Time minimal control of batch reactors

TIME MINIMAL CONTROL OF BATCH REACTORS

B. BONNARD AND G. LAUNAY

Abstract. In this article we consider a control system modelling a batch reactor in which three species^X1X2X3 are reacting according to the scheme ^{X1 ! X2 !X3}, each reaction being irreversible. The control is the temperature ^T of the reactions or the derivative of this temperature with respect to time. The terminal constraint is to obtain a given concentration of the product^X2 at the end of the batch. The objective of our study is to introduce and to apply all the mathematical tools to compute the time optimal control as a closed-loop function.

This work can be used to optimize the yield of chemical batch reactors.

1. Introduction

Until now, the chemical batch reactors are mainly operating at constant temperature. Substantial gain in the yield of such reactors can obviously be obtained by controlling the temperature during the batch. This leads to optimal control problems. Recent developments in geometric optimal control allow to handle the mathematical complexity of such problems.

In 3], the authors consider the time minimal problem for a batch reactor in which three species

X

¹

X

²

X

³are reacting according to the scheme

X

^1!

X

^{2 !}

X

³, each reaction being irreversible and of the rst order, while the nal constraint is to obtain a given ratio of the two concentrations of

X

¹

X

² and the control is the derivative of the temperature with respect to time.

Due to symmetry the problem can be reduced to a time minimal control problem for a planar system and the time minimal control is computed as a closed-loop function. The objective of this article is to generalize this analysis for a network of the form

X

^{1 !}

X

^{2 !}

X

³, each reaction being irreversible but of any order, while the terminal constraint is now to obtain a desired production of the intermediate specie

X

². If the control is the derivative of the temperature with respect to time, this leads to a time minimal control problem for a system in ^R3 much more complex than a planar one.

The main objective of our study is to present general techniques and results to handle the problem they are mainly threefold:

Bernard Bonnard: Universite de Bourgogne, Laboratoire de Topologie, UMR 5584, BP 400, 21011 Dijon Cedex, France. Email: bbonnard@satie.u-bourgogne.fr.

Genevieve Launay: INRIA Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France. Email: launay@pascal.inria.fr.

Received by the journal November 18, 1997. Accepted for publication August 24, 1998.

c

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

rst we compute the time optimal control as a closed-loop function near the terminal constraint. This analysis is intricated it has motivated and it uses the mathematical developments of 5], 14],

secondly we need global estimates concerning the number of switchings of an optimal control,

endly we have to introduce the concept of focal points and give an algorithm to compute such points. This part is related to techniques introduced in 4].

The combination of the three previous analysis with numerical simulations allows to compute the optimal synthesis in many situations. Also it can be applied to more general reaction scheme to get closed-loop sub-optimal control laws.

This work was motivated by practical control problem and is currently implemented on a batch reactor located at Caen, France. Preliminary simu- lations indicate that in our experiment the gain of the optimal law is about 15% with respect to a constant operating temperature.

In order to present our main results we need to introduce precisely our problem.

1.1. Statement of the problem

1.1.1. Chemical kinetics. In this article, we shall restrict our study to a batch reactor in which three species

X

¹

X

²

X

³ are reacting according to the scheme

X

^{1 !}

X

^{2 !}

X

³, where the two reactions are irreversible and of the

n

i-th order with respect to the species

X

i,

i

= 1

2. Assuming that the reactions are at constant volume, denote by

c

j, 1

j

3 the molar concentration of

X

j and by

T

the temperature of the reactions from the laws of chemical kinetics 7] we know that they satisfy:

8

>

<

>

:

c

_¹ =^;

k

¹

c

ⁿ¹¹

c

_² =

k

¹

c

^{n11 ;}

k

²

c

ⁿ²²

c

_³ =

k

²

c

^{n22 (1.1)}

where _

c

denotes

d dt

⁽

c

)

The parameters

k

i are depending on

T

according to Arrhenius law

k

i=

A

iexp(^;

E

i

=RT

)

i

= 1

2 (1.2) where

A

i

E

i are respectively the frequency factor and the activation energy of the

i

-th reaction and

R

is the gas constant all these parameters are positive. Similar equations can be obtained when dealing with a network of two simple reactions

n

¹

X

^{1 !}

n

²

X

^{2 !}

n

³

X

³, where the

n

j's are the stoichiometric coecients of the reactions.

Our optimal control problem is the following: minimize the batch time when a desired production quantity is xed for a batch. The desired product can be

z

=

c

²

=c

¹ (problems of index 1) or

c

² (problems of index 2). Hence we shall solve time minimal problems with nal constraint

z

(

t

F) =

d

¹ (resp.

c

²(

t

F) =

d

²), where

t

F is the batch time, and with initial condition

z

(0)

< d

¹ (resp.

c

²(0)

< d

²).

1.1.2. Controls. Two mathematical controls are possible:

T

, subject to constraints

T

m

T

T

M the associated problems are^P1 (target

z

=

d

¹) and^P2 (target

c

² =

d

²)

or

u

= _

T

, subject to constraints

u

^;

u

u

⁺ with

u

^;

<

0

< u

⁺ the associated problems are _^P1 (target

z

=

d

¹) and _^P2 (target

c

²=

d

²).

The choice of the temperature as control is clearly related to the chemical process. If we introduce

v

=

k

¹

=

E

²

=E

¹

=

A

²

=A

¹ (1.3) then system (1.1) becomes:

8

>

<

>

:

c

_¹ =^;

vc

ⁿ¹¹

c

_² =

vc

^n11;

v

c

ⁿ²²

c

_³ =

v

c

^{n22 (1.4)}

and

v

can be taken as the control, since

v

is an increasing one-to-one function of

T

, the constraints

T

m

T

T

M being equivalent to

v

m

v

v

M. Observe that system (1.4) is ane with respect to

v

if and only if

= 1 in this particular case the ratio

dc

²

=dc

¹ does not depend on

v

and the system is not controllable. Since in general the system is not ane with respect to the input it may happen that there is no admissible time optimal control law this is due to a relaxation phenomenon analyzed in 6].

If we cannot directly track the optimal temperature prole in the previous system (1.4), we choose

u

= _

T

as the control and we get the following system (which is ane with respect to

u

):

8

>

>

>

<

>

>

>

:

c

_¹ =^;

vc

ⁿ¹¹

c

_² =

vc

^{n11 ;}

v

c

ⁿ²²

c

_³ =

v

c

ⁿ²²

v

_ =

h

(

v

)

u

with

h

(

v

) = (

Rv=E

¹)ln²(

v=A

¹)

(1.5) By considering

u

= _

T

as a control law, we compute an optimal law related only to the chemical network this choice is an idealization and the tem- perature can become negative. To handle this diculty we must introduce constraint on the state- coordinate

v

this problem will not be considered in our study.

1.1.3. Physical space. The physical space

P

of our problem is dened as follows:

8

>

<

>

:

c

¹

>

0

c

²0

c

²(

t

)

>

0 if

t

is not the initial time]

c

³0

z < d

¹ or

c

²

< d

² except at nal time,

0

< v < A

¹ since

v

=

k

¹ =

A

¹exp(^;

E

¹

=RT

) with

T >

0

:

^(1.6) Then the following inequality:

h

(

v

)

>

0 (1.7)

comes from (1.5) and (1.6).

1.1.4. An example. The main properties of our study can be well under- stood by considering the following example. Notice that according to system (1.1),

z

=

c

²

=c

¹ satises:

z

_= (

v

+

vz

)

c

^n11;1;

v

z

ⁿ²

c

^n12;1

Therefore _

z

is independant of

c

¹if and only if

n

¹ =

n

² = 1. In this case

n

¹ =

n

² = 1 the structure of ^P1 and _^P1 is particular and allows simplications.

This introduces the lightning analysis of the next section.

1.2. Solving problem ^P1 in the case

n

¹ =

n

² = 1

1.2.1. Computations. According to section 1.1, problem ^P1 is the time minimal problem of reaching the target (

z

=

d

¹) from an initial value

z

^{0 2} 0

d

¹ of

z

for the dynamical system (1.4) which becomes in the coordinates

c

¹

zc

³ when

n

¹=

n

² = 1:

8

>

<

>

:

c

_¹ =^;

vc

¹

z

_ =

v

+

vz

^;

v

z

c

_³ =

v

c

^{2 (1.8)}

the variable

v

such that

v

m

v

v

M being considered as the control.

Note that the optimal law is: maximize _

z

subject to

v

m

v

v

M. Note also that the target (

z

=

d

¹) is not accessible from

z

^{0 2} 0

d

¹ such that max_vt

z

_ 0, where max_vt

z

_ is the maximum of _

z

for any

v

²

v

m

v

M] and for any time t.

Consider then

H

(

c

¹

zv

) = _

z

. From (1.8) it comes

H

= 0 when

v

= 0 (1.9)

@H @v

^{= 1 +}

z

^;

v

^;1

z

(1.10)

@

²

H

@v

^{2 =;}

(

^;1)

v

^;2

z

(1.11) So if

⁶= 1,

H

is a non-linear function of

v

such that there exists a single positive value ^

v

of

v

satisfying

@H

@v

= 0 and so corresponding to an extremum of

H

, which is from (1.11):

a maximum (positive since (1.9) holds) if

>

1,

a minimum (negative since (1.9) holds) if

<

1.

Let us introduce

G

(resp.

S

) the set of points (

v >

0,

z >

0) such that

H

= 0 (resp.

@H

@v

= 0). From (1.8),

G

is dened by

z

(

v

^;1;1) = 1, and from (1.10),

S

is dened by

z

(

v

^;1;1) = 1: so if

⁶= 1,

G

^\

S

= . 1.2.2. Results. Figure 1 shows

G

and

S

in the cases

<

1,

>

1 and a careful observation of this gure gives the optimal trajectories. In order to describe these optimal trajectories, let us denote by

m (resp.

M

^

) any arc satisfying (1.8) with control

v

m (resp.

v

M

^

v

). Then, for

a

b

c

in ^f

m

M

^

^g,

a

b

c denotes the concatenation of a

a arc followed by a

b arc and then ended by a

c arc where each arc of the sequence may be empty.

O

v H>0

@H@v ^>0

H<0 z

d

1

vA

O

() 1

;1

1

;1 S(@H@v ^{=0) G(H =0)}

@H@v ^<0

H<0

z v=vM

@H@v ^>0

z

d

1

H>0

^ v

1 vm

O

H>0 G(H=0) S(@H@v ⁼⁰⁾

@H@v ^>0 @H@v ^<0

1

;1

() 1

;1

^ zm

^ zM

v=^v v=vm

H<0

v=vM

z v

@H@v ^<0

vM

O

Figure 1 (i) :

<

1 Figure 1 (ii) :

>

1 Figure 1

Case

<

1. For any

z

⁰ in 0

d

¹,

if

v

M

< v

A, where

v

A =

d

¹ 1 +

d

¹

1

1; is the

v

-coordinate of

G

^\(

z

=

d

¹), the target (

z

=

d

¹) is not accessible,

if

v

M

v

A, optimal trajectories are

M. Case

>

1. For any

z

⁰ in 0

d

¹,

if

v

M

v

^¹, where ^

v

¹=

d

¹ 1 +

d

¹

1

1; is the

v

-coordinate of

S

^\(

z

=

d

¹), optimal trajectories are

M,

if

v

m

v

^¹

< v

M, optimal trajectories are

M

^,

M being empty if and only if

z

⁰

> z

^M where ^

z

M = 1

=

(

v

_M^;1;1) is the

z

-coordinate of

S

^\(

v

=

v

M),

if ^

v

¹

< v

m

<

^1;1 , optimal trajectories are

M

^ m,

M being empty if and only if

z

⁰

> z

^M, and

M

^ being empty if and only if

z

⁰

> z

^m

where ^

z

m = 1

=

(

v

_m^;1;1) is the

z

-coordinate of

S

^\(

v

=

v

m),

if

v

m

^1;1 , the target (

z

=

d

¹) is not accessible.

1.2.3. Conclusion. This example shows the following phenomenon. The value

= 1 is a bifurcation value for the parameter

which is the ratio of activation energies

E

¹

=E

²(cf. (1.3) in section 1.1.2) the optimal law appears to be radically dierent in the case

<

1 and

>

1. More precisely, the optimal policy when the target is accessible is:

when

<

1, to apply the maximal temperature

T

M

whereas if

>

1, there exists an intermediate optimal heat law in some domain of the state space which is neither

T

m nor

T

M. Such a law is called a singular optimal control, see 3] it plays an important role in our problem it is related to the convexity of the function

v

^7!

H

= _

z

and all the analysis relies on the behavior of

H

.

1.3. Maximum Principle and Extremals

In order to make this article self-contained and to give the results as clearly as possible, let us recall some basic concepts and results about Pontryagin's Maximum Principle (in brief PMP) and extremals.

1.3.1. Statement of PMP. Consider a system of the form

x

_(

t

) =

f

(

x

(

t

)

u

(

t

))

x

(

t

)^2Rn

u

(

t

)^2Rm (1.12) where

f

is an analytic mapping from ^RnRto ^Rn and where the set ^U of admissible controls is the set of measurable mappings

u

() dened on an interval

t

⁰(

u

)

0] of ^R;and taking their values in

u

¹

u

²]. Let

N

be a regular analytic submanifold of^Rn. The PMP asserts that if

u

(

t

),

t

²

t

⁰

0]

is an optimal control for the time minimal control problem with terminal manifold

N

, then there exists a so-called adjoint vector

p

(

t

) ^{2 Rnnf}0^Rng with

p

absolutely continuous, such that the following equations are satised almost everywhere on

t

⁰

0]:

dt d

⁽

x

) =

@H

@p

⁽

x

p

u

)

ddt

⁽

p

) =^;

@H

@x

⁽

x

p

u

) (1.13)

H

(

x

p

u

) =

M

(

x

p

) (1.14)

where

H

(

xpu

) =^h

pf

(

xu

)ⁱ,^h

ⁱbeing the canonical inner product in^Rn and where

M

(

xp

) = max_u

2u¹u^2]

H

(

xpu

). Moreover we have:

t

^7!

M

(

x

(

t

)

p

(

t

)) is constant and non-negative (1.15) and at the nal time 0 the so-called transversality conditions are satised:

x

(0)²

N p

(0) is orthogonal to

T

x⁽⁰⁾

N

(1.16) where

T

x

N

denotes the tangent space to

N

at

x

.

1.3.2. Definitions.

System (1.13) is called the hamiltonian lift of (1.12), and

H

is called the Hamiltonian.

Any (

xpu

) solution to (1.13), (1.14) and (1.15) is called an extremal (sometimes the adjoint state

p

will be omitted).

Any extremal (

xpu

) that satises the transversality conditions (1.16) is called a BC-extremal.

Any extremal (

xpu

) such that

M

(

xp

) = 0 is called exceptional.

An extremal (

xpu

) is called regular if and only if for almost all

t

,

u

(

t

) is equal to

u

¹ or to

u

² if moreover

u

is piecewise constant, the extremal is called bang-bang.

An extremal (

xpu

) is called singular if and only if for every time

@H @u

⁽

xpu

) = 0

Let (

xpu

) be an extremal. A time

is called a switching time if and only if

belongs to the closure of the set of times where

x

is not^C1 the corresponding point

x

(

) is then called a switching point.

1.3.3. Notation. As in section 1.2.2, if

a and

b are two arcs solutions to system (1.12),

a

b denotes the concatenation of

a followed by

b (this corresponds to the concatenation of the controls associated to

a and then to

b).

1.3.4. Singular extremals. By denition a singular extremal (

xpu

) is a solution to (1.13), (1.14) and (1.15) satisfying

@H

@u

= 0 from (1.14) it has to satisfy the Legendre condition

@

²

H

@u

^{2 j(xpu)}0 if this inequality is strict for every time

t

, it is called the strong Legendre condition. Then by the implicit function theorem the singular control can be locally computed as a function of (

xp

) by solving

@H

@u

= 0 equivalently the singular control can be obtained as solution to the Cauchy problem for the dierential equation

d

dt @H

@u

= 0, the strong Legendre condition implying the existence of such a solution.

Recall that, in order to be admissible, the singular control has to belong to

u

¹

u

²]. Any point of a singular extremal where the singular control is equal to

u

¹ or to

u

² is called a saturation point.

1.4. Statement of the main results

We shall give the main results of our article. To simplify the presentation we shall assume that each reaction is of rst order, i.e. that

n

¹=

n

² = 1.

1.4.1. Problem^P_¹ in the case

n

¹ =

n

² = 1. According to section 1.1, _^P1 is the time minimal control problem of reaching the target (

z

=

d

¹) from an initial value

z

^{0 2}0

d

¹ of

z

for the dynamical system (1.5) and for control

u

such that

u

^;

u

u

⁺ with

u

^;

<

0

< u

⁺. When

n

¹ =

n

² = 1, system (1.5) becomes in the state-coordinates

c

¹

zc

³

v

:

8

>

>

>

<

>

>

>

:

c

_¹ =^;

vc

¹

z

_ =

v

+

vz

^;

v

z c

_³ =

v

c

²

v

_ =

h

(

v

)

u

with

h

(

v

) = (

Rv=E

¹)ln²(

v=A

¹)

(1.17) We shall prove subsequently (cf. section 3.1) that we can restrict our study to the planar system:

(

z

_ =

v

+

vz

^;

v

z

v

_ =

h

(

v

)

u

with

h

(

v

) = (

Rv=E

¹)ln²(

v=A

¹) (1.18) Let us discuss the corresponding optimal synthesis.

We introduce the following notations:

Let

u

s be the singular control, and let us denote by

^; (resp.

⁺

s) any arc satisfying (1.18) with control

u

^; (resp.

u

⁺

u

s).

Let

A

be the set of the points of the target (

z

=

d

¹) that are accessible from

(

z < d

¹).

Let

G

be the set of the points (

z >

0,

v

²]0

A

¹) such that _

z

= 0.

Let

S

be the set of the points (

z >

0,

v

²]0

A

¹) such that

d dt @H

@u

^{= 0,} where

H

denotes the Hamiltonian corresponding to system (1.18) and let

S

¹ =

S

^\

N

. From section 1.3.4,

S

contains all the singular extremals.

Computations show that the singular control

u

s is negative and that there is on

S

at most one saturation point

S

^sat, where

u

s=

u

^;.

In the following gures (Figures 2 and 3) we assume that

v

A

< A

¹, where

v

A denotes the v-coordinate of

G

^\(

z

=

d

¹) if

A

¹

v

A, these gures have to be restricted to (

v < A

¹).

Case

<

1. Near the target, every optimal trajectory is of the form

⁺ and the local synthesis is given by Figure 2.

S G(z_ =0)

S

1

A

1 vA

+

+ z

d

1

v

A, empty if and only if

A

¹

v

A

Figure 2

Case

>

1. In this case we can describe the global synthesis. Let

z

^sat denote the

z

-coordinate of the saturation point

S

^sat. We have two situations:

if

z

^sat

d

¹, optimal trajectories are

⁺

^;and the synthesis is represented on Figure 3(i),

if

z

^sat

< d

¹, optimal trajectories are

⁺

^;

s and the synthesis is given by Figure 3(ii)

where each arc of these sequences may be empty

1.4.2. Problem ^P2 in the case

n

¹ =

n

² = 1. According to section 1.1,

P

2 is the time minimal problem of reaching the target (

c

² =

d

²) from an initial value (

c

²)^{0 2} 0

d

² of

c

² for the system (1.4) where the control is

v

such that

v

m

v

v

M.

We shall prove subsequently (cf. section 3.1) that we can restrict our study to the planar system:

(

c

_¹ =^;

vc

¹

c

_² =

vc

^1;

v

c

^{2 (1.19)}

Let us denote by ^

v

the singular control and by

m (resp.

M

^

) any arc satisfying (1.19) with control

v

m (resp.

v

M

^

v

).

Let

A

be the set of the points of the target (

c

² =

d

²) that are accessible from (

c

²

< d

²).

Let

E

m (resp.

E

M) be the point of target such that _

c

² = 0 when

v

=

v

m

(resp.

v

=

v

M).

A

1

v G(z_=0)

d

1

G(z_=0)

; S

S

+

+

+

+

;

;

G(z_=0) S

S

A

1

v d

1

;

G(z_=0)

s

+

+

+

+

+

+

;

;

A, O

O z z

S

1

S

1 v

1

v

A

v

1

v

A

S z

sat

S

sat

v

sat

S

sat z

sat

v

sat

+ Figure3(i):z

sat d

1

Figure3(ii):z

sat

<d

1

setofswitchingp ointsfor

+

; ,

setofrstswitchingp ointsfor+;s

Figure 3

Let

S

m (resp.

S

M) be the point of the target such that 0 =

@H

@v

^jv=vm, i.e.

v

m = ^

v

^jt⁼⁰ (resp. 0 =

@H

@v

^{jv=vM, i.e.}

v

M = ^

v

^jt⁼⁰) where

H

is the Hamiltonian (recall that, from section 1.3.4, the singular control ^

v

is solution to

@H

@v

^{= 0).}

Case

<

1. Each optimal trajectory is of the form

M and the synthesis is represented on Figure 4.

A M

EM

M M

c

1 d

2 c

2

Figure 4 Case

>

1. Results are threefold:

Near the target, each optimal policy has at most one switching.

Each optimal trajectory is of the form

M^

m, where each arc of the sequence may be empty, and each open-loop control is^C0.

Numerical simulations(cf. section 8) show that the optimal synthesis is given by Figure 5.

A

Em ^Sm ^SM

^ v=vM

^

^

c

1

^ v=vM

M

M

M

M

M

m

^ v=vm

^ v=vM

^

m

d

2 c

2

Figure 5

1.4.3. Problem^P_² in the case

n

¹ =

n

² = 1. According to section 1.1, _^P2 is the time minimal problem of reaching the target (

c

²=

d

²) from an initial value (

c

²)^{0 2} 0

d

² of

c

² for the control system (1.5) where the control

u

satises

u

^;

u

u

⁺ with

u

^;

<

0

< u

⁺.

We shall prove subsequently (cf. section 3.1) that we can restrict our study to the three dimensional system:

8

>

<

>

:

c

_¹ =^;

vc

¹

c

_² =

vc

^1;

v

c

²

v

_ =

h

(

v

)

u

with

h

(

v

) = (

Rv=E

¹)ln²(

v=A

¹) (1.20)

Let us denote by

u

s the singular control and by

^; (resp.

⁺

s) any arc satisfying (1.20) with control

u

^; (resp.

u

⁺

u

s).

The set

A

of the points of the target (

c

² =

d

²) that are accessible from (

c

²

< d

²) is included in the set ^A of the points of the target such that

c

_²0, the frontier of which is the set^E of the points of the target such that

c

_²= 0.

Let

S

² be the set of the points of the target such that

d dt @H

@u

^{= 0, where}

H

denotes the Hamiltonian (from section 1.3.4 any singular extremal satises this condition). It turns out that at the nal time the singular control is neg- ative and that there is on

S

² one saturation point

S

^sat, where

u

^;=

u

s^jt⁼⁰. On Figure 6 we describe the stratication of the target by the optimal control in the (

vc

¹) coordinates it is deduced from the equations of the Maximum Principle, except near the set ^E where the situation is intricated because of accessibility problems.

non-accessible points: and on^E, with^E+ non-accessible

u=u

+ u=u

+

u=u

;

on^E, with^E; accessible

A

1 v u

;

<us^jt⁼⁰

S

2

v

sat

E c

1

O

E

; S

sat

Figure 6 (ii) :

>

1 non-accessible points: and

us^jt^=0<u;

A

1 c

1 S

2 E

E

+

O

S

2 E

v

Figure 6 (i) :

<

1

Figure 6 : stratication of the target

Case

<

1. It turns out that near the target the optimal policy is

u

=

u

⁺. Moreover there exists one point ^E+ on ^E such that

d

²

dt

²⁽

c

²) = 0 near this point the local synthesis is given by Figure 7

and

non-accessible points: on^E, with^E+ non-accessible

v E

+

+ E

+ c

1 E

+

A

1

Figure 7 : local synthesis near ^E+ when

<

1

O

Case

>

1. We got the following local and global results:

Lo cal synthesisNear the target the optimal synthesis is described as fol- lows:

For optimal trajectories arriving near

S

², the synthesis can be topologically described by an invariant foliation (

v

=

v

⁰). Let

v

^satdenote the

v

-coordinate of the saturation point

S

^sat the syntheses in each leaf (

v

=

v

⁰) are shown by Figure 8:

if

v

⁰

< v

^sat, optimal trajectories are of the form

⁺

^;

s(see Figure 8 (i)) if

v

⁰

v

^sat, optimal trajectories are of the form

⁺

^; (see Figure 8 (ii)) where each arc of these sequences may be empty.

For optimal trajectories arriving near^E, the situation is intricated. Near the target, close enough to ^E, the optimal policy is

u

=

u

^;. Moreover there exists one point ^E; on ^E where

d

²

dt

²⁽

c

²) = 0 let

v

^;denote the

v

-coordinate of ^E;. Near a point ^{E0 6}=^E; of ^E, the optimal synthesis is described by a

C

0 invariant foliation^F: (

v

=

v

⁰) the leaves of which are given by Figures 9 (i) and (ii) near ^E; there is no such foliation and the synthesis is given by Figure 9 (iii).

Global switching rulesEach optimal control law has at most two switch- ings and each optimal trajectory is of the form

⁺

^;

s, where each arc of this sequence may be empty.

1.5. Summary

This paper is organized as follows. In section 2 we recall extremality results for time minimal problems with single input ane systems. In section 3 we introduce the concepts of projected and reduced problems, which are straightforward but important tools in our study. The use of projected problems is related to symmetry properties of our systems. The concept of

c

1

c

2

E S

2

E (c

2

=d

2 )

;

+

+

+

+

+

;

s

(c

2

=d

2 )

+

+

S

2

+

;

set of switching points for

⁺

^; set of rst switching points for

⁺

^;

s

Figure 8 (ii) :

v

⁰

v

^sat Figure 8 (i) :

v

⁰

< v

^sat

Figure 8 : local synthesis near

S

² in the leaf

v

=

v

⁰ when

>

1 reduced problem is more acute it is related to Goh's transformation and lightens the relationship between problems ^Pi(with control

T

) and _^Pi(with control _

T

),

i

= 1

2. In section 4 we analyze problem ^P2 rst we solve ^P2 near the target when

⁶= 1 secondly we nd global bounds on the number of switchings. In section 5 (resp. 6) we apply results of 5] and 14] to compute in the case_

⁶= 1 the closed-loop optimal control near the target for problem

P

1 when

n

¹ =

n

² = 1 (resp. problem _^P2 for any

n

¹

n

²) secondly in the case

>

1, we nd global switching rules for _^P1 when

n

¹ =

n

² = 1 (resp.

_

P

2, except when

n

² = 1 and

n

¹ 2). Section 7 deals with the concept of conjugate and focal points. Endly, section 8 is an appendix regrouping commented numerical experiments.

2. Extremals for time minimal problems with single input affine systems

2.1. Definitions and elementary results

2.1.1. Statement of the problem. Throughout this section 2, the con- sidered system is:

x

_(

t

) =

X

(

x

(

t

)) +

u

(

t

)

Y

(

x

(

t

))

x

(

t

)^2Rn and

u

(

t

)^2R (2.1) where

X

and

Y

are analytic vector elds. As in section 1.3.1, the set ^U of admissible controls is the set of measurable mappings

u

() dened on an interval

t

⁰(

u

)

0] of ^R; and taking their values in

u

¹

u

²] and again

N

denotes a regular analytic submanifold of^Rn, which is the target to be reached within minimal time.

This problem is said to be at if and only if

Y

is everywhere tangent to

N

. Note that problems _^P1 and _^P2 dened in section 1.1 match the above

v E

;

;

;

; c

1

Figure 9 (i):

v

⁰

< v

^;

E

0

;

;

Figure 9 (ii):

v

⁰

> v

^;

E

0 no optimal

trajectory

local synthesis near^E0 =^E\(

v

=

v

⁰) in the leaf

v

=

v

⁰ when

>

1

non-accessible points: and on ^E, with ^E; accessible Figure 9 (iii): local synthesis near ^E; when

>

1

;

A

1 v

;

;

;

E

;

Figure 9 conditions and are both at.

NotationsLet us denote by

u

s the singular control and by

¹ (resp.

²

s) any arc satisfying (2.1) with control

u

¹ (resp.

u

²

u

s).

2.1.2. Switching function. For any extremal (

xpu

) the Hamiltonian is

H

(

xpu

) =^h

pX

(

x

)+

u Y

(

x

)ⁱ(from section 1.3.1 applied to system (2.1)).

Hence the equation

@H

@u

= 0 (which is satised by any singular extremal) is equivalent to^h

pY

(

x

)ⁱ= 0. The mapping:

:

t

^7!h

p

(

t

)

Y

(

x

(

t

))ⁱ (2.2) evaluated along (

xp

) is called the switching function.

Computing we get:

_(

t

) =^h

p

(

t

)

XY

](

x

(

t

))ⁱ (2.3) and along any smooth extremal:

(

t

) =^h

p

(

t

)

ad²

X

(

Y

)(

x

(

t

))^;

u

(

t

)ad²

Y

(

X

)(

x

(

t

))ⁱ (2.4)

where the Lie bracket of two analytic vector elds

Z

¹

Z

² is computed with the convention:

Z

¹

Z

²](

x

) =

@Z

²

@x

⁽

x

)

Z

¹(

x

) ^;

@Z

¹

@x

⁽

x

)

Z

²(

x

), and where ad

Z

¹(

Z

²) is the mapping dened by ad

Z

¹(

Z

²) =

Z

¹

Z

²].

2.1.3. Elementary regular cases.

Proposition 2.1. Let

z

⁰ = (

x

⁰

p

⁰) ^{2 Rn} (^Rnnf0^Rng) be such that

h

p

⁰

Y

(

x

⁰)^{i 6}= 0. In a neighborhood of

z

⁰, any extremal is a

¹ (resp.

²) arc if and only if ^h

p

⁰

Y

(

x

⁰)ⁱ is negative (resp. positive).

Proof. From PMP any extremal maximizes the Hamiltonian.

Corollary 2.2. Let (

xpu

) be an extremal with switching time

then necessarily^h

p

(

)

Y

(

x

(

))ⁱ= 0, i.e. (

) = 0.

Proposition 2.3. Let

z

⁰ = (

x

⁰

p

⁰)^2Rn(^Rnnf0^Rng) be such that

Y

(

x

⁰)⁶= 0, ^h

p

⁰

Y

(

x

⁰)ⁱ = 0 and ^h

p

⁰

XY

](

x

⁰)^{i 6}= 0: such a point

z

⁰ is called a normal switching point. Then close enough to

z

⁰ there is only one extremal containing

z

⁰, which is

²

¹ (resp.

¹

²) if and only if ^h

p

⁰

XY

](

x

⁰)ⁱ is negative (resp. positive),

x

⁰ being the switching point.

Proof. Consider (

xpu

) an extremal that contains

z

⁰, and let

⁰ be the time such that

z

⁰ = (

x

(

⁰)

p

(

⁰)). Since ^h

p

⁰

XY

](

x

⁰)^{i 6}= 0 i.e. (from (2.3)) _(

⁰)⁶= 0, close enough to

z

⁰ the considered extremal has no singular arc (indeed, from section 2.1.2, any singular arc satises 0). Then the result comes from the maximization of the Hamiltonian (cf. PMP) after expanding at

⁰ up to the rst order.

2.1.4. Singular extremals: generalities.

Definitions and notations. A point (

xp

) is called ordinary if and only if ^h

p

ad²

Y

(

X

)(

x

)ⁱ⁶= 0 let

be the set of non-ordinary points. A singular extremal (

xpu

) such that (

x

(

t

)

p

(

t

)) ^{2 R2nn}

is said to be of order 2.

Let ! be the variety ^f(

xp

) ^{2 Rn}(^Rnnf0^Rng) ^h

pY

(

x

)ⁱ = 0^g and let

!⁰=^f(

xp

)^2Rn(^Rnnf0^Rng) ^h

pY

(

x

)ⁱ=^h

p XY

](

x

)ⁱ= 0^g included in

!.Let

H

s be the restriction to !⁰ⁿ

of the mapping: (

xp

) ^{7! h}

pX

(

x

) +

u

s(

xp

)

Y

(

x

)ⁱ, where:

u

s(

xp

) = ^h

p

ad²

X

(

Y

)(

x

)ⁱ

h

p

ad²

Y

(

X

)(

x

)ⁱ (2.5) Proposition 2.4. Any singular extremal (

xpu

) is such that (

x

(

t

)

p

(

t

))²

!⁰. Moreover the singular extremals (

xpu

) of order 2 are dened by:

u

(

t

) =

u

s(

x

(

t

)

p

(

t

)) where

u

s(

xp

) is dened in (2.5), and (

xp

) is a solution to the ordinary dierential analytic system:

dx dt

⁼

@H

s

@p

⁽

xp

) and

dp

dt

^=;

@H

s

@x

⁽

xp

) with the following admissibility constraints:

u

¹

u

s(

xp

)

u

² Proof. See 4].