Set-Valued Analysis, Viability Theory and Partial Differential Inclusions

Working Paper

Set-Valued Analysis, Viability Theory and Partial Differential

Inclusions

Jean- Pierre Au bin He'line Frankowska

WP-92-60 August 1992

HIIASA

International Institute for Applied Systems Analysis o A-2361 Laxenburg Austria Telephone: +43 2236 715210 n Telex: 079 137 iiasa a o Telefax: +43 2236 71313

Set-Valued Analysis, Viability Theory and Partial Differential

Inclusions

Jean- Pierre Au bin H i l i n e Frankowska

WP-92-60 August 1992

Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute or of its National Rlember Organizations.

EIIIASA

International Institute for Applied Systems Analysis A-2361 Laxenburg Austria Telephone: +43 2236 715210 Telex: 079 137 iiasa a Telefax: +43 2236 71313

Set-Valued Analysis, Viability Theory and Partial Differential Inclusions

Jean-Pierre Aubin & Hblkne Frankowska CEREMADE, Universitk de Paris-Dauphine

&

IIASA August 26, 1992

FOREWORD

Systems of first-order partial differential inclusions - solutions of which are feedbacks governing viable trajectories of control systems ^- are derived. A vari- ational principle and an existence theorem of a (single-valued contingent) solution t o such partial differential inclusions are stated. To prove such theorems, tools of set-valued analysis and tricks taken from viability theory are surveyed.

This paper is the text of a plenary conference t o the World Congress on Nonlinear Analysis held a t Tampa, Florida, August 19-26, 1992.

Set-Valued Analysis, Viability Theory and Partial Differential Inclusions

Jean-Pierre Aubin & HClGne F'rankowska

1 Introduction

We explain how systems of first-order partial differential inclusions arise in the case of the simplest economic example one can think of: one consumer, one commodity.

Let K := [0, b] the subset of a scarce commodity x. Assume t h a t the consumption rate of a consumer is equal t o a

>

0, so that, without any further restriction, its exponential consumption will leave the viability subset [O,b]. Hence its consumption is slowed down by a nonnegative price which is regarded as a control. We assume that a bound c is set t o inflation. In summary, the consumption and the price evolve according t o the simple system of differential inclusions:

(1) i) for almost all t 2 0, xl(t) = ax(t) - u(t) ii) and - c

5

ul(t)

<

^c

subjected to the viability constraint

A subregulation map R : [0, b]

- ^R+

for this problem is a set-valued map R satisfying the viability property: from any xo E Dom(R), uo E R(xo), starts a solution (x(.), u(-)) to the above control system satisfying

By using tools of set-valued analysis, and specifically, the concept of contin- gent derivative of a set-valued map, we shall see that such subregulation maps are solutions t o the first-order partial differential inclusion

In particular, single-valued solutions ^T to

regarded as feed backs in control theory or planning procedures in economics, are of special interest. In this example, we can exhibit (at least) two of them.

For t h a t purpose, let us introduce t h e functions p! and p; defined on [O,

4

by

(

^{i ) p:(u) :=} 5(e-'"Ic - 1

+

^{:u) ~j -} uZ ^2c

I

^{i i ) p!(u) :=} -cea(u-ab)lc/a2

+

^{u / a}

+

^{e / a 2}

and t h e functions r! and rk defined on [0, b] by i ) r k ( x ) = u if and only if x = pL(u)

i i ) r ! ( x ) = 0 if x E [0, pL(0)] ( p ! ( ~ ) = s ( 1 - e-a2b/c

1)

[

ⁱⁱⁱ⁾ r i ( x ) = u if and only if r = p!(u) when x E [ p : ( 0 ) , b]

Then one can show t h a t both rk and r! are such single-valued solutions t o (3).

But, instead of looking for examples of solutions, one can look for the largest1 subregulation map, which can be shown t o exist and computed in this particular example: T h e subregulation m a p RC defined by

is the largest subregulation map.

Indeed, set ub(t) := uo

+

ct and u b ( t ) := uo - ct and denote by x u ( - ) and x b ( - ) the solutions starting a t zo t o differential equations x' = a x - un(t) and x' = a x

-

u b ( t ) respectively. Then any solution ( x ( . ) , u ( - ) ) t o the system (1) satisfies u b ( . )

<

^{u ( . )}

<

u f ( . ) and thus, xu(.)

5

x(.)

<

x b ( - ) because

1.) We first observe t h a t t h e equations of the curves t ^H ( x b ( t ) , u b ( t ) ) and t ^H ( x b ( t ) , u b ( t ) ) passing through ( x o , u o ) are solutions t o t h e differential equations

1 _{b l b}

dp! = --(aP! - u ) d u & dpc = --(apc - u ) d u

C

'in the sense that every subregulation map R satisfies Graph(R) C Graph(RC).

the solutions of which are

1

ii) &u) = ea(u~-u)/c(zO - uo/a

+

^c/a2)

+

^{u / a - c/a2}

Let

pb,

be the solution passing through (O,O), which is equal t o Pk(u) = +(e-au/c - 1

+

:u) and P!(u) = - ~ e ~ ( " - " ~ ) / ~ / a ~

+

^u/a

+

c/a2 be the solution passing through the pair (ab, b).

2.) If uo

>

rL(zo), then any solution (x(.), u(.)) starting from (zO, uO) satisfies

z ( t )

5

z b ( t ) = p:(ub(t))

<

^pi(u(t))

because pL(.) is nondecreasing. Hence, when z ( t l ) = 0, we deduce that u(tl)

>

0, SO that such solution is not viable.

If 0

5

uo

<

r!(z0), any solution (z(.), u(.)) satisfies inequalities

Therefore, when x(tl) = b for some time t l , its velocity z f ( t l ) = ab - u ( t l ) is positive, so that the solution is not viable.

3.) It remains to show that from any initial pair (zo, uo) where uo E RC(zo) starts a t least a solution. Actually, we shall construct a heavy so- lution, i.e., a solution for which the prices evolve with minimal velocity.

Assume for instance that uo

<

azo.

Since we want t o choose the price velocity with minimal norm, we take u'(t) = 0 as long as the solution z(.) t o the differential equation z' = ax - uo yields a consumption z ( t )

<

P!(uO). When for some time t l , the consump- tion z ( t l ) = &uo), it has t o be slowed down. Otherwise (z(tl

+

^{E ) , uO)}

will be below the curve p! and we mentioned that in this case, any solution starting from this situation will eventually cease to be viable. Therefore, prices should increase t o slow down the consumption growth. The idea is t o take the smallest velocity u' such that the vector (zf(tl), u') takes the state inside the graph of Rc: they are the velocities u' 2 z ' ( t l ) / p ~ ( u o ) . By construction, it is achieved by the velocity of xfl(.), which is the highest one allowed to increase prices. Therefore, by taking

and u(t) := uo+c(t - t l ) f o r t E [ t l , t l +(ab-uo)/c], we get a solution which ranges over the curve z n ( t ) = P!(uu(t)). According t o the above differential

equation, we see that z(t) increases t o b where it arrives with velocity 0 and the price increases linearly until it arrives a t the equilibrium price ab. Since (blab) is an equilibrium, the heavy solution stays there: we take z ( t )

=

^b

and u(t)

=

ab when t 2 t l

+

^uo/c.

One last remark: Quincampoix proved in [38,4:L] the semipermeability property of the part of the boundary of the graph of RC contained in the interior of [0, b] x R+: The solutions which reach this boundary cannot come back to it, and have t o remain on its boundary.

Looking for both single-valued and set-valued solutions t o systems of first-order partial differential inclusions is then the topic of this paper. We present it in the framework of control of systems under state constraints, which provided the motivation for studying this class of problems in the first place.

We shall review

1. The Tools coming from Set-Valued Analysis 2. The Tricks taken from Viability Theory

3. The Theorems dealing with single-valued and set-valued solutions t o systems of first-order partial differential inclusions

2 The Tools

2.1 Upper Limits of Sets

In this paper, X, Y, Z denote finite dimensional vector-spaces. The unit ball is denoted by B (or

Bx

if the space must be mentioned). Let Ii

c

^{X ,}

we denote by

dK(x) := d(x, K ) := inf 112 - yll

Y€K

the distance from x t o K, where we set d(x,0) := +oo. Upper Limits of sets have been introduced by PainlevC and popularized by Kuratowski in his famous book Topologie, so that they are often called Kuratowski upper limits of sequences of sets.

Let (Kn)nEN be a sequence of subsets of X . W e say that the subset

Limsupn+, K n := ^{X E X

I

^{lim n+m} inf d ( ~ , Kn) = 0)

is t h e upper limit o f the sequence K n 2

.

Upper limits are obviously closed and Limsupn,,Kn is t h e set o f cluster points o f sequences x, E K,, i.e., o f l i m i t s o f subsequences xnt E h',t.

2.2 Contingent Cones

Let K

c

X b e a subset o f a normed vector space X ^and x E h'. T h e contingent3 cone TK(x) is t h e upper limit o f t h e subsets ( K

-

x ) / h

so t h a t TK(x) is always a closed cone of "tangent directions" (which is convex when K is convex or, more generally, when the contingent cone is lower semicontinuous, a vector space when K is a smooth manifold).

2.3 Graphical Convergence of Maps

Let us consider a sequence o f set-valued maps Fn : X ^-c, Y. T h e set-valued m a p FU := ~iml,,,~, f r o m X t o Y defined by

is called t h e (graphical) upper limit o f t h e set-valued maps F,. Even for single-valued maps, this is a weaker convergence than the pointwise con- vergence: if f, : X ^H Y converges pointwise t o f , then, for every x € X , f (x) E fn(x). If the sequence is equicontinuous, then fn(x) = { f (x)).

The following result justifies the introduction of this concept of conver- gence:

Theorem 2.1 (Convergence Theorem) Let F, be a sequence of nontriv- ial set-valued maps from K

c

X to Y with uniform linear growth: there exists c

>

0 such that, for any n 2 0,

'and that the subset

is its lower limit. We shall use only upper limits in this paper, but symmetric definitions based on lower limits can be introduced as well.

3introduced by G . Bouligand in the 30's.

Let us consider measurable functions x, and y, from R to X and Y re- spectively, satisfying y, (w) E Fm (x,(w)) for almost all w E R.

If

I

^i, x,(.) converges almost everywhere t o a function x(.)

I

ii) y,(-) ^E L1(R; Y ) and converges weakly in L1(R; Y )

(

to a function y(.) E L1(R;Y) then for almost all w E 51, y(w) E T5Fn(x(w)).

2.4 Contingent Derivatives of Maps We introduce the differential quotients

of a set-valued map F : X ^?-. Y a t (x, y) E Graph(F).

T h e contingent derivative D F ( x , y) of F at (x, y) E Graph(F) is the graph- ical upper limit of differential quotients:

We deduce the formula

Graph(DF(x, Y)) = T G ~ ~ ~ ~ ( F ) ( ~ , Y ) Indeed, we know that the contingent cone

is the upper limit of the differential quotients graph(^)-(=,y)

h when h -t O+.

It is enough t o observe that

and t o take the upper limit t o conclude.

2.5 Weak Derivatives: Distributional and Contingent Deriva- t ives

Let us consider a single-valued map f : X ^H Y and its differential quotients V h f ( ~ ) ( v ) := (' ⁺ hV) h

-

('). The function f is Giteaux differentiable if these differential quotients converge for the pointwise convergence topol- ogy. This strong requirement can be weakened in (at least) two ways, each way sacrificing different groups of properties of the usual derivatives.

The distributional derivative is the limit of the difference-quotients x ^H V h f (z)(v) := f ( x ⁺ hv) - f ( x ) (when h ³ 0) in the space of dis-

h

tributions, and the limit is a vectorial distribution

D,

f E V1(X; Y) (and no longer necessarily a single-valued function).

Furthermore, one can define differential quotients of any vectorial dis- tribution T

c

V1(X;Y) and define the derivative of a distribution as their limit (when h + 0) in the space of distributions.

The contingent derivative is the upper graphical limit of the difference- quotients v ^H Vh f ( x ) ( v ) := f ( x ⁺ h y ) - f ( x ) (when h ^{I L}

-

^{0+), and}

the limit is a set-valued map D f (x) : X ^2~ Y (and no longer necessarily a single-valued function).

Furthermore, we have defined differential quotients of any set-valued map F : X ^2~ Y and defined the contingent derivative of a set-valued map as their limit the upper graphical limit (when h + 0+).

In both cases, the approaches are similar: they use (different) conver- gences weaker than the pointwise convergence for increasing the possibility for the difference-quotients t o converge, a t the price of losing some properties by passing t o these weaker limits (the pointwise character for distributional derivatives, the linearity of the differential operator for graphical deriva- tives).

2.6 Epilimits

For reasons motivated both by optimization theory and Lyapunov stability, we involve the order relation on R by characterizing extended functions V ^: X ^H R U {f w } by their epigraphs instead of their graphs.

T h e epigraph of the lower epilimit

of a sequence o f extended functions Vn : X ^H R ^U { t o o ) is the upper limit of the epigraphs:

One can check that

2.7 Contingent Epiderivatives

Let V : X H R U {f oo) be a nontrivial extended function and x belong t o its domain.

We associate with it the differential quotients

V(x t hu) - V(x)

?.t VhV(x)(u) :=

h

The contingent epiderivative D t V ( x ) of V at x E Dom(V) is the lower epilirnit of its differential quotients:

The contingent cone to the epigraph of V at ( x , V ( x ) ) is the epigraph of contingent epiderivative:

Indeed, we know that the contingent cone

is the upper limit of the differential quotients & P ( ~ ) - ('7 V(x)) when h ⁺ O+. It is enough t o observe that h

t o conclude.

We refer t o Set-Valued Analysis ([14]) for further details on these concepts and their properties.

3 The Tricks

Let us consider a control system (U, f ) defined by a feedback set-valued map U : X

-

^Z

a map f : Graph(U) ^H X describing the dynamics of the system governing the evolution

(5) i ) for almost all t , xf(t) = f (x(t), u(t)) ii) where u(t) E U(x(t))

Let us remark that when we take for controls the velocities, i.e., U(x) :=

F ( x ) and f (x, u) := u, we find the usual differential inclusion x' E F ( x ) . Conversely, the above system is the differential inclusion x' E F ( x ) in dis- guise where F ( x ) := f (x, U(x)).

We say that a closed subset K C Dom(U) is viable under (U, f ) if from any initial state xo E K starts a t least one solution on [0, co[ t o the control system (5) viable in K (in the sense that for all t

>

^{0, x(t) E K ) .}

We associate with any subset K

c

Dom(U) the regulation m a p RK :

K ^--t Z defined by

where TK(x) is the contingent cone to K a t x E K.

We say that K is a viability domain o f (U, f ) if a n d only if t h e regulation m a p RK is strict (has nonempty values).

The Viability Theorem holds true for the class of M a r c h a u d systems, which satisfy the following conditions:

i ) Graph(U) is closed ii) f is continuous

iii) the velocity subsets F ( x ) := f (x, U(x)) are convex iv) f and U have linear growth

Theorem 3.1 ( V i a b i l i t y Theorem) Let us consider a Marchaud control system (U, f ) . Then a closed subset K

c

Dom(U) is viable under (U, f ) if and only if it is a viability domain of (U, f ) .

Furthermore, any "open loop" control u(.) regulating a viable solution x(-) in the sense that

for almost all t, ~ ' ( t ) =

f

( ~ ( t ) , ~ ( t ) )

obeys the regulation law

(7) for almost all t , u ( t ) E R K ( x ( t ) )

Otherwise, i f K is not a viability domain of the control system ( U , f ) ,

there exists a largest closed viability domain of ( U , f ) contained i n h (possibly ' empty), denoted Viab(h7), called the viability kernel of K , and equal to the set of states xo E K from which starts a solution of the control system viable i n K .

Finally, the upper limit of closed viability domains K n of control systems ( U n , f,) satisfying uniform linear growth is a viability domain of ~ F u ( x ( - ) ) , where F M is the graphical upper limit of the maps defined by

F,(x)

:=

fn(x7 U n ( x ) ) .

What we are aiming a t , now, are closed loop or feedback controls ^{T ,} which are single-valued selections of the regulation map R K : V x E K , ^{T ( X )} E

R K ( x ) .

One can naturally use selection procedures of the regulation map. (See Chapter 6 of Viability Theory, [5, Aubin]). This raises some problems because the graph of the regulation map is not closed whenever inequality constraints

are involved in the definition of K . )

The idea we propose here is t o find systems of first-order partial differ- ential inclusions the solutions of which are such feedbacks.

The trick is then t o set a bound t o the velocities o f the controls: we asso- ciate with the control system and with any nonnegative continuous function

( 2 , u ) + ~ ( x , U ) with linear growth4 the system of differential inclusions ( 8 ) i ) ~ ' ( t ) = f ( x ( t > , 4 1 ) )

ii) ~ ' ( t ) E ~ ( x ( t ) , u ( t ) ) B

and we regard the condition u ( t ) E U ( x ( t ) ) as a new viability constraint defined on the state-control pairs by:

Observe that any solution ( x ( . ) , u ( . ) ) t o ( 8 ) viable in Graph(U) is a an absolutely continuous solution to the control system ( 5 ) .

- - - - - -

'which can be a constant p , or the function (z, u) -r c11uJ(, or the function ( z , u ) + c(11uII

+

^llzll+ 1). One could also take other dynamics u' E @ ( z , u) where is a Marchaud map.

From now on, we assume that K := Dom(U) ( b y setting U ( x ) := 8 when x

4

K i f needed.)

W e are looking for closed set-valued feedback maps R contained in R K (and thus, in U ) , called subregulation maps, the graph o f which i s made o f the initial state-control pairs yielding viable solutions t o the control system.

Among these subregulation maps, we shall be particularly interested by single-valued subregulation maps - which are closed loop controls we are looking for.

4 The Theorems

This is naturally possible thanks t o T h e Viability Theorem.

Theorem 4.1 Let us assume that the control system (5) satisfies

(9) i ) Graph(U) is closed

i i ) ^f is continuous and has linear growth

Let ( 2 , u ) --, p ( x , u ) be a nonnegative continuous function with linear growth and R : Z

-

^X a closed set-valued map contained in U . Then the two following conditions are equivalent:

a) ^- R is a subregulation map: from any initial state xo E D o m ( R ) and any initial control uo E R ( x O ) , there exists a state-control solution ( x ( - ) , u ( . ) ) to the control system (5) starting at ( x o , uo) and viable in the graph of R : Q t >_ 0 , u ( t ) E R ( x ( t ) )

b, ^- R is a solution to the system o f first-order partial differential inclusions

( 1 0 ) ^Q ( x , u ) E Graph(R), 0 E D R ( x , u ) ( f ( x , u ) ) - c p ( x , ~ ) B satisfying the constraint: Q x E K , R ( x ) C U ( x ) .

Such a subregulation map R is actually contained in the regulation map R K . The law regulating the evolution of state-control solutions viable in the graph of R takes the form of the system of differential inclusions

i ) x l ( t ) = f ( x ( t > , u ( t > ) ( 1 1 )

ii) u l ( t ) E G R ( x ( ~ ) , ~ ( t ) ) where the set-valued map G R defined by

is called the metaregulation map associated with R.

Furthermore, there exists a largest subregulation map denoted RV con- tained in U.

In the case of single-valued regulation maps, the system of first-order partial diflerential inclusions (10) can be written in the form

If r is differentiable and if we set B := [-I, +lIm, it boils down t o

In this case, it is a "viable manifold" of the characteristic system (8).

4.1 Heavy Viable Evolution

Assume that a subregulation map R is given. We introduce its minimal selec- tion gh associating with each state-control pair (2, u) the element gh(x, u) of minimal norm of DR(x, u)( f (x, u ) ) (which also minimizes the norm of elements of GR(x, u)).

We shall say that the solutions t o the closed loop differential system i) xl(t) = f ( x ( t ) , 4 t ) )

ii) ul(t) = g&(x(t), u(t))

are heavy viable solutions to the control system (U, f ) associated with R.

This minimal selection can be regarded as an instance of dynamical closed loop control.

Theorem 4.2 (Heavy Viable Solutions) Let us assume that U is closed and that f , cp am continuous and have linear growth. Let R(.)

c

U(.) be a subregulation map such that the associated metaregulation map is lower semicontinuous with closed convex images. Then from any initial state- control pair (xo, uo) in Gmph(R), there exists a heavy viable solution to the control system (U, f ) associated with R.

The case when the growth cp is equal t o 0 is particularly interesting, because the inverse ^{N O} of the 0-growth regulation map ^RO determines the

areas NO(u) regulated by the constant control u, called the viability cell or niche of u. A control u is called a punctuated equilibrium if and only if its viability cell is not empty. Naturally, when the viability cell of a punctuated equilibrium is reduced to a point, this point is a n equilibrium. So, punctuated equilibria are constant controls which regulate the control systems (in its viability cell).

Any heavy viable solution (x(.), u(-)) to the control system (U, f ) satisfies the inertia principle: "keep the controls constant as long as they provide viable solutions".

Lndeed, set

CR(U) := {x E K

1

0 E DR(x, u)(f(x, u)))

We observe that if for some time t l , the solution enters the subset CR(u(tl)), the control u(t) remains equal t o u(tl) as long as x(t) remains in CR(u(tl)).

Since such a subset is not necessarily a viability domain, the solution may leave it.

If for some tf

>

0, u ( t f ) is a punctuated equilibrium, then u(t) = ut, for all t 2 tf and thus, x(t) remains in the viability cell NP(u(tf)) for all t 2 t f .

This approach has been used in the regulation of AUV (autonomous underwater vehicles) by Nicolas Seube, when neural networks are introduces to learn in an adaptive way the feedbacks regulating viable evolutions of a tracking problem. See [47, Seube] and [7, Aubin] for further details.

We refer t o Viabililty Theory ([5]) for an exhaustive presentation of these concepts and their properties.

We shall derive the existence of a feedback control from a Variational Principle.

We denote by C ( K , X ) the space of continuous single-valued maps T :

A

'

^H Z . A closed (convex) process is a set-valued map whose graph is a closed (resp. convex) cone. Closed convex processes share most of the properties of continuous linear operators, and in particular can be transposed (see Chapter 2 of Set-Valued Analysis, [14]). The transpose of a closed process A : X ^-v, Y is the closed convex process A* : Y*

-

X * defined by

p E A*(q) if and only if V u, V v E A(u), (p, u)

<

^{(q, v)}

Since the contingent derivative DT(x) : X

-

Z is a closed process, we can define and use its transpose DT(x)* : Z*

-

^{X*. If T} is differentiable,

we obtain the usual transpose of the linear operator rl(x): Dr(x)* = rl(x)*.

We introduce the functional ^i@ defined by

This functional is lower semicontinuous on C(K, X ) (supplied with the com- pact convergence topology), even though this functional involves the "deriva- tives" of T. This basic nontrivial property of implies t h e following existence theorem:

Theorem 4.3 Let

R

C C ( K , Y) be a nonempty compact subset of selections of the set-valued map U (for the compact convergence topology.)5 Suppose that the functions f and (P are continuous and that

c := inf @(T)

<

+m ER

Then there exists a single-valued solution T(.) to the partial differential in- clusion

v

^{5 E} K , 0 E D r ( x ) ( f (x, r(x))) - ( ~ ( 2 , T(x))

+

^{c ) B}

which is a closed-loop control of the system (U, f ) , i.e., a continuous map satisfying r(x) E U(x) for every x E K such that from every xo E Ii' starts a viable solution to the diflerential equation xl(t) = f (x(t), r(x(t))) and

In the case of partial differential inclusions, this variational principle is related t o the concept of viscosity solutions. Naturally, this may force us t o change the initial bound on the growth of the velocity control by adding this constant c.

Another way to proceed is t o modify the bound on the velocity of the controls by replacing (8)ii) by

) = f(x(t), u(t>>

(12)

ii) ul(t) - Au(t) E cp(x(t), u(t))B

5Let us recall that the Michael Theorem implies that every lower semicontinuous map with closed convex values from a metric space to a Banach space has continuous selections.

where A E C ( Z , Z ) is a linear operator with X := i n f l l z l l = l ( A x , x )

>

0 large enough.

Then t h e associated single-valued subregulation maps r are closed set- valued solutions t o system of first-order partial differential inclusions

Theorem 4.4 Assume that the map f : X

x

Y ^H X is Lipschitz, that cp ^: X

x

Y ^H Y as Lipschitz with nonempty convex compact values and that

Let A E C ( Z , Z ) such that X

>

max(y,4ll f ) ) A ) ) c p l l A ) (where

( 1

^f

) I A

^denotes

the Lipschitz constant of f ) . Then there exists a bounded Lipschitz contin- gent solution to the partial differential inclusion (13), which is a closed-loop control of the system ( U ,

f),

i.e., a continuous map satisfying r ( x ) E U ( x ) for every x E K such that from every xo E K starts a viable solution to the differential equation x ' ( t ) = f ( x ( t ) , r ( x ( t ) ) ) and

References

[I] AUBIN J.-P. (1981) Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and diflerential inclusions, Advances in Mathematics, Supplementary studies, Ed. Nachbin L., 160-232

[2] AUBIN J.-P. (1982) Comportement lipschitzien des solutions de probltrnes de minimisation convexes, Comptes-Rendus de l'Acad6mie des Sciences, PARIS, 295, 235-238

[3] AUBIN J.-P. (1987) Smooth and Heavy Solutions to Control Problems, in Nonlinear and Convex Analysis, Eds. B-L. Lin & Simons S., Proceedings in honor of K y Fan, Lecture Notes in pure and applied mathematics, June 24-26, 1985,

[4] AUBIN J.-P. (1990) A Survey of Viability Theory, SIAM J. on Control and Optimization, 28, 749-788

[5] AUBIN J.-P. (1991) Viability Theory, Birkhguser, Boston, Basel, Berlin [6] AUBIN J.-P. ( t o appear) Static and Dynamical Economics: A Viability A p

proach,

[7] AUBIN J.-P. (to appear) Neural Networks and Qualitative Physics: A Viability Approach

[8] AUBIN J.-P., BYRNES C. & ISIDORI A. (1990) Viability Kernels, Con- trolled Invariance and Zero Dynamics for Nonlinear Systems, Proceedings of t h e 9th International Conference on Analysis and Optimization of Sys- tems, Nice, June 1990, Lecture Notes in Control and Information Sciences, Springer-Verlag

[9] AUBIN J.-P. & CELLINA A. (1984) Differential Inclusions, Springer-Verlag, Grundlehren der Math. Wiss.

(101 AUBIN J.-P. & DA P R A T O G. (1990) Stochastic Viability and Invariance, Annali Scuola Normale di Pisa, 27, 595-694

[ l l ] AUBIN J.-P. & DA PRATO G. ( 1990) Solutions contingentes de l'e'quation de la varie'te' centrale, Comptes-Rendus de 1'AcadCmie des Sciences, Paris, 311, 295-300

[12] AUBIN J.-P. & DA PRATO G. (1992) Contingent Solutions to the Cen- ter Manifold Equation, Annales de 1'Institut Henri PoincarC, Analyse Non LinCaire, 9, 13-28

[13] AUBIN J.-P. & EKELAND I. (1984) Applied Nonlinear Analysis, LViley- Interscience

[14] AUBIN J.-P. & FRANKOWSKA H. (1990) Set-Valued Analysis, Birkhauser, Boston, Basel, Berlin

[15] AUBIN J.-P. & FRANKOWSKA H. (1990) Inclusions auz de'rive'es partielles gouvernant des contr6les de re'troaction, Comptes-Rendus d e 1'AcadCmie des Sciences, Paris, 311, 851-856

[16] AUBIN J.-P. & FRANKOWSKA H. (1991) Syst6mes hyperboliques d'incluaions auz de'rive'es partielles, Comptes-Rendus de 1'Acadkmie des Sci- ences, Paris, 312, 271-276

[17] AUBIN J.-P. & FRANKOWSKA H. (1991) Feedback control for uncertain systems, in Modeling, estimation and control of systems with uncertainty, Di Masi, Gombani & Kurzhanski Eds, 1-21

[18] AUBIN J.-P. & FRANKOWSKA H. (1992) Hyperbolic systems of partial differential inclusions, Annali Scuola Normale di Pisa, 18, 541-562

[19] AUBIN J.-P. & FRANKOWSKA H. ( t o appear) Partial differential inclu- sions governing feedback controls, IIASA WP-90-028

[20] CARDALIAGUET P. (1992) Conditions sufisantes de non vacuitd du noyau de viabiliti, Comptes-Rendus de 1'AcadCmie des Sciences, SCrie 1, Paris,

[21] C A R O F F N. (to appear) Generalized solutions of linear partial differential equations with discontinuous coeficients, J. Diff. Int. Eq.

[22] C A R O F F N. & FRANKOWSKA H. (1 paraitre) Optimality and Shocks for Hamilton-Jacobi-Bellman equation, in Proceedings of the First Franco- Romanian Conference on Optimization. Optimal Control, Partial Differential Equations, Birkhiuser, Boston, Basel, Berlin

[23] DORDAN 0. (1990) Algorithme de simulation qualitative d t n e e'quation diffdrentielle sur le simpleze, Comptes-Rendus d e 1'AcadCmie des Sciences, Paris, 310, 479-482

[24] DORDAN 0. (to appear) Analyse qualitative, Masson, Paris

[25] DORDAN 0. (1992) Mathematicalproblems arising in qualitative simulation of a differential equation, Artificial Intelligence

[26] DOYEN L. (to appear) Inverse function theorems and shape optimization, [27] FORESTIER A. & LE FLOCH P. (to appear) Multivalued Solutions t o some

Nonlinear and Nonstrictly Hyperbolic Systems, Preprint CMA, 212, 1990 [28] FRANKOWSKA H. (1987) L'e'quation dJHomilton-Jawbi contingente,

Comptes Rendus de 1'Acaddmie des Sciences, PARIS, Sdrie 1, 304, 295-298 [29] FRANKOWSKA H. (1989) Optimal trajectories associated to a solution of

contingent Hamilton-Jawbi Equations, Applied Mathematics and Optimiza- tion, 19, 291-311

[30] FRANKOWSKA H. (1989) Nonsmooth solutions of Hamilton-Jacobi- Bellman Equations, Proceedings of the International Conference Bellman Continuum, Antibes, France, June 13-14, 1988, Lecture Notes in Control and Information Sciences, Springer Verlag

[31] FRANKOWSKA H. (1993) Lower semicontinuous solutions of Hamilton- Jacobi-Bellman equation, SIAM J. on Control and Optimization,

[32] FRANKOWSKA H. (1993) Control of Nonlinear Systems and Differential In- clusions, Birkhsuser, Boston, Basel, Berlin

[33] FRANKOWSKA H., PLASKACZ S. & RZEZUCHOWSKI T. (to appear) The'orBmes de Viabilite' Mesurable et l'e'quation dlHamilton- Jacobi-Bellman, Comptes-Rendus d e lYAcaddmie des Sciences, SCrie 1, Paris,

[34] FRANKOWSKA H. & QUINCAMPOIX M. (1991) L'algorithme de viabilite;

Comptes-Rendus d e 1'AcadCmie des Sciences, Sdrie 1, Paris, 312, 31-36 [35] FRANKOWSKA H. & QUINCAMPOIX M. (1991) Viability kernels of dif-

ferential inclusions with constraints: algorithm and applications, J. Math.

Systems, Estimation and Control, 1, 371-388

[36] FRANKOWSKA H. & QUINCAMPOIX M. ( t o appear) Value-functions for diflerential games with feedback stmtegies

[37] LIONS P.-L. (1982) Generalized solutions of Hamilton-Jacobi equations, Pit- man

[38] QUINCAMPOIX M. (1990) Frontikres de domaines d'invariance et de vi- abilit6 pour des inclusions difirentielles avec contmintes, Comptes-Rendus de 1'Acadkmie des Sciences, Paris, 311, 411-416

[39] QUINCAMPOIX M. (1990) Playable diflerentiable games, J . Math. Anal.

Appl., 194-211

[40] QUINCAMPOIX M. (1991) Target problems and viability kernels, in Mod- eling, estimation and control of systems with uncertainty, Di Masi, Gombani

& Kurzhanski Eds, 351-373

[41] QUINCAMPOIX M. (1992) Diflerential inclusions and target problems, SIAM J. Control and Optimization, 30, 324-335

[42] QUINCAMPOIX M. (1992) Enveloppes d'invariance pour des inclusions dif- fe'rentielles Lipschitziennes et applications auz problkmes de cibles, Comptes- Rendus d e 1'Acadkmie des Sciences, Paris, 314, 343-347

[43] SAINT-PIERRE P. (1990) Approzimation of slow solutions to diflerential inclusions, Applied Mathematics & Optimisation

[44] SAINT-PIERRE P. (to appear) Newton's method for set-valued maps, Cahiers de MathCmatiques de l a Decision

[45] SAINT-PIERRE P. ( t o appear) Viability property of the boundary of the viability kernel,

[46] SEUBE N. (1991) Apprentisage de lois de contr6le re'gulant des contraintes sur l'e'tat par re'seauz de neurones, Comptes-Rendus d e 1'AcadCmie des Sci- ences, Paris, 312, 446-450

[47] SEUBE N. (1992) Rkgulation de systkmes contr6lCs avec contraintes sur l'ktat par rCseaux de neurones, Thkse Universitk d e Paris-Dauphine [48] SEUBE N. ( t o appear) A Neural Network Approach For Autonomous Un-

derwater Vehicle Control Based On Viability Theory,

[49] SEUBE N. ( t o appear) A neural network learning scheme for viable feedback control laws,

[SO] SEUBE N. ( t o appear) Neural Network Learning Rules For Control: Appli- cation To A UV Tracking Control,

[51] SMOLLER J. (1983) Shocks waves and reaction-diffusion equations, Springer -Verlag