that c3= 0 and it results
x3(µ) =sinhµp
1¡"2:
Using (8) we have u2= ˙x1,u1= ˙x3¡u2x3= ˙x2¡u2x2and by direct computation, we obtain the control
variables
u2(µ) =sinhµ
coshµ¡"; u 1(µ) =1p
1¡"21¡"coshµ
coshµ¡":
If"= 0 we obtain the case of driftless control affine systems with quadratic cost with the solution
x1(t) = ln cosh t; x2(t) = sinh t+ cosh t; x3(t) = sinh t;
and control variables
u2(t) = tanh t; u 1(t) = sech t:
Conclusions . In this paper we treat some topics of dynamical systems using Lie geometric methods.
In the case of driftless control affine systems with holonomic distribution and positive homogeneous cost
we proved that the framework of Lie algebroids is better than cotangent bundles in order to apply the
Pontryagin Maximum Principle and find the optimal solution. As futher developments, we try to use
the framework of Lie algebroids in the case of nonholonomic distribution (in particular, strong bracket
generating) and characterize the abnomal solutions using the geometry of Lie algebroids.
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