On the optimal control of implicit systems

ON THE OPTIMAL CONTROL OF IMPLICIT SYSTEMS

PHILIPPE PETIT

Abstract. In this paper we consider the following problem, known as implicit Lagrange problem: nd the trajectoryxargument of

min

Z

1

0

L(xx_)dt

where the constraints are dened by an implicit dierential equation F(xx_) = 0

with dimF=n^;q <dimx=n. We dene the geometric framework of aq--submanifold in the tangent bundle of a surrounding manifold X, which is an extension of the-submanifold geometric framework dened by Rabier and Rheinboldt for control systems. With this geometric framework, we dene a class of well-posed implicit dierential equations for which we obtain locally a controlled vector eld on a submanifold W of the surrounding manifold X by means of a reduction procedure.

We then show that the implicit Lagrange problem leads locally to an explicit optimal control problem on the submanifold W, for which the Pontryagin maximum principle is naturally apply.

1. Introduction

We consider for the state

x

of^Rnthe implicit dierential equation

F

(

x x

_) = 0

:

(1.1)

In this equation the control

u

does not appear explicitly, but only because there are less equations than unknowns, namely

F

: ^{Rn Rn ! Rn;q}, where

q < n

(see 6].) Here, the control variable

u

belongs to ^Rq. The cost function is the Lagrangian

L

(

x x

_) of

T

^Rn. A process is a trajectory

x

() belonging to

C

¹(0

1]

^Rn) the set of continously dierentiable functions (resp.

KC

¹(0

1]

^Rn) the set of continuous and piecewise dierentiable functions,

AC

(0

1]

^Rn) the set of absolutely continuous functions, see the footnotes of the subsection 5.2.) A trajectory

x

() is admissible if

x

(0) =

a

,

x

(1) =

b

and

F

(

x

(

t

)

x

_(

t

)) = 0

⁸

t

²0

1] (resp

:

a

:

e

:

on 0

1])

:

For any admissible trajectory

x

() the cost is

J

(

x

()) =

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt:

Institut de mathematiques, UMR C9994 du CNRS, Universite Paris 7, UFR de math- ematiques, CASE 7012, 2, place Jussieu F-75251 Paris Cedex 05, France. E-mail:

petit@labomath.univ-orleans.fr.

Received by the journal April 29, 1997. Revised September 1997. Accepted for publi- cation January 5, 1998.

c

Societe de Mathematiques Appliquees et Industrielles. Typeset by L^ATEX.

An admissible trajectory

x

() belonging to

C

¹(0

1]

^Rn) (resp.

KC

¹(0

1]

^Rn),

AC

(0

1]

^Rn)) is a weak minimum (resp. strong minimum) of

J

if

J

(

x

())

J

(

x

())

for any admissible trajectory

x

() belonging to

C

¹(0

1]

^Rn) (resp.

KC

¹(0

1]

^Rn),

AC

(0

1]

^Rn)). An admissible trajectory

x

() belonging to

C

¹(0

1]

^Rn) (resp.

KC

¹(0

1]

^Rn),

AC

(0

1]

^Rn)) is a weak local mini- mum (resp. strong local minimum) of

J

if there exist an

" >

0 such that for any trajectory

x

() belonging to

C

¹(0

1]

^Rn) (resp.

KC

¹(0

1]

^Rn),

AC

(0

1]

^Rn)) such that ^jj

x

()^;

x

() ^jj1

< "

(resp. ^jj

x

()^;

x

() ^jj0

< "

) where

jj

x

()^jj1= max_t

20^1]max^fj

x

(

t

)^j

^j

x

_(

t

)^jg (resp

:

^jj

x

()^jj0= max_t

20^1]j

x

(

t

)^j) then

J

(

x

())

J

(

x

())

:

Remark 1.1. a) An admissible trajectory

x

() belonging to

C

¹(0

1]

^Rn) which is a strong (local) minimum is also a weak (local) minimum, mean- while a trajectory

x

() belonging to

C

¹(0

1]

^Rn) can be a weak (local) minimum without to be a strong (local) minimum.

b) The necessary conditions for the weak local minimum are also necessary conditions for the strong local minimum, and the suciency conditions for the strong local minimum are also the suciency conditions for the weak local minimum.

We will subsequently turn our attention to the geometry of the implicit dierential equation (1.1). More precisely, we will extend the denitions of

-submanifold, reducible and completely reducible

-submanifold in 16]

to our situation (see also 15].) Let us consider the manifold

X

= ^Rn and its tangent bundle

TX

=

T

^Rn=^{Rn Rn}. Let us assume that the subset

M

=

F

^;1(0) is a submanifold of

TX

(it is the case when

F

is a submersion).

A trajectory

x

() is admissible if (

x

(

t

)

x

_(

t

)) belongs to

M

for any

t

²0

1].

The implicit Lagrange problem¹ is

P

0 min

(x⁽⁾x^_())2M x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt:

If

x

() is an admissible trajectory then

x

(

t

) has to belong to the set

W

=

(

M

) for any

t

²0

1]. Let us assume that

W

is a submanifold of

X

, then

x

_(

t

) has to belong to the subspace

T

x⁽t⁾

W

of

T

x⁽t⁾

X

for any

t

in 0

1] and thus (

x

(

t

)

x

_(

t

)) belongs to the set

M

¹ =

TW

^\

M

for any

t

in 0

1]. In other

1When q=n, the constraint F(xx_) = 0 being absent, this is the simple problem of the calculus of variations (see 3, 4, 5, 10, 11])

min

x(0)=a

x(1)=b Z

1

0

L(x(t)x_(t))dt:

ESAIM: Cocv, March 1998, Vol. 3, 49{81

words, any admissible trajectory for the implicit Lagrange problem (^P0) is an admissible trajectory for the following implicit Lagrange problem

P

1 min

(x⁽⁾x^_())2M¹ x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt:

Thus, if

x

() is a solution of ^P0 then it is a solution of ^P1 and conversely.

Moreover, the startpoint

a

and the endpoint

b

have to belong to

W

. This replacement of the submanifold

M

of

TX

by the submanifold

M

¹ of

TX

is the reduction procedure and

M

¹ is called the reduction of

M

. Let us assume that we are able to do with

M

¹, what we have done with

M

, then we construct a submanifold

W

¹ =

(

M

¹) of

X

and a submanifold

M

²of

TX

. If

x

() is an admissible trajectory for the problem ^P1 then it is an admissible trajectory for the problem

P

2 min

(x⁽⁾x^_())2M² x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt:

Let us assume that we construct by induction a sequence of implicit Lagrange problem

Pk min

(x⁽⁾x^_())2M^k x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt

such that

W

k =

(

M

k) is a submanifold of

X

and

M

k⁺¹ =

M

k ^\

TW

k

is a submanifold of

TX

, then any admissible trajectory

x

() of ^Pk is an admissible trajectory of^Pk⁺¹. Therefore any admissible trajectory of^P0 is an admissible trajectory of ^Pk for any

k

. Thus any admissible trajectory of

P

0 is an admissible trajectory of the following implicit Lagrange problem

Pc min

(x⁽⁾x^_())2C⁽M⁾ x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt

with

C

(

M

) =^\k⁰

M

k. Clearly, the strong (resp. weak) minimum of^P0 are the strong (resp. weak) minimum of^Pc and the points

a

et

b

have to belong to

(

C

(

M

)). Furthermore, if the sequence^f

M

k^gk⁰is stationary then

C

(

M

) is a submanifold of

TX

and the smallest integer

such that

M

k =

M

⁸

i

will be called the index. Then we can wonder if ^Pc is equivalent to an explicit control problem (see the subsection 5.2 of the Appendix). This will be the case for the class of well-posed implicit dierential equations.

2. Definitions and Main results

Using the geometric framework of

q

-

-submanifold of the section 3, we are able to dene a well-posed implicit dierential equation.

Definition 2.1. An implicit dierential equation (1.1) is well-posed if the

set

M

=

F

^;1(0)

is a completely reducible

q

-

-submanifold of

T

^Rnsuch that the core

C

(

M

) is not empty.

Definition 2.2. The index of a well-posed implicit dierential equation (1.1) is the maximum over the index of the non-empty connected component of the core

C

(

M

).

Now, we consider only well-posed implicit dierential equations. As in the introduction, for each

q

-

-submanifold

M

k of

T

^Rnof the chain of reduction we consider the implicit Lagrange problem

Pk min

(x⁽⁾x^_())2M^k x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt

and for

C

(

M

) the core of

M

we consider the implicit Lagrange problem

Pc min

(x⁽⁾x^_())2C⁽M⁾ x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt:

The sequence^fPk^gk⁰ is called the chain of reduced implicit Lagrange prob- lems of the well-posed implicit dierential equation (1.1) and ^Pc is called the central implicit Lagrange problem. Evidently, the points

a

and

b

have to belong to

(

M

k) for any

k

and thus to belong to

(

C

(

M

)).

Definition 2.3. Any point

x

of^Rnis consistent with a well-posed implicit dierential equation

F

if it belongs to the projection of the core

C

(

M

).

According to the denitions we can formulate the following theorems (proofs are given in the subsection 5.1 of the Appendix).

Theorem 2.4. Let

F

be a well-posed implicit dierential equation (

q < n

) of

T

^Rn,^f

P

k^gk⁰ its chain of reduced implicit Lagrange problems and ^Pc its central implicit Lagange's problem. Then, any admissible trajectory

x

() of

P

0 is an admissible trajectory of^Pk for any

k

. In particular, any admissible trajectory

x

() of ^P0 is an admissible trajectory of^Pc and any strong (resp.

weak) minimum

x

() of^Pc is a strong (resp. weak) minimum of ^P0. Theorem 2.4 shows that the strong (resp. weak) minimum are living in the core

C

(

M

). According to the theorem 3.31, the

q

-

-submanifold

C

(

M

) is locally the image of a controlled vector eld. Thus, we are able to show that locally the trajectories of

C

(

M

) are in bijection with the trajectories of the controlled vector eld .

Theorem 2.5. (Local equivalence) Let

F

be a well-posed implicit dier- ential equation,

C

(

M

) its core, (

x

⁰

p

⁰) a point belonging to

C

(

M

) and

W

=

(

V

) a local projection of

C

(

M

) at (

x

⁰

p

⁰),

O

an open set of ^Rq, a controlled vector eld given by the theorem 3.31. If

x

() is a local tra- jectory of

C

(

M

) such that (

x

(

t

)

x

_(

t

)) belongs to

V

for any

t

, then there

ESAIM: Cocv, March 1998, Vol. 3, 49{81

exists an unique continuous (resp. piecewise continuous) control

u

() taking its value in

O

such that (

x

(

t

)

x

_(

t

)) = (

x

(

t

)

u

(

t

))² for any

t

. Conversely, for any initial condition

x

⁰ belonging to

W

and for any continuous (resp.

piecewise continuous) control

u

() taking its value in

O

there exists a unique local trajectory of

C

(

M

) such that (

x

(

t

)

x

_()) belongs to

V

for any

t

.

Then, on the one hand, we have shown that the strong (resp. weak) minimum of the implicit Lagrange problem ^P0 are the strong (resp. weak) minimum of the central implicit Lagrange problem ^Pc (theorem 2.4), and on the other hand that (locally) the admissible trajectories of^Pc are in bijection with the admissible trajectories of the controlled vector eld (theorem 2.5).

Now, let us consider a strong minimum

x

() of the central implicit Lagrange problem ^Pc. Let

be a point in the set

T

and

W

=

(

V

) a local projection of

C

(

M

) at (

x

(

)

x

_(

)),

O

an open set of ^Rq, a controlled vector eld given by the theorem 3.31. There exists

" >

0 such that for any point

t

belonging to the interval

I

" =

^;

"

+

"

] then (

x

(

t

)

x

_(

t

)) belongs to

V

. We naturally consider the following local implicit Lagrange problem

Pc" min

(x⁽⁾x^_())2V x^(;"⁾⁼x^(;"⁾ x⁽⁺"⁾⁼x⁽⁺"⁾

Z ⁺"

^;"

L

(

x

(

t

)

x

_(

t

))

dt:

Theorem 2.6. (Local optimality) Let

F

be a well-posed implicit dieren- tial equation. If

x

() is a strong minimum of the central implicit Lagrange problem ^Pc then for any

belonging to

T

there exists

" >

0 such that the trajectory _

x

^j_I^"() is a strong minimum of the implicit Lagrange problem^Pc"

Let us also consider the following local explicit optimal control problem

Pe" min

(x⁽⁾x^_())=(x⁽⁾u⁽⁾⁾ u⁽⁾²O x^(;"⁾⁼x^(;"⁾ x⁽⁺"⁾⁼x⁽⁺"⁾

Z ⁺"

^;"

L

( (

x

(

t

)

u

(

t

))

dt:

Theorem 2.7.

x

() is strong (local) minimum of the implicit Lagrange prob- lem ^Pc"if, and only if, the corresponding admissible process(

x

()

u

()) is a strong (local) minimum for the explicit control problem ^Pe".

This leads to consider the following local implicit Lagrange problem. Let

W

=

(

V

) be a local projection such that there exists an open set

O

in ^Rq and a controlled vector eld :

W O

^!

TW

given by the theorem 3.31

PV min

(x⁽⁾x^_())2V x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

(

x

(

t

)

x

_(

t

))

dt

where the points

a

and

b

belong to

W

. The admissible trajectories (resp.

strong minimum) of^PV are in bijection with the admissible processes (resp.

2In a local coordinate systemxofW,(xu) takes the form (xf(xu)).

strong minimum) of the explicit optimal control problem

Pe min

(x⁽⁾x^_())=(x⁽⁾u⁽⁾⁾ u⁽⁾²O x⁽⁰⁾⁼a x⁽¹⁾⁼b

Z

1

0

L

( (

x

(

t

)

u

(

t

))

dt:

Finally, we choose a local coordinate system

x

= (

x

¹

x

^r) of

W

and then apply the Maximum Principle to the problem ^Pe with the pseudo- Hamiltonian

H

⁰ :

T

^?

W O

^!R (

xu

)^7!Xr

i⁼¹

i

f

ⁱ(

xu

) +

⁰

L

(

xf

(

xu

)) and the controlled vector eld

;!

H

⁰ :

T

^?

W O

^!

T

(

T

^?

W

) (

xu

)^7!;!

H

⁰(

xu

) =^Xr

i⁼¹

@H

⁰

@

i (

xu

)

@

@x

^i;

r

X

i⁼¹

@H

⁰

@x

^{i (}

xu

)

@

@

i

where

⁰ = 0

1.

Remark 2.8. Obviously, the necessary conditions of optimality are invari- ant by bundle isomorphism

h

. Let us consider a bundle isomorphism

h

x

~= ~

X

(

x

)

u

~= ~

U

(

xu

) with inverse

x

=

X

(~

x

)

u

=

U

(~

x u

~)

:

In the new coordinates (~

x u

~) the controlled vector eld is

f

~(~

x u

~) =

@ X

^~

@x

⁽

X

(~

x

))

f

(

X

(~

x

)

U

(~

x u

~))

the Lagrangian ~

L

(~

x u

~) =

L

(

X

(~

x

)

U

(~

x u

~)) and the pseudo-Hamiltonian

H

~(~

x

^~ ~

u

) =^Pr_i⁼¹

~i

f

~ⁱ(~

x u

~)^;

~⁰

L

^~(~

x

~

u

))

:

The extremals (~

x

()

^~()) are the projection of a triplet (~

x

()

^~

()

~

u

()) such that(~

a

): (~

x

()

^~()

~

u

()) is a trajectory of the controlled vector eld ( ~

⁰ =

0

1)

;!

H

~^~0 :

T

^?

W

^~

O

^{~ !}

T

(

T

^?

W

^~) (~

x u

^~ )^7!;!

H

~^~0(~

x

^~ ~

u

) =^Xr

i⁼¹

@ H

^~~0

@

^~i (~

x

^~

u

~)

@

@ x

~^{i ;}

n

X

i⁼¹

@ H

^~~0

@ x

~^{i (~}

x

^~

u

~)

@

@

^~i

(~

b

): for any

t

belonging to 0

1] (resp. a.e. on 0

1])

H

~^~0(~

x

(

t

)

~

u

(

t

)

^~(

t

)) = max

u

~²

O

^~

H

^~~0(~

x

(

t

)

u

^~(

t

))

:

ESAIM: Cocv, March 1998, Vol. 3, 49{81

Clearly, the extremals (~

x

()

^~

()) are in bijection with the extremals (

x

()

()) via the relationship

(~

x

()

^~()) = ( ~

X

(

x

()

^t

@X

@

~

x

^{( ~}

X

(

x

()))

())

:

For any triplet (

x

()

()

u

()) such that (

a

) and (

b

) are satised, the triplet (~

x

()

^~()

~

u

()) = ( ~

X

(

x

())

^t

@X

@

~

x

^{( ~}

X

(

x

()))

()

U

^~(

x

()

u

())) satises (~

a

) and (~

b

).

Example 2.9. The controlled rigid pendulum. A mass

m

is attached at the extremity of a rigid massless wire of length

l

and xed at the origin.

is the tension of the wire,

g

the gravity constant and the control

u

= (

u

¹

u

²) acts on the mass. The equations of the system are

m x

¹ = ^;

lx

¹⁺

u

¹

m x

² = ^;

lx

²⁺

mg

+

u

²

0 =

x

²¹+

x

^22;

l

²

:

In order to return to an implicit dierential equation and to use the reduction procedure we consider the following mapping

F

⁰ :

T

^R7!R5

F

⁰(

xp

) =

0

B

B

B

B

@

p

^1;

x

³

p

^2;

x

⁴

p

³+

x

¹

p

^7;

p

⁵

p

⁴+

x

²

p

^7;

p

^6;

g

x

²¹+

x

^22;

l

²

1

C

C

C

C

A

p

= _

x

where _

x

⁵ =

u

¹

=m

, _

x

⁶ =

u

²

=m

and _

x

⁷=

=ml

and the submanifold, of

T

^R7,

M

⁰ =

F

^0;1(0) .

M

⁰ has dimension 9. The equation of the set

W

⁰ =

(

M

⁰)

is

x

²¹+

x

^22;

l

²= 0

it is a submanifold of ^R7 of dimension 6.

TW

⁰ =

G

^;10 (0) where

G

⁰ is the mapping

G

⁰(

xp

) =

x

²¹+

x

^22;

l

²

p

¹

x

¹+

p

²

x

²

:

Thus the reduction of

M

⁰ is

M

¹ =

TW

^{0 \}

M

⁰ =

F

^1;1(0), where

F

¹ is the mapping

F

¹(

xp

) =

0

B

B

B

B

B

B

@

p

^1;

x

³

p

^2;

x

⁴

p

³+

x

¹

p

^7;

p

⁵

p

⁴+

x

²

p

^7;

p

^6;

g

x

²¹+

x

^22;

l

²

x

¹

x

³+

x

²

x

⁴

1

C

C

C

C

C

C

A

it is a submanifold of dimension 8. The equations of the set

W

¹ =

(

M

¹)

are

x

²¹+

x

^22;

l

² = 0

x

¹

x

³+

x

²

x

⁴ = 0

and it is a submanifold of ^R7 of dimension 5.

TW

¹ =

G

^;11 (0), where

G

¹ is the mapping

G

¹(

xp

) =

0

B

B

@

x

²¹+

x

^22;

l

²

x

¹

x

³+

x

²

x

⁴

p

¹

x

¹+

p

²

x

²

p

¹

x

³+

x

¹

p

³+

p

²

x

⁴+

x

²

p

⁴

1

C

C

A

:

Thus

M

² =

TW

^1\

M

¹ =

F

^2;1(0), where

F

² is the mapping

F

²(

xp

) =

0

B

B

B

B

B

B

B

B

@

p

^1;

x

³

p

^2;

x

⁴

p

³+

x

¹

p

^7;

p

⁵

p

⁴+

x

²

p

^7;

p

^6;

g

x

²¹+

x

^22;

l

²

x

¹

x

³+

x

²

x

⁴

x

²³+

x

²⁴+

x

¹

p

⁵+

x

²

p

⁶+

x

²

g

^;

l

²

p

⁷

1

C

C

C

C

C

C

C

C

A

is a submanifold of dimension 7. Finally the set

W

² =

(

M

²) is in fact

W

¹ and thus

M

³ =

TW

^2\

M

² =

TW

^{1 \}

M

² =

M

² (since

M

²

TW

¹), then

C

(

M

⁰) is the submanifold

M

² and

W

=

(

C

(

M

⁰)) is the submanifold

W

². Moreover

C

(

M

) = (

W

^R2) where (

xv

) = (

xf

(

xv

)) is the vector eld of the state

x

²

W

depending on

v

= (

v

¹

v

²)^2R2such that

f

(

xv

) =

0

B

B

B

B

B

B

B

B

B

B

@

x

³

x

⁴

;

x

¹

l

²⁽

x

²³+

x

²⁴) +

x

²

l

²⁽

x

²

v

^1;

x

¹(

v

²+

g

))

;

x

²

l

²⁽

x

²³+

x

²⁴)^;

x

¹

l

²⁽

x

²

v

^1;

x

¹(

v

²+

g

))

v

¹

v

²

l

1²⁽

x

²³+

x

²⁴+

x

¹

v

¹+

x

²(

v

²+

g

))

1

C

C

C

C

C

C

C

C

C

C

A

and

v

¹ =

u

¹

=m

et

v

² =

u

²

=m

. On the other hand, from the relation

x

¹ =

l

sin

and

x

² =

l

cos

(

²]^;

) we obtain

x

³ =

l

^_cos

x

⁴ = ^;

l

^_sin

8

>

<

>

:

x

_³ = ^;

l

^_2sin

+

l

cos

x

_⁴ = ^;

l

^_2cos

^;

l

sin

x

_⁷ = _

²+

v

^1sin

l

⁺

v

^2cos

l :

Therefore,

satisfy the following second order implicit dierential equation

l

=

v

¹cos

^;(

v

²+

g

)sin

(2.1) Thus, we take for

W

the parameterization

x

=

X

(

z

) = (

l

sin

l

cos

l#

cos

^;

l#

sin

y

⁵

y

⁶

y

⁷)

where

z

= (

#y

⁵

y

⁶

y

⁷)²]^;

^R4

the controlled vector eld

(

zv

) = (

zg

(

zv

)) =

# @@

^{+ (}

v

^1cos

l

^;(

v

²+

g

)sin

l

⁾

@@#

⁺

v

¹

@y @

⁵ +

v

²

@y @

^{6 + (}

#

²+

v

^1sin

l

^{+ (}

v

²+

g

)cos

l

⁾

@y @

⁷

ESAIM: Cocv, March 1998, Vol. 3, 49{81

and the Lagrangian ~

L

(

zv

) =

L

(

X

(

z

)

@X @z

⁽

z

)

g

(

zv

))

:

Then the problem^Pc

is equivalent to the explicit optimal control problem min

(z⁽t⁾z^_(t⁾⁾⁼⁽z⁽t⁾v⁽t⁾⁾⁸t^201]

v⁽t^)2R2 z^(0)2fa^gR3 z^(1)2fb^gR3

Z

1

0

L

~(

z

(

t

)

v

(

t

))

dt

for which we obtain the necessary conditions of optimality with the pseudo- Hamiltonian

H

⁰(

zv

) =

¹

#

+

²(

v

^1cos

l

^;(

v

²+

g

)sin

l

^{) +}

³

v

¹+

⁴

v

² +

⁵(

#

²+

v

^1sin

l

^{+ (}

v

²+

g

)cos

l

^);

⁰

L

^~(

zv

)

:

Remark 2.10. For this system, the kinetic energy is

T

(

^__{) = 12}

ml

²

^_2, the potential energy is

V

(

) =^;

mgl

cos

and the Lagrangian is

L=

T

^;

V

_{= 12}

ml

²

^_2+

mgl

cos

:

The virtual work of the control

u

is

W

u =

Q

= (

u

¹

l

cos

^;

u

²

l

sin

)

and for the tension it is zero. The Lagrange equation

dt d @

^L

@

^_;

@

^L

@

⁼

Q

gives the second order dierential equation 2.1.

3. Geometry of Implicit Differential Equations For the problem ^P,

M

is a submanifold of

T

^Rn it is obvious that the reduction procedure that we present in the introduction is not applicable to any submanifold

M

of

T

^Rn, especially the submanifolds

M

for which

^jM

admits singularities. In this section we will dene the class of submanifolds of

T

^Rnthat will be allowed for the problem ^P. For this class of submanifolds we will be able to apply locally the reduction procedure. First of all, let us make some comparisons with the denition of

-submanifolds given in 16]. The authors' concern is to answer to the problem of the existence and uniqueness of solutions, namely to put

M

in the form

M

=

(

Y

) (3.1)

for a section

:

Y

^!

TY

of a connected submanifold

Y

of^Rnwith a dimen- sion equal to that of

M

. In this situation,

M

is equivalent to an ordinary dierential equation and, thus, the problem of existence and uniqueness is solved. Here, it is not our purpose to obtain the existence and uniqueness of the solutions (since in this case the optimal control problem admits an obvious solution, namely the trajectory which (possibly) goes from

a

to

b

) but to have the existence and uniqueness of a family of solutions, in other words to nd a submanifold

Y

of ^Rn, an open set

U

of ^Rq and a mapping

:

Y U

^!

TY

such that the diagram

Y U

^!

TY

T

^Rn

Pr

^{& #}

Y

^Rn switches and such that

M

= (

Y U

)

:

(3.2)

This occurs for the submanifold

C

(

M

) in the example of the controlled rigid pendulum. Even though in 16] the equality (3.1) suggest that the submanifold

M

is locally embedded in a tangent bundle

TY

of a submanifold

Y

of ^Rnwith the same dimension as that

M

, in our situation the equality (3.2) suggests that the submanifold

M

is locally embedded in a tangent bundle

TY

of a submanifold

Y

of ^Rnof dimension less than or equal to the dimension of

M

.

3.1.

q

-

-submanifold

For our geometric framework we will consider separable, Haussdorf man- ifold

X

with nite dimension and, for reasons of convenience, they are as- sumed to be smooth (although they could be of class

C

^k,

k

2). Let us recall some elements of dierential geometry. The dimension of a manifold

M

is the maximal dimension among the dimension of the connected compo- nents of

M

. A pure manifold

M

is a manifold such that all the connected components have the same dimension. For any manifold

X

, the points belonging to the tangent bundle

TX

are denoted by (

xp

) with

x

belonging to

X

and

p

belonging to

T

x

X

. The canonical projection

:

TX

^!

X

is the mapping such that

(

xp

) =

x

. For the manifold ^Rn, the tangent bundle is identied with^{Rn Rn}and the projection

is identied with the projection onto the rst factor. Moreover, for any submanifold

Y

of

X

and any point belonging to

Y

, the subspace

T

x

Y

is identied with a subspace of

T

x

X

and, thus,

TY

is identied with a submanifold of

TX

. Subsequently, the follow- ing notation

f

: (

Xa

)^!(

Yb

) means that the mapping

f

is dened in an open neighborhood

U

of

a

in

X

and

b

=

f

(

a

). As in the case of manifolds, all the mappings are assumed to be smooth (once again they could be of class

C

^k,

k

2). For any mapping

f

:

X

^!

Y

and any point

x

belonging to

X

the linear tangent mapping is denoted by

T

x

f

. Now let us give the denition of subimmersion and the subimmersion theorem

Definition 3.1. (subimmersion) Let

X

,

Y

be manifolds and a mapping

f

:

X

^!

Y

.

(a):

f

is a subimmersion at

x

²

X

if

r

= rank

T

x

f

is constant in an open neighborhood of

x

in

X

.

(b):

f

is a subimmersion on

X

if it is a subimmersion at

x

for all points

x

of

X

. In particular, for each connected component of

X

the rank

r

has a constant value on , we shall call it the rank of

f

on . Theorem 3.2. (subimmersion theorem) Let

X

be connected manifolds and

f

:

X

^!

Y

a subimmersion with rank

r

. Then, the following statements holdESAIM: Cocv, March 1998, Vol. 3, 49{81

(a): for any

y

belonging to

f

(

X

) the set

M

=

f

(

y

) is a submanifold of dimension

m

^;

r

and

T

x

M

=

KerT

x

f

.

(b): for any point

x

belonging to

X

there exists an open neighborhood

V

of

x

in

X

such that the set

W

=

f

(

V

) is a submanifold of dimension

r

and

T

y

W

=

ImT

x

f

for any point

y

belonging to

W

. Moreover, if

N

is any submanifold of

X

of dimension

r

such that

x

belongs to

N

and

T

x

N

^\ker

T

x

f

=^f0^g then the mapping

f

^j_N is a local dieomorphism of some open neighborhood of

x

in

N

onto an open neighborhood of

y

in

f

(

V

).

Proof. see 9]

For an implicit dierential equation (1.1) the following proposition gives a criterion for the projection

^j_M to be a subimmersion.

Proposition 3.3. Let

G

:^{Rn Rn!Rn;q} be a mapping with 0

q

n

such that

DG

(

xp

) has full rank

n

^;

q

in an open neighborhood of a point (

x

⁰

p

⁰) belonging to

G

^;1(0) and

U

an open neighborhood of this point in

Rn Rn such that the set

M

=

U

^\

G

^;1(0) is a submanifold of ^{Rn Rn}of dimension

n

+

q

. Then, the mapping

^j_M where

is the projection onto the rst factor is a subimmersion at (

x

⁰

p

⁰) of rank

r

=

+

q

if, and only if, rank

D

p

G

(

xp

) =

n

^;

q

is constant in an open neighborhood of (

x

⁰

p

⁰) in

M

.

Proof. On the one hand, for any point (

xp

) belonging to

M

, the tangent space

T

⁽xp⁾

M

is equal to ker

DG

(

xp

)and his dimension is equal to

n

+

q

. On the other hand, the linear tangent mapping

T

⁽_xp⁾(

^j_M) :

T

⁽_xp⁾

M

^!

T

x^Rnis the restriction of the canonical projection to the subspace

T

⁽_xp⁾

M

. Namely, the mapping

(

xp

)²

T

⁽xp⁾

M

^7!

x:

Then, for any point (

xp

) belonging to

M

ker

T

⁽xp⁾(

^jM) =^f(

xp

)²

T

⁽xp⁾

M=x

= 0^g=^f0^g ker

D

p

G

(

xp

)

:

(3.3) Clearly, the mapping

^jM has constant rank

r

in an open neighborhood of (

x

⁰

p

⁰) in

M

if, and only if, dim Im

T

⁽xp⁾(

^jM) =

r

in an open neighborhood of (

x

⁰

p

⁰) in

M

therefore if, and only if,

dimker

T

⁽xp⁾(

^jM) = dim

T

⁽xp⁾

M

^;dimIm

T

⁽xp⁾(

^jM) =

n

+

q

^;

r

(3.4) in an open neighborhood of (

x

⁰

p

⁰) in

M

. According to (3.3), (3.4) holds if, and only if, dimker

D

p

G

(

xp

) =

n

+

q

^;

r

=

n

^;

in an open neighborhood of (

x

⁰

p

⁰) in

M

that is if, and only if, dimIm

D

p

G

(

xp

) =

in an open neighborhood of (

x

⁰

p

⁰) in

M

.

Now we give the denition of a

q

-

-submanifold

M

of

TX

.

Definition 3.4. (

q

-

-submanifold) Let

X

be a manifold,

q

a xed integer less than or equal to the dimension of

X

,

M

a submanifold of

TX

and (

xp

) a point of

M

.

M

is a

q

-

-submanifold of

TX

at (

xp

) (in an neighborhood of (

xp

) in

M

) if the following conditions hold

(a): there exists a connected open neighborhood

U

of (

xp

) in

M

and a submanifold

Y

of

X

such that dim

Y

+

q

= dim

U

and

U

is a subset of

TY

.

(b): the mapping

^jU :

U

^!

X

is a subimmersion in the neighborhood of (

xp

).

M

is a

q

-

-submanifold of

TX

if for any point (

xp

) belonging to

M

,

M

is a

q

-

-submanifold at (

xp

).

Remark 3.5. a) If

M

is a

q

-

-submanifold at a point (

xp

) of

M

, then we can assume that the mapping

^jU :

U

^!

X

is a subimmersion on

U

, even if this means shrinking

U

. Moreover, for any point (

xp

) belonging to

U

, the rst condition of the denition holds (It is enough to take

U

and

Y

).

Thus, for any point (

xp

) belonging to

U

,

U

is a

q

-

-submanifold at (

xp

) in other words

U

is a

q

-

-submanifold of

TX

.

b) When

M

is not a

q

-

-submanifold of

TX

we can consider the set, possibly empty, of points (

xp

) of

M

such that

M

is

q

-

-submanifold of

TX

at (

xp

).

If it is a non-empty set, according to a), it is an open set of

M

and a

q

-

- submanifold of

TX

.

The denition of a

q

-

-submanifold can be formulated in the following wayDefinition 3.6. (bis) Let

X

be a manifold and

q

an integer less than or equal to the dimension of

X

. A submanifold

M

of

TX

is a

q

-

-submanifold of

TX

if for any connected component of

M

the following conditions hold.

(a): for any point (

xp

) of there exists an open neighborhood

U

in of (

xp

) and a submanifold

Y

of

X

such that dim

Y

+

q

= dim and

U

is a subset of

TY

.

(b): the mapping

^j : ^!

X

is a subimmersion in a neighborhood of any point (

xp

) of .

Remark 3.7. This denition extends the denition of a

-submanifold in 16], which is the case

q

= 0 of our denition. The rst condition means exactly that

M

is locally embedded in the tangent bundle

TY

of a subman- ifold

Y

of

X

of dimension less than or equal to the dimension of

M

. For the second, according to the subimmersion theorem, for any point (

xp

) of

M

there exists an open neighborhood

V

of (

xp

) in

M

such that

W

=

(

V

) is a submanifold of

X

of dimension the rank of the mapping

^j at (

xp

).

This is the local analogous of the condition,

W

=

(

M

) is a submanifold of

X

, supposed in the global reduction procedure. For any connected com- ponent of

M

the inequality 2

q

dim is satised (dim 2dim

Y

, dim

Y

+

q

= dim). We shall use this inequality to prove a property of the index.

Example 3.8. Let :

X U

^!

TX

be a smooth controlled vector eld with dim

U

=

q

. Let us assume that

(a): the mapping ^@_@u(

xu

) has full rank

q

.

(b): for any (

xp

) belonging to

TX

either the equation (

xu

) = (

xp

) has a unique solution or it does not have any solution.

(c): the mapping (

xu

)^7! (

xu

) is proper.

ESAIM: Cocv, March 1998, Vol. 3, 49{81