Spray control

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International Journal of Control

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Spray control

A. El Jai , K. Kassara & O. Cabrera Published online: 08 Nov 2010.

To cite this article: A. El Jai , K. Kassara & O. Cabrera (1997) Spray control, International Journal of Control, 68:4, 709-730, DOI:

10.1080/002071797223307

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Spray control

A. EL JAI

, K. KASSA RA

and O. CABRERA

The aim of this paper is to introduce the original concept of spray control. Owing to environmental motivations we have developed the concept of spreadability (see for example El Jai and Kassara 1994, 1996 ) . In this paper we investigate the associated control theory for linear distributed systems. The spray control problem concerns the determination of controls which generate the spreadability of the system. For that purpose, we propose a ® rst approach leading to an adaptive spray control based on regional control results and a second approach driven from linear quadratic control theory. Simulation results illustrate the second approach and show that convenient controls can transform a non-spreadable system into a spreadable one.

1. Introduction

The spreadability of distributed parameter systems (see El Jai and Kassara 1994 and references therein ) is motivated by the phenomena of expansion that one can usually observe in the environmental processes, such as vegetation dynamics, pollution or medical applications. The idea is related to the fact that the subdomains where the state of a distributed system is enforced to obey a spatial property are non- decreasing. Our purpose in this paper is to explore both theoretical and implementa- tion approaches related to these ideas but not to develop an application to a concrete problem which needs a huge amount of work on the ecological or physical aspects of the system. In this paper we shall develop the associated control theory, which will be called spray control theory. More precisely, we search for controls making a linear distributed parameter system spreadable. In El Jai and Kassara (1994 ) , we have considered the transport equation because of the role that it plays in modelling matter transfer processes. Although some characterizing conditions have been obtained in the homogeneous case, the problem is more complicated in the general one where the velocity ® eld is not constant or the sink/source term is taken into account.

In the introductory paper (El Jai and Kassara 1994 ) , we considered the problem for abstract linear distributed parameter systems. This has been formulated by quadratic equations in Hilbert spaces, and the results obtained have a theoretic interest, whereas the implementation aspect leads to some technical di culties.

In this paper, using optimal control techniques, we propose overcoming the above di culties by considering two di€ erent approaches. The ® rst approach consists of a time-discretized formulation of the spray control problem, and leads to a sequence of regional control problems for which we apply results of regional control theory developed by El Jai (see for example Amouroux et al., 1994 and El Jai

0020-7179/97$12.00Ñ 1997 Taylor & Francis Ltd.

Received 12 August 1996. Revised 7 April 1997.

²

IMP/CNRS-University of Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, France. e-mail: eljai@univ-perp.fr.

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et al. 1995 ) . The second approach is inspired by linear quadratic control theory, (see for example Balakrishnan 1981, Banks and Kunish 1984 ) , and consists of minimizing a criterion which takes into account the spreadability property and the energy and the measure of the complement of the spreadable zones. While the obtained solutions may not be necessarily an exact spray control, they are supposed to generate a `weak’

spreadability with satisfactory energy.

This paper is organized as follows. Section 2 concerns preliminaries on spreadability and the statement of spray control problem. Section 3 is devoted to weak spreadability and in section 4 we shall explore the adaptive spray control problem, by means of regional control theory. Section 5 is devoted to the statement of an optimization framework in order to solve the optimal spray control problem.

2. Spray control theory

Before introducing spray control theory we give a brief presentation of the concept of spreadability.

2.1. Preliminaries on spreadable systems

Let

be an open bounded domain in the euclidean space

and I = ] ^{0 , T} [ ^{a time}

interval. Consider a distributed parameter system (S) whose state is denoted by z(t) = z( . , ^t)

^{® t} Î ^I

Assume P to be a given property which may describe a spatial constraint on the state of the system (S) and let

x

= { ^x Î

| P ^{z( x} , ^t) } ^t Î ^{I (2 . 1)}

be the zones where the state obeys the property P , then we have the following de® nition.

De® nition 2.1: The system ( S) is P -spreadable during the time interval I along the ( x

)’s if the family ( x

)

Î

de® ned in (2.1 ) is non-decreasing.

Figure 1 illustrates an example of vegetation dynamics in the case of spreadability (the subdomains ( x

) are growing in space ) .

As regards the property P , various cases may be considered, the general one consists of the following constraint situation

P ^{z( x} , ^t) Û ^{( x} , ^t , ^{z( x} , ^t)) Î ^K

where K Ì

´ I ´ is a set of spatial state constraints. In our case we shall consider

P ^{z( x} , ^t) Û ^z( x , ^{t) =}

^{( x} , ^{t) (2 . 2)}

with

´ I ® as a desired target trajectory to be tracked during the time interval I, and we say that the system ( S) is

-spreadable. In the case where

= 0, the system is said to be null-spreadable and this holds, for instance, for deserti® cation processes where the zones ( x

) stand for the desert zones. Similarly, in an air quality process (see for example Friedman and Littman 1994 ) , the polution may be described by the same statement where

P ^{z( x} , ^t) Û ^{z( x} , ^{t) £ z}

where z is the concentration of pollution, and the zones ( x

) may be regarded as the

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unpolluted zones. Naturally it is easy to see, from the system theory point of view, that the above statement concerns only processes described by the distributed parameters systems de® ned on spatial domains. In El Jai and Kassara (1994, 1996 ) we dealt with spreadable dynamics in the case of the transport equation

¶ z

¶ t ⁺ v . Ñ z =

(z) + q

´ I (2 . 3)

It has been proven that, under certain conditions, system (2.3 ) is spreadable if a geometric condition involving the velocity ® eld and the initial zone holds. The reader may ® nd various examples of spreadable systems in El Jai and Kassara (1994 ) . 2.2. What is spray control?

Let us suppose that the system (S) is excited by a control v to be precisely de® ned later and denote the associated state by z(x , ^{t) = z(x} , ^t , ^{v) .} Furthermore let P ^{be as in} the previous subsection and ( x

) the subregions de® ned by

x

= { ^x Î

| P ^z( x , ^t , ^v) } ^t Î ^{I; v} Î ^{V (2 . 4)}

then we can de® ne a spray control as follows.

De® nition 2.2: The control v is a P -spray control if the family x

= ( x

)

Î

is non- decreasing (that is to say, the excited system is P -spreadable along ( x

) ) . In the case of (2.2 ) with

= 0, we say that v is a null-spray control.

An example of null-spray control is developed considering a transport equation in El Jai and Kassara (1996 ) . Obviously, the interest of the notion is to make spreadable any system by considering a convenient feedback spray control. The above de® nition clearly leads to various control problems related to spreadability.

(a ) The spray control problem concerns the existence and the determination of spray controls. No attention is given to the energy or the areas of spreadable zones.

(b ) In the case where the set of spray controls is not empty, it is possible to derive spreadability to the whole domain

in minimum time? What about the energy associated with these controls?

(c ) Given a non-decreasing family of subregions ( s

)

Î

, do controls leading to expanding a property along the subregions s

exist?

Figure 1. An example of a spreadable distributed system.

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Remark 2.3: Without loss of generality we may restrict (2.2 ) to the case where

= 0

(null-spreadability ) .

2.3. Spray control problem

Let us consider a linear distributed parameter system governed by the following di€ erential equation

Ç

z( t) = Az(t) + Bv ( t) 0 < t < T (2 . 5) with the following hypotheses.

(1 ) the Hilbert state space is assumed to be Z = L

(

);

(2 )

is the control space with

Ì L

( I ,

^);

(3 ) A is a linear operator which is densely de® ned on its domain

( A) and generates a strongly continuous semi-group (

(t))

on the state space Z;

(4 ) B is a bounded linear operator mapping the control space

into the state space Z.

For an initial condition given by

z(0) = z

Î

^{( A) (2 . 6)}

the solution z( t) = z( t , v) of the system ((2.5 ) , (2.6 )) is therefore given by

z( t , ^{v ) =}

^{( t) z}

⁺ ò

(t - ^{s) Bv ( s) ds t} Î ^{I; v} Î

^{(2 . 7)}

which may be written in the form

z( t , ^{v ) =}

^{(t) z}

^{+ G}

^{v v} Î

where G

is the bounded linear operator de® ned by

G

v = ò

( t - ^{s) Bv( s) ds t} Î ^{I; v} Î

Furthermore we consider, for any subregion x , the linear bounded operator

g

f Î ^L

⁽

) ® f / ^x Î ^L

⁽ x ) (2 . 8)

where f / ^x is the restriction of f to x . Let us consider the zones

x

= { ^x Î

| ^{z( x} , ^t , ^{v ) = 0} } ^t Î ^I , ^v Î

^{(2 . 9)}

where the state of the system is equal to 0 and assume that the initial zone

x

= { ^x Î

| ^z

^{= 0} } ^{/ =} » ^{(2 . 10)}

For the consistency of the problem we assume x

to be non-empty, which only means that we wish to spread a property that already exists on a non-empty set. The spray control problem may then be stated as follows.

Find v Î

^{such that:}

( 1) the family ( x

)

Î

is non-decreasing, ( 2) the functional constraint (2.5 ) is satis® ed

ü ïï ý ïï þ

(2 . 11)

This problem is very di cult and

4 and 5 are devoted to its solution by means of a

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simpli® cation hypothesis. The ® rst solution is based on discretizing the time interval I and considering an adaptive spray control problem based on a given non- decreasing family ( x

)

Î

. The second one consists of an extended tracking problem.

In both approaches the only possible achievement is to ® nd controls which make the system weakly spreadable. This weaker concept is developed in the next section.

3. Weak spreadability

3.1. De® nition of weak spreadability

It is well known that most distributed linear systems cannot be steered to reach a given desired pro® le (exact controllability ) . Consequently, it is clear that the stronger concept of spreadability, which is more constraining, is harder to achieve. This was noticed by Ucinski and El Jai (1997 ) , and has motivated the introduction of a weaker idea of spreadability. Various approaches of weak spreadability may be considered.

The most natural one consists of trying to steer the system as close as possible to a given pro® le along increasing subregions. So we consider the following de® nition.

De® nition 3.1:

(1 ) Given

> 0 and a pro® le p Î Z, the system (2.5 ) ± (2.6 ) is weakly p-spread- able with the tolerance

if there exists a family of subdomains ( x

) , ^{( x}

^{) Ì P(}

^{) (where P(}

) holds for the set of parts of

) such that (a ) x

É x

,

(b ) x

Ì x

; " t , ^s such that 0 £ t £ s £ T ,

(c ) x

=

,

(d ) i g

[ ^{z( . , t) - p( . )} ] i

£

meas ( x

) " t , ⁰ < ^t £ T where meas ( x

) is the measure of x

(i.e. simply the surface of x

) .

(2 ) The system is said to be weakly null-spreadable if it is weakly spreadable

with the pro® le p = 0.

Let us notice the following.

(1 ) The above de® nition may be relaxed by removing condition (c ) . (2 ) Condition (d ) may be rewritten in the form

i g

[ ^{z( . , t) - p( . )} ] i

meas ( x

) £

which means that, at any time t, an average deviation of the state z from the desired pro® le p over the current zone x

must not exceed a tolerance

. A weaker condition consists of replacing (d ) by (d  ^{) given by}

( d  ^{) ImG}

^{= L}

⁽ x

) " t , ^{0 < t £ T (3 . 1)}

which means that, for all t, the reached state on x

will be as close as possible to p (X holds for the closure of X ) . In what follows the weak spreadability is considered with the condition (d  ^{) .}

(3 ) Obviously, if a system is spreadable then it is weakly spreadable with the tolerance

= 0.

(4 ) Notice that if the system is weakly spreadable, the family ( x

) is not unique and each of such choices will correpond to a di€ erent value of

.

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De® nition 3.2: Any family of subregions verifying the conditions (a ) and (b ) of de® nition (3.1 ) is called a spread. We denote by

the set of spreads

S

= { ^{s = ( s}

⁾

Î

Ì P(

) | ^{( s}

⁾

^{and s}

^{É x}

} ^{(3 . 2)}

u

For technical reasons it is useful to consider the set of spreads verifying the condition (c ) of de® nition (3.1 ) and denoted by

Â

S

 ^{= {} s Î

| ^s

⁼

} ^{(3 . 3)}

An example of spread s Î

in a one-dimensional case is presented in Fig. 2.

3.2. Characterization of weak spreadability We have the following result.

Proposition 3.3: A spread s = ( s

)

Î

is such that s Î

 if and only if

s

= { ^x Î

|

^{( x) £ t} } ^{; t} Î ^{I (3 . 4)}

where

Î

^with

T

= {

Î ^L

⁽

) | ^{0 £}

^{£ T and}

/ ^x

⁼ 0 } ^{(3 . 5)}

Proof: Let s be in

 and consider (see Fig. 2 )

¿

( x) = inf { ^s Î ^I | ^x Î s

} ^x Î

Then one can easily see that, for all t Î ^I

x Î s

Þ

^{( x)} £ t

and the converse is true since the family s is supposed to be non-decreasing. Finally we have

s

= { ^x Î

|

^{( x) £ t} } ^t Î ^I

and therefore we obtain (3.4 ) .

Conversely, for s ³ t, we have, with (3.4 )

{ ^x Î

|

^{( x) £ t} } ^{= s}

^Ì { ^x Î

|

^{( x) £ s} } ^{= s}

and, with (3.5 ) , s

É x

and s

=

. Thus s = ( s

) Î

 ^.

Corollary 3.4: The system (2.5 ) ± (2.6 ) is weakly spreadable if a spread verifying (3.4 ) with (3.5 ) exists and is such that the condition ( d  ⁾ of de® nition (3.1 ) is satis® ed.

For more details on weak spreadability, see Ucinski and El Jai (1997 ) . In the next sections the spray control problem to be solved is derived from (2.11 ) assuming that, on the subdomains ( x

) the objective is to achieve the weak null-spreadability.

Figure 2. A spread in

(left ) and a spread in

 ^{(right ) .}

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4. Adaptive spray control problem 4.1. Problem formulation

We ® rst rewrite the problem (2.11 ) in a discretized version and then we develop its solution in the case where:

(i ) the subdomains are supposed given and increasing,

(ii ) the pro® le to be spread is p = 0 (case of weak null-spreadability ) . Let us consider a sequence (t

)

of the time interval I such that

0 = t

< t

< ´´´ < t

= T and denote by ( x

) the associated subdomains de® ned by

x

= { ^x Î

| ^z(x , ^t

, ^v

^{) = 0} }

Then the discrete version of (2.11 ) may be stated as follows:

® nd v = ( v

, ^v

, ^{. . .} , ^v

^{) where v}

Î

^{such that:}

(1) the sequence ( x

)

is non-decreasing;

(2) the functional constraint (2.5 ) is satis® ed.

ü ï ý ï þ ⁽⁴

. 1)

It immediately follows that any solution v of the above problem satis® es

z( t

, ^v

) = 0 on x

0 £ i £ m - ^{1 (4 . 2)}

which is equivalent to the inclusion

x

Ì x

(4 . 3)

Thereby, in the case where the subregions x

are assumed to be increasing and given, the adaptive null-spray control problem may be seen as a sequence of regional control problems (see for example Amouroux et al. 1994, El Jai et al. 1995c and Zerrick 1993 ) . For given and increasing (t

)

and ( x

)

, the problem becomes:

® nd v = ( v

, ^v

, ^{. . .} , ^v

^{) where v}

Î

^{such that:}

(1) z( t

, ^v

^{) = z( t}

, ^v

, ^z

, ^t

^{) = 0 on x}

^.

(2) the functional constraint (2.5 ) is satis® ed.

ü ïï ý ïï þ

(4 . 4)

The above problem (4.4 ) will be solved in the next section considering regional analysis theory.

Figure 3. The map

.

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4.2. Preliminaries on regional controllability

Consider again the linear system (2.5 and let x Ì

be such that meas ( x ) > 0 and 0 £ t

£ t

£ T for all i = 0 , ¹ , ^{. . .} , m. Then we have the following de® nition (see El Jai et al. 1995c, and Zerrick, 1993 ) .

De® nition 4.1: The system is said to be x -regionally controllable on the time interval I = [ ^t

^{, t}

] ^{if for all z}

^{= z( t}

^{) Î}

^{( A) and y}

^{Î L}

^{( x} ), there exists a control v Î ^L

^{( t}

, ^t

,

) such that

z(t

, ^v , ^z

, ^t

) / ^{x =} y

(4 . 5) where z( t

, ^v , ^z

, ^t

^{) = z(t}

, v) denotes the solution of the system (2.5 ) at the ® nal time t

from the initial condition z

at time t = t

.

Remark 4.2:

(i ) In the above de® nition we have pertinently added the time interval. In fact, contrary to the case of linear lumped parameter systems, the property of controllability depends upon the time interval for distributed parameter systems. This is the case for ® nite propagation velocity systems (for examples see Curtain and Zwart 1995 and El Jai and Pritchard 1988 ) .

(ii ) If the system is controllable (i.e.

-regionally controllable ) then it is x -

regionally controllable for any x Ì

.

Now the solution of the system (2.5 ) at time t

, from the initial state z

at t = t

, is given by

z(t

, ^{v ) =}

^{( t}

- ^t

^{) z}

^{+ Gv v} Î

where the bounded linear operator G is de® ned by Gv = ò

(t

- ^{s) Bv(s) ds v} Î

Considering the operator g

de® ned in (2.8 ) , the relation (4.5 ) may be expressed as follows

g

Gv = y

- g

(t

- ^t

^{) z}

^{(4 . 6)}

Consequently, according to De® nition 4.1 the system is x -regionally controllable on the time interval [ ^t

^{, t}

] if and only if

Im ( g

G) = L

( x ) (4 . 7)

Remark 4.3: It is convenient to mention that regional controllability as de® ned above is very strong. In fact every compact system (see for example Curtain and Pritchard 1978 ) is not controllable. Thus, weak regional controllability is de® ned by Im ( g

G) = L

( x ) despite (4.7 ) and most distributed parameter systems are only weakly x -controllable. Various examples and counterexamples are developed in Curtain and Pritchard (1978 ) , El Jai et al. (1983, 1995 ) .

4.3. Solution of the adaptive spray control problem

Given a sequence of increasing subregions ( x

)

, ^x

^Ì

^{for all i, the}

adaptive spray control problem may be stated as a sequence of weak x

-regional controllability problems. For i , ^{0 £ i £ m, let y}

Î ^L

⁽ x

) and consider the following minimum energy ® nal constraint control problem

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min

Î

ò

^{| v |}

^dt

where

W

= { ^v Î

| ^z(t

, ^v , ^z

, ^t

^{) / x}

^{= y}

ü ïïïï ý ïïïï þ

(4 . 8)

Then we have the following lemma.

Lemma 4.4: Suppose that the linear system (2.5 ) is x

-regionally controllable on the time interval [ ^t

^{, t}

] then for all y

the problem (4.8 ) has a unique solution v * ^{given by}

v

= G

z (4 . 9)

where

z = y

-

^{( t}

- ^t

^{) z}

^{on x}

0 otherwise

{ ^{(4 . 10)}

and G

denotes the pseudo-inverse operator of G.

Proof: We shall use pseudo-inverse theory (see for example Rall 1970 ) , in Hilbert spaces. First, if the system (2.5 ) is x

-regionally controllable, then for all y

, the set of admissible controls

is not empty. Now according to (4.6 ) , let us consider the operator H = g

G which is a linear bounded operator mapping the Hilbert space

onto L

( x

). The constraints set

in (4.8 ) may be given as follows

W

= { ^v Î

| ^{Hv = y}

- g

( t

- ^t

^{) z}

⁾ }

Therefore, using pseudo-inverse theory, the solution of problem (4.8 ) may be expressed as follows,

v

= H

(y

- ^g

^(t

- ^t

^)z

⁾

where H

denotes the pseudo-inverse of the operator H and it is well known that H

= G

g

whereas one can easily see by (2.8 ) that for all y Î ^L

⁽ x

)

g

y = g

y = y on x

0 otherwise

{

u

It is well known that, in this case, we have

G

= G

( GG

) ^-

(4 . 11) where

( G

)( . ) = B

( t

- ^{. )}

Î ^L

⁽

); t Î ^I

and

GG

z = ò

( t

- ^{t) BB}

y

^{( t}

- ^{t)z dt}

which is invertible when the controllability hypothesis holds.

Remark 4.5: If the system is assumed to be only weakly x

-regionally controllable then one has to assume that

is not empty, nevertheless the control expression (4.9 )

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holds and allows the system to reach approximately the desired state y

on x (weak x -controllability ) .

In (4.9 ) and (4.10 ) the optimal control v

has an open loop structure and only depends on the initial and the ® nal states. So we denote v

= G

z by

v

( t) = H ( t , ^z

, ^t

, ^y

, ^t

, ^x

^{) (4 . 12)}

where z

Î ^L

⁽

) , ^y

Î ^L

⁽ x

) and 0 £ t

< t

£ T . Then we have the following result.

Theorem 4.6: Assume the system (2.5 ) to be weakly controllable on I = [ ^{0 , T} ] ^{, then}

the adaptive null-spray control problem (4.4 ) has at least one solution.

Proof: The proof is based on a recursive scheme. Let 0 = t

< t

< ´´´ < t

= T

be a given sequence of times and consider (as in

4 . 1) the following subsets x

= { ^x Î

| ^{z( x} , ^t

, ^{v) = 0} } ^v Î

^{; k = 0} , ^{. . .} , ^m

Now since the system is x

-regionally controllable (because it is controllable ) then by lemma (4.4 ) the control

v

= H ( . , ^z

, ^t

, ⁰ , ^t

, ^x

^{) (4 . 13)}

steers the system to approach the zero state on x

z( t

, ^v

, ^z

, ^t

) / ^x

= 0 (4 . 14) Let z

= z( t

, ^v

, ^z

, ^t

) and consider the following set

x

= { ^x Î

| ^z

^{= 0} }

hence by (4.3 ) x

Ì x

. Therefore, one can follow the same procedure from the level k - 1 to the level k considering

v

= H ( . , ^z

-

, ^t

-

, ⁰ , ^t

, ^x

-

) z

= z( t

, ^v

, ^z

-

, ^t

-

)

x

= { ^x Î

| ^z

^{= 0)}

ü ïïï ý ïïï þ

(4 . 15)

Now consider the control

v such that

~

v / [ ^t

^-

^{, t}

] ^{= v}

^{k = 1 , . . . , m (4 . 16)}

Then it can be easily seen that the control

v solves the adaptive spray control.

Theorem 4.6 ensures that it is possible to achieve the weak null-spreadability of the system with a certain (unknown ) tolerance margin

using an adaptive control scheme.

Remark 4.7:

(i ) Clearly the control given by (4.16 ) and (4.15 ) is not unique, because at each level k there are many possibilities to choose the control v

. Nevertheless, from Lemma 4.4 the control (4.9 ) is a minimum energy control (see Curtain and Pritchard 1978, El Jai and Pritchard 1988, El Jai and Amouroux 1990 ) . (ii ) Clearly the initial set x

= { ^x Î

| ^z

^{= 0} } is not empty. Then, with the

spreadability hypothesis, the ( x

) are increasing and non-empty.

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4.4. Algorithmic approach

Theorem 4.6 leads to a directly implementable algorithm which can be considered for the general case of

-spreadability as de® ned earlier in (2.2 ) . Let

x

= { ^x Î

| ^z(x , ^t

, ^v

^{) =}

} ^v

Î

^{; k = 0} , ^{. . .} , ^m

where

is given in L

(

); k = 0 , ^{. . .} , ^m , then one can obtain a solution of the adaptive spray control problem replacing respectively the relations (4.13 ) and (4.15 ) by v

= H ( . , ^z

, ^t

,

, ^t

, ^x

^{) (4 . 17)}

v

= H ( . , ^z

-

, ^t

-

,

, ^t

, ^x

-

) z

= z(t

, ^v

, ^z

-

, ^t

-

)

x

= { ^x Î

| ^z

⁼

}

ü ïï ý ïï þ

(4 . 18)

Consequently this leads to the following algorithm, alg

Step 1. Initial data k = 0; z

Step 2. Consider x

= { ^x Î

| ^z

⁼

}

Step 3. Compute v

= H ( . , ^z

, ^t

,

, ^t

, ^x

^{) and z}

= z( t

, ^v

, ^z

, ^t

⁾

Step 4. Test if k < m go to 2

Step 5. Solution (4.16 ) gives the solution

v

The above algorithm may be simply implemented as Step 3 is easy to execute. For this step, one has to use Lemma 4.4 and expression (4.12 ) which de® nes the operator H involving the pseudo-inverse operator of G, given in (4.11 ) .

5. Linear quadratic spray control problem (LQS )

In this section we come back to the more general problem (2.11 ) where the spread along which the system will grow is unknown. So the purpose is to solve the problem which consists of ® nding a convenient control v which makes a linear system weakly spreadable. For that purpose we consider a quadratic criterion J which depends on v and the spread s along which the system will expand and will characterize the optimal control, and the associated optimal trajectory which minimizes J. The considered system is described by the linear equation

Ç

z( t) = Az(t) + Bv( t) 0 < ^t < ^T

z(0) = z

Î

^{( A)} } ^{(5 . 1)}

and is assumed to evolve in a given open bounded set

. We consider the same hypothesis as in the previous sections and denote x

= { ^x Î

| ^z

^{= 0} } ^.

5.1. Problem statement

The basis LQS problem consists of ® nding an optimal control which makes the system (5.1 ) weakly null-spreadable. It means that the control to be implemented will only achieve weak null-spreadability. In fact, as the spread along which the system will expand is also unknown, the problem will consist of ® nding both a control v Î

and a spread s = ( s

) Î

such that a quadratic functional is minimized.

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We consider the criterion de® ned by J

( v , ^{s ) =} ò

ò

z

( x , ^t , ^{v) dx dt +} a ò

^{| v |}

^{dt +} b F( s ) (5 . 2) where a and b are positive coe cients. The cost function (5.2 ) is a combination of three terms. Minimizing J

will lead to an optimal couple ( v , ^s ) which steers the state z as close to 0 as possible on the subdomains s

while the transfer energy is minimized. In the third term the functional F

® will be chosen in such a way that the system is spread over the widest area possible. For that purpose F will be such that the more F decreases the more the spread s enlarges.

In summary, the LQS control problem may be stated by min J

(v , ^{s )}

v Î

, ^s Î

} ^{(5 . 3)}

The solution of the above problem, when it exists, will be a couple (v

, ^s

^{) such}

that the spray control v

makes the system (5.1 ) weakly null-spreadable along s

if we restrict the spread s to vary in

 . Moreover, the solution of the problem (5.3 ) ensures the weak null-spreadability of the system (5.1 ) with a certain tolerance margin.

Remark 5.1:

(1 ) At this level one has to notice that the functional F has no explicit form, whereas the set

is not supplied with any suitable topological structure, enabling us to use the classical optimization results. These di culties will be removed in the next section.

(2 ) In the case of

-spreadability (2.2 ) , it su ces to replace z in (5.2 ) by z -

involving a linear tracking problem (see Curtain 1983 ) . The criterion J becomes

J

( v , ^{s ) =} ò

ò

[ ^{z(x , t , v ) -}

^{( x)} ]

^{dx dt + a} ò

^{| v |}

^{dt +} b F( s ) (5 . 4) According to remark (5.1 ) we shall begin by reformulating the problem (5.3 ) . 5.1.1. Choice of F. First notice that if F is such that

meas ( s

) £ ^kF( s ) s Î

^{; t} Î ^{I (5 . 5)}

hence minimizing F( s ) may lead to smaller meas ( s

) and then larger s

( s

holds for the complementary of s

with respect to

) , see Fig. 4. The functional F in (5.2 ) may then be chosen by considering

F( s ) = F(

) = ò

¿²

( x) dx (5 . 6)

where s and

are given by (3.4 ) and (3.5 ) . This choice satis® es the condition (5.5 ) . Indeed for

Î

^and s = ( { ^x Î

|

^{(x) £ t} } ⁾

Î

in

 then, applying Markov inequality, we have

meas ( s

) = meas ( { ^x Î

|

^{( x)} > ^t } ^{) £ 1} _t ò

¿²

( x) dx 0 < ^t < ^T

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If we denote J( v ,

^{) = J( v} , ^{s ) with s and}

satisfying (3.4 ) and (3.5 ) and apply the Fubini formula to (5.2 ) one can see that the criterion J

may be rewritten in the form J(v ,

^{) =} ò

( ^ò

^z

^{( x , t , v) dt} ) ^{dx + a ò}

^{| v |}

^{dt + b ò}

^{( x) dx (5 . 7)}

It follows from the propositions (3.3 ) and remark (5.1 ) that the optimal spray control problem (5.3 ) may be formulated as follows

min J( v ,

⁾

v Î

,

Î

} ^{(5 . 8)}

Moreover, it turns out that the constraints set

´

is now a closed, convex and bounded subset of the Hilbert space L

( I ,

^{) ´ L}

⁽

^).

Remark 5.2:

(1 ) The functional J is not quadratic with respect to the couple (v ,

^{) . Never-}

theless it is coercive in the product space

´

, that is, due to

J( v ,

^{) ³ min (} a , ^{b )} _i ⁽ v ,

⁾ _i

⁽⁽ v ,

⁾ Î

´

) (5 . 9) (2 ) For ® xed

the formulation (5.8 ) , with (5.7 ) , leads to a classical linear

quadratic control framework (see for example Lions 1968 ) .

5.2. Existence of L QS solution

This subsection will be devoted to the analysis of the question of the existence of a solution (v ,

) to the optimal spray control problem (5.8 ) . For this purpose we have the following result.

Theorem 5.3: L et us assume that: v ® z = z(v) maps bounded sets of

into bounded sets of L

(

´ ^I) with r ³ 4, then the optimal spray control problem (5.8 ) has at least one solution.

Proof: As already mentioned, the constraints set

´

is a closed convex subset of

W

=

´ L

(

) and the objective functional J is coercive. Hence, if J is a weakly lower semi-continuous functional, there exists at least one optimal solution. Let w

= ( v

,

) be a sequence in

which converges weakly to w = ( v ,

⁾ Î

^{. Writing}

J( w

) = ( J( v

,

⁾ - ^{J( v}

,

^{)) + J( v}

,

^{) n ³ 1 (5 . 10)}

and with (5.7 ) we obtain

Figure 4. The functional F.

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J(w

) = r

+ b ( i

i

- i

i

^{) + J( v}

,

^{) (5 . 11)}

where

r

= ò

( ^ò

^z

^{(x , t , v}

^{) dt} ) ^{dx - ò}

( ^ò

^z

^{( x , t , v}

^{) dt} ) ^dx

which can be written in the following form

r

= ò ò

^q

^{( x , t)z}

^{( x , t , v}

^{) dx dt (5 . 12)}

with q

( x , ^{t) = c} [

,

] ^{( t)} - ^c [

,

] ^{( t) x} Î

, ^t Î ^{I (5 . 13)}

Now, since for all

, the functional J( . ,

) is convex and GaÃteaux di€ erentiable, then it is weakly lower semi-continuous and so

lim inf J( v

,

^{) ³ J( v} ,

^{) (5 . 14)}

Moreover, since the sequence

converges weakly to

, we have

lim inf i

i ³ i

i

^{(5 . 15)} On the other hand, by (5.13 ) and the Fubini formula, we have

i ^q

i

^{= 2} ò

max (

,

^{) dx} - ò

¿_n

dx - ò

¿

dx

and so q

converges to 0 in L

(

´ I). Now, since the sequence v

is bounded (because weakly convergent ) , it follows from the hypothesis of the theorem that the sequence z( v

) is bounded in L

(

´ I) , ^{r ³ 4, then z}

^{( v}

) is bounded in L

(

´ I) and therefore (5.12 ) gives

n

lim

¥ r

= 0 (5 . 16)

Finally, the relations (5.14 ) , (5.15 ) and (5.16 ) combined with (5.11 ) imply that lim inf J( w

) ³ J(w) and therefore the functional J is weakly lower semi-

continuous.

5.3. The optimality system

Hence we are concerned with deriving the equations which characterize optimal solutions of the spray control problem (5.8 ) . Since the constraints set is convex, any solution (v ,

) of the problem (5.8 ) satis® es the following inequalities (see for example Lions 1968 )

J Â ^{(v ,}

^{) ´ ( u - v ,} x -

^{) ³ 0 ( u} , ^{x )} Î

´

(5 . 17) where the objective functional is supposed to be GaÃteaux di€ erentiable on the set

V

´

. By considering partial derivatives the relation (5.17 ) becomes

¶ J

¶ v ´ ⁽ u - ^{v ) ³ 0 u} Î

¶ J

¶

^{´ (} x -

^{) ³ 0 x} Î

ü ïï ý ïï þ

(5 . 18)

But it is convenient to notice that the ® rst inequality in (5.18 ) is equivalent to

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¶ J

¶ v ( v ,

^{) = 0 (5 . 19)}

Now, let P( t , ^{x )} , ^t Î ^I , ^x Î

, be de® ned as follows. For given x Î

the operator Q = P( . , ^x ) stands for the solution of the following Riccati equation

d

d t k ^{Q( t)z}

, ^z

l ⁺ k ^{Q( t)Az}

, ^z

l ⁺ k ^{Q( t) z}

, ^Az

l ⁼ a ¹ k ^Q(t)BB

^{Q( t) z}

^{. z}

l - ò

^z

^z

^{d x}

Q( T) = 0

ü ï ý ï þ

(5 . 20)

where z

and z

are in

( A). Then one can show the following result.

Lemma 5.4: Assume that, for

Î

, the Riccati equation (5.20 ) has a unique solution P( . ,

⁾ then the Euler equation (5.19 ) is equivalent to

v + 1

a ^B

^{P( .} ,

^{) z( .} , ^{v ) = 0 (5 . 21)}

Proof: It su ces to show that we are in the linear quadratic control setting (see for example Banks and Kunish 1984 ) . For this purpose, let

J

(v) = ò

( ^ò

^z

^{( x , t , v) dt} ) ^{dx + a ò}

^{| v |}

^{dt v Î}

which may be expressed as follows

J

( v) = ò

k ^{R ( t) z(t} , ^v) , ^{z( t} , ^v) _l ^{dt + a} ò

^{| v |}

^{dt v} Î

with R ( . ) denoting the operator de® ned, for all

Î ^L

⁽

) and x Î

, by ( R( t)

)( x) =

( x) if

( x) £ t

0 if

( x) > t Finally notice that {

¶ J

¶ v ⁼ J Â

^{( v) ( v} Î

^{) (5 . 22)}

and that (5.21 ) is exactly the Euler equation J Â

^{( v) =} 0 which is associated with the linear quadratic regulator problem

min

Î

J

( v)

Therefore, we obtain the required result by (5.22 ) .

As regards the GaÃteaux di€ erentiability of the objective functional J with respect to x we have the following result.

Lemma 5.5: The functional J = J( u , ^{x )} is GaÃteaux di€ erentiable with respect to

x Î

^{for all u} Î

^{such that}

z( u) Î ^L ¥ (

´ I) (5 . 23)

In this case we have

¶ J

¶ x ^{( u} , ^{x ) z =} _k ^{2b x} - ^z

^{( .} , ^{x ( . )} , ^{u); z} _l ^z Î ^L

⁽

) (5 . 24)

Proof: Let u Î

satisfy (5.23 ) . It turns out from the expression (5.7 ) of the criterion J( u , ^x ) that the major di culty for computing ¶ J/ ¶ x comes from the ® rst term. Let

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