How Set-Valued Maps Pop Up in Control Theory

Working Paper

How Set-Valued Maps Pop Up in Control Theory

Halina Frankowska *

WP-96- 1 16 December 1996

5 2 11ASA

International Institute for Applied Systems Analysis A-2361 Laxenburg Austria

.

L A .

.IMI. Telephone: +43 2236 807 Fax: +43 2236 71313 E-Mail: info@iiasa.ac.at

How Set-Valued Maps Pop Up in Control Theory

Halina Frankowsku *

WP-96-116 December 1996

"CIVRS, LIRA

749,

CEREMADE, UniversitC Paris-Dauphine F-75775 Paris Cedex 16

and

International Institute for Applied Systems Analysis 2361 Laxenburg, Austria

T l

orklng P a p e r s

are interim reports on work of the International Institute for Applicd Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its Kational Member Organizations, or other organizations supporting the work.

FfJllASA

International Institute for Applied Systems Analysis A-2361 Laxenburg Austria

.L .A

...

Telephone: +43 2236 807 Fax: +43 2236 71313 E-Mail: info@iiasa.ac.at

How Set-Valued Maps Pop Up in Control Theory

H. Frankowska

CNRS URA 749, CEREMADE Universitk Paris-Dauphine 75775 Paris Cedex 16 France

Abstract

We describe four instances where set-valued rnaps intervene either as a tool to s t a t e t h e results or as a technical tool of the proof. T h e paper is composed of four rather independent sections:

1 . Set-Valued Opti~rial Synthesis and Differential I ~ ~ c l u s i o n s 2. Viability Kernel

3. Nonsmooth Solutions to Hamilton-Jacobi-Bellnlar~ Equations 4. Interior and Boundary of Reachable Sets

1 Optimal Synthesis

We define optimal synthesis in two cases: for the Mayer problem with locally Lipschitz value function and for the time optillla1 control problem with lower serniconti~luous time optill-la1 function.

1.1 Mayer Problem with Lipschitz Value Fuiiction

Consider a complete separable metric space I J , a continuous f : Rn x

U

H

R n , a locally LipsChitz p : Rn ^H R a n d the nlil~irnization problem inin (p(z(1))

1

^{x is solves} ( I ) , ^z(0) =

< o )

( 1 ) xl(t) = f(x(t),u(t)), 7 4 t ) E

U

T h e value function of this problem is defined by

V(to,

z O ) = inf (p(x(1))

1

^z solves ( I ) , x(t0) = xo)

T h e value function generates the optimal sy~lthesis since it is constant along optimal trajectories. I t is well known t h a t in general it is nonsmooth. If

V R

>

0, 3 ^{C R}

>

0, V u ,

f(.,

u ) is C R - Lipschitz on B R ( )

{ ii)

^{3 k}

^>

^0,

^v

x , supu,u

11

^{f ( x , u),I}

5

k(1

+

^11x11)

where B R denotes the closed ball of center zero and radius R, then the value function is locally Lipschitz (see for instance [25, FLEMING & RISHEL]).

T h e opt.ima1 feedback set-valued map is given by

denotes the directional derivative of V in the direction ( 1 , v ) .

where -

a ( l , v )

T h e sets U ( t , x ) may be empty a t points where V is not differentiable. T h e

"optimal" control system can be described then in t h e following way

A natural question arises : W h a t are t h e solutions o f t h e above closed loop system? A possible answer comes from the theory of differential iiiclusioiis:

Solutio~ls are absolutely continuous functions such t h a t

Let us introduce t h e set-valued m a p of "optimal dynamics"

Theorem 1.1 ([30, H.F.]) Asstrine that V zs locally Lzprchttz. T h e n the followzr~g two state~rlents are equivalent:

2) x solver the dzfferentzal zncluszon

ii) x is optimal: V ( t O , x o ) = p ( x ( 1 ) )

Proof - T h e proof is extremely s i n ~ p l e . Fix a trajectory x of (3j and set $ ( t ) = V ( t , x ( t ) ) . Then $ is absolutely coritiriuous a n d for almost all t

If i ) holds true, then $ ~ ' ( t ) = 0 a.e. and thus Ij) is constant equal t o p ( x ( 1 ) ) . If i i ) is satisfied, then, y'll = 0 and thus &(t , x ( t ) ) = 0 a.e..

It was proved in [12, CANNARSA & FRANKOWSKA] t h a t for smooth problems G is upper semicontinuous but its values are not convex. We recall next t h e definition of upper semicontinuous maps.

Let X , Y be metric spaces a n d F : X ^-u Y be a set-valued m a p , i.e., V x E X , F ( x ) C Y . T h e (Painlevh-I<uratowski) upper limit is defined by

If

Y

is compact, then F is u p p e r s e m i c o n t i n u o u s on X if a n d only if

When the d a t a f , p are srnooth enough, then the value function V has

"regular" directional derivatives a n d therefore the i m p C; inherits upper selnicontinuity, b u t in the same time the function

is concave. If it is both concave ant1 convex, then

V

is differentiable a t ( t , ~ ( t ) ) . So t h e values of G' may be nonconvex a t points where

V

is riot differentiable. We would like t o underline t h a t qualitative theory of dif- ferential inclusions is build for upper semicontinuous set-valued maps with convex values. Most of its results are not valid wi.thout convexity assump- tions. Because of t h a t one should not expect optimal trajectories t o have a nice s t r u c t u r e when

V

is nonsmooth.

1.2 Time Optimal Feedback

We describe next t h e problem for which t,he value function is in general dis- colitil~uous. Consider a colnplet,e s e p a r a l ~ l e lr~et,ric space lT and a cont.illuous map f :

R"

x lJ ⁺⁺

R".

Let y(.; L:, ,u.) denote the solution tlo

where u ( t ) E U a.lmost everywhere and let I<

c Rn

be a closed set. Consider the tiine optimal coiltrol problem f o r system

(4)

,with target I<:

T ( z ) = inf i n f i t >_ 0

I

y ( t ; x , u ) E I { } .

U

By the usual convention T ( x ) = ^+m when no trajectory s t a r t i n g a t x reaches Ir'. A vector p E

Rn

is called a (proximal) i ~ o r i n a l t o S

c Rn

a t a point x E

s

^if

d i s t s ( x

+

P) = JlpJl

Proxiinal n o r m a l s were introduced ill [9, BONY]. Tlie Hamiltonian associ- a t e d t o t h e above control systenl is defined by H ( z , p) = ( P , f (z., 1 1 ) ) .

T o definr t i m e o p t i m a l synthesis we need t h e following extension of directional derivative. For p : Rn ^k R U {+m) t h e u p p e r contingent derivative of p a t xo in t h e direction v is defined by

D L p ( x o ) ( v ) = l i m s u p ^{~ ( X O}

+

^{hv'j -} ~ ( x o )

h-0+, v'-ti 11

See [3, AUBIN & F R A N K O W S K A ] for properties of contingent derivatives.

In t h e two results below we impose a s s u m p t i o n s ( 2 ) a n d t h a t for all x E Rn, f ^{( i : ,} U ) is closed a n d convex.

Theorem 1.2 ([13, CANNARSA, H . F . & SINESTRARI]) Let G ( . ) be a fixed control such that the corresponding trajectory y ( . ) = y ( . ; X O , u) satisfies

for so7~te norrrial ^u to R"\I< at y ( T ) . T h e n ⁱⁱ is tinie 0pti7t)al ~f and only if, f o r every t E [0, To[,

y(.s) - y(1) V .u E D y ( t ) := Limsup,,,

s - t

T h e proof is n o t as straightforward as in t h e pl.evious section, since T ( . ) m a y b e merely lower semicontinuous. I t is shown first t h a t t h e co-state p ( . ) of Pontryagin's m a x i m u m principle verifies a n adjoint inclusion. T h e n t h e Viability T h e o r e i n f r o m [2, AUB[N] is applied t o sllow t h a t t ^H V ( t , y ( t ) ) is Lipschitz even when 1' is discontinuous. T h e above result suggests t o define t h e t i m e o p t i m a l synthesis in t h e following way:

T h e associated set-valued m a p of "optiinal dyrlamics" is

In view of T h e o r e m 1.2 it is n a t u r a l t o expect optimal trajectories t o solve t h e followiilg closed loop s y s t e m

Consider t h e differential inclusion

T h e t i m e o p t i m a l function T ( . ) being in general discontinuous, t h e argu- m e n t s f r o m t h e proof of T h e o r e m 1.1 are not valid any longer. For this reason we have t o change t h e notion of solution.

D e f i n i t i o n 1.3 ( [ 3 7 , M A R C H A U D ] ) A continuous m a p y : [O,To]

-

Rn is a contingent solution of (5) if

We a l r e a d y know t h a t every t i m e o p t i m a l solution is a contingent solution of (5) u n d e r all a s s u m p t i o n s of T h e o r e m 1.2. Conversely,

Tlleore~n 1.4 ( [ 1 3 , C A N N A R S A , H.F. & S I N E S T R A R I ] ) Suppose that y(.) i s a contingent solutton of ( 5 ) ill [O,To] salisfyzng

T h e n y i s t i m e optimal.

2 Viability Kernel

We provide n e x t t h r e e e x a m p l e s leading t o t h e notion of viability kernel.

E x a n l p l e 1: I l l l p l i c i t C o n t r o l S y s t e n i

T h e way t o m a k e it "explicit" is t o defirie t h e set-valued m a p F ( x ) = {,u 13 u E U, f ( x , v , u ) = 0 ) a n d t o s t u d y t h e differential inclusion

B u t in general F is n o t defined o n tlie whole s p a c e b u t only o n a subset, D o m ( F ) := {x

I

F ( x )

#

0 ) . F u r t h e r m o r e t h e r e a r e xo E D o r n ( F ) f r o m where n o t r a j e c t o r y defined over R+ of tlie control s y s t e m s t a r t s .

Exaliiple 2: Control Systeln w i t h State Co~istraillts

T o "get rid" f r o m t h e c o n s t r a i n t s let u s i n t r o d u c e t h e set-valued m a p

T h e new control s y s t e m is

Again U may be defined only over a subset I< = {x

1

U(x)

# 0).

Further- more, there are xo € I< from where no trajectory of the control system satisfying state constraints starts.

Exaniple 3: Bounded Chattering

The problem is t o find solutions to the control system

That is u has to be absolutely continuous and, in particular, it call not have jumps. Define a new dynamical system

= f ( x ( t ) , u ( t ) ) , 4 0 ) = xo1 u(t) E U(x(t)) ul(t) E BM

{

Naturally there may exist xo E Dorn(U) from where no trajectory of the above system starts.

In all three cases we reduced the control system under investigation to the following so called viability problem

xl(t) E F ( x ( t ) ) , forallnost all t 2 0 , x(t) E K , V t

2

0,

x(0) = xo E K

The viability kernel Viab(1i') of K (under F ) is the set of all initial conditions xo E Ii' from which starts a t least one solution (defined over

R + )

of the differential inclusion (6). T h e notion of viability kernel was introduced in [ I , ACTBIN']. If

F is upper semicontinuous with closed convex images ii)

>

0, s " p u ~ ~ ( r )

l l v l l ⁵

^/i(llxll

+

¹⁾

then the viability kernel Viab(K) is closed and enjoys some stal~ilit~y prop- erties. Algorithms were obtained to compute the viability kernel which for low dimensions run on PC's. See [36, FRANKOWSKA & QUIN- CIAMPOIX], [38, SAINT-PIERRE], [It5,16,17, CARDALIAGUET, QUIN- CAMPOIX & SAINT-PIERRE]. These (global) algorithms were inspired by "zero-dynamics" of [ l o , BYRNES & ISIDORI].

Since the notion of viability kernel revealed t o be very useful in "comput- ing" Lyapunov functions, time-optimal function, solving the target prob- lem and in some applied problems (see [8, BONNEUIL & MULLERS], [19, CARTELIER & MCTLLERS], [22, DOYEN & GABAY], [23, DOYEN, GABAY & HOURCADE] and also [2, AUBIN] and its bibliography) the research is carried out in Universitd Paris-Dauphine by P. Cardaliaguet, L. Doyen, M. Quincampoix, P. Saint-Pierre and N. Seube to perfection algorithms for computing the viability kernel.

3 Solutions t o HJB Equations

We address here the Hamilton-Jacobi-Bellnlan equation of optimal control.

Consider again the Mayer Problem from Section 1.1. As we already ob- served for locally Lipschitz d a t a the value function is locally Lipschitz. It is easy to understand how t h e generaIized (bilateral) solutions arise in the Lipschitz case.

Defi~litiorl 3.1 ([21]) Let

4

: X ^H R U {+m) be an extended function and xo E X be such that ~ ( x o )

#

m. The subdifferential of

4

at xo is :

T h e value function V is nondecreasing along solutions of the control system and is constant along optimal solutions. For this reason the following state- ment follows easily by classical arguments (see for instance [25, FLEMING

& RISHEL]): If V is differentiable a t ( t , x ) , then

T h e Hamiltonian H ( x , .) being convex, we have

where a V ( t , x ) denotes Clarke's generalized gradient.

On the other hand V is a viscosity solution t o the H J B equation (see [21, CRANDALL, EVANS & LIONS]). In part,icular,

But 8 - V ( t , x ) C a V ( t , x ) a n d therefore V is a bilateral solution:

This notion of solution is valid as well for lower seinicontinuous f i ~ n c t i o n s

[ 7 , BARRON & J E N S E N ] , [32,33, FRANKOWSKA]. I11 [7] the authors ex-

tended t h e maximum principle of PDE's to lower semicontinuous functions.

T h e alternative approach proposed in [33] is based on viability theory.

T h e "geometry" behind the method of proving uniqueness of nons- mooth solutions is the following one (details can be found in [4, AUBlN &

FRANKOWSKA]): Consider the reachable set R ( t ) a t time t of

for all possible choices of x l . Then R ( t ) is the epigraph of V ( t , .)

T h e semigroup properties of reachable sets are used t o investigate tangents to the epigraph of V . Namely consider the differential inclusion

where F is a locally Lipschitz set-valued m a p with convex compact images, and define its reachable m a p

R ( t , x o ) = { x ( t )

1

x solves the above inclusion) T h e n by well known results from [24, FILIPPOV]

where B denotes t h e closed unit ball. T h e fact t h a t V is a bilateral solution follows from monotonicity properties of the value function. Proofs of the converse are based on Viability Theory [2, AUBIN].

In conclusion, we have t o underline t h a t for problems with lower serni- continuous cost function, it is natural t o use subdifferentials rather t h a n upper directional derivatives, because subdifferentials are related t o tangents t o the epigraph of V : E p i ( V ) = { ( t , x , r )

I

^r

>

^{V ( t ,} x ) ) which is closed.

On t h e other hand t o construct optimal synthesis via subdifferentials one needs e x t r a assumptions which may be difficult to check. We used here upper directional derivatives, related to tangents t o the hypograph of V : { ( t , x , r )

1

r

5

V ( t , I ) ) not closed in general.

Next we discuss briefly the method of characteristics of H J B equations.

Since t h e Hamiltonian of Mayer's p r o b l e n ~ is not differentiable at, ( x , O ) , wr considel. t h e Bolza problem:

( P ) ^minimize

lr

L ( x ( t ) , u ( t ) ) d t

+

p ( x ( T ) ) over solution-control pairs ( 2 , u ) of control system

T h e Hamiltonian H in this case is defined by

arid t h e value function is given by

t o , x = i ; f l r L ( x ( t ; to, x o , ~ ) , u ( t ) ) d t

+

p ( x ( ~ ; t o , ~ o , u ) ) where x ( . ; to , x o , ^{U )} denotes t h e solution t o (7) corresponding t o t h e control u. T h e H J B equation is

If H is s m o o t h , t h e n t h e characteristic s y s t e m of this equation is t h e fol- lowing Hamiltonian s y s t e m

It is well known t h a t for nonconvex problems i t is n a t u r a l t o expect shocks for such system:

3 x l ( T )

#

z 7 ( T ) , 3 t o

<

T such t h a t z l ( t O ) = x 2 ( t 0 )

T h i s i m p l i e s t h a t f o r s o m e i n i t i a l c o n d i t i o n s a n d some i n i t i a l t i m e t o w e have m u l t i p l e o p t i m a l trajectories o r equivalently

3 z 0 such t h a t Lirns~p,,,~ { g ( t o l X I }

is n o t a singleton.

In t h e two results below we impose t h e following assumptions:

H I ) f , g a r e locally Lipschitz;

f ,

g , L ( . , u ) are differentiable, cp E C1 H 2 ) V ( t o , 2 0 ) E

[O,

T ] x Rn a n optimal solution of ( P ) does exist a n d V : [0, T ] x R" R is locally Lipschitz

H 3 ) L ( x , .) is continuous, convex and 3 c

>

0 such t h a t L ( x . u )

>.

c

1 1 ~ 1 1 ~

H4) For all r

>

0 , there exists k,

>.

⁰ such t h a t

V u E R m , L ( . , u ) is k, - Lipschitz on B,(O)

H 5 ) H' is locally Lipschitz a n d the Hamiltonian s y s t e m is complete.

Tlleorexn 3.2 Every solution ( x , p ) t o the Han~iltoilian system

\

- p l ( t ) = P to) E -Limsup,-,,,V~(to,~) is so that z is optimal.

Tlieore~n 3.3 (BYRNES & H.F.) .4ssu~r~e that H ( x , .) is strictly con- vex and let (?,I) be a trajectory-control paw. If

c

is an optinlal trajectory of the Bolza problem, then for all t € ] t o , TI, V is dzfferentrable at ( t , c ( t ) ) . Tlie above extends earlier results of [14, CANNARSA & SONER]) of cal- culus of variations. Further s t u d y of shocks is continued in [18, C A R O F F

& FRANKOWSKA].

4 Interior and Boundary of Reachable Sets

4.1 Local Controllability

Consider the control system

where f verifies (2) and f ( x ,

U )

are closed a n d convex. Its reachable set a t time t

>

0 is given by

R ( t ) = { x ( t )

1

x is solves ( 8 ) )

We address the following q ~ e s t ~ i o n : W h e n J:O E I n t , ( R ( t ) ) f o r a l l t

>

^{0 ?}

Let us first recall t h e Graves theoreln (1047): if f : ,A' ^r

Y

is

C"

and f ' ( x o ) is surjective, then V ^E

>

0 , f ( z 0 ) E I n t ( f ( B , ( a : o ) ) ) , where B , ( x o ) denotes the closed ball of center x o and radius E .

A very similar result holds true also for set-valued maps. Here we apply it to the reachable rnap R ( . ) . But in order to get such extension of Graves' the or en^, one needs to differentiate set-valued malls on metric spaces. Rrcall first the notion of Painlevk-Kuratowski lower l i m i t o f sets. Let F : S ?- Y be a set-valued m a p . T h e lower limit is given by

I,iminf,,,o F ( x ) :=

{

^r lim

^-

^{r o} y,

I

^11, E F ( x )

1

We introduce k-order v a r i a t i o n s o f reachable sets:

Notice t,hat for all k

>

^{1 ,} R ~ ( X O ) C R ~ + ' ( X ~ ) .

Tlieore~n 4.1 ([31, H . F . ] ) If 0 E f ( x 0 , U ) and for some v l , ... up E R k ( 0 ) 0 E Int co {vl, ..., u p }

then xo E I n t ( R ( t ) ) for all t

>

0 . Furthermore there exist L

>

0 , E

>

0 such that for all small t

>

0 , all yl E B , ( x o ) and y E R ( t ) there exists t 1 such that

Y1 E R ( t i ) & It1 - tl _< L

v m

4.2 Lipschitz Behavior of Controls

Consider again the control system (8) and let ( z , Z ) be its trajectory control pair. We impose assumptions (2) and t h a t f (., u ) E C' for all u . T h e linearized control system is given by

and the corresponding reachable set by R ~ ( T ) = { w ( T )

I

w solves ( 9 ) ) . Theorem 4.2 ([31, H.F.]) Assume that 0 E Int ( R L ( T ) ) . Then r ( T ) E I n t ( R ( T ) ) and there ezist E

>

0, L

>

0 such that for all b E B , ( z ( T ) ) we can find a control u(.) satisfying

4.3 Nonsmooth Maximum Principle

Consider the control system (1) a n d assume ( 2 ) . Let g : Rn

-

^{R~ be a}

locally Lipschitz function a n d Iio, I<, C Rn be closed. We impose t h e following end-point constraints:

Define t h e reachable set a t time one : R ( l ) = ( ~ ( 1 ) ^11: solves ( I ) , (10)).

Let r be a trajectory of ( 1 ) , ( 1 0 ) . It is well known (see for instance [20, CLARKE], [39, WARGA] etc.) t h a t if g ( r ( 1 ) ) is a boundary point of g ( R ( l ) ) , then a maximum principle holds true. T h e a i n ~ of this section is t o make evident t h a t behind there is an "alternative" inverse mapping theorem, which is much more t h a n the characterization of boundary of reachable sets. Recall t h a t generalized Jacobian of a locally Lipscllitz func- tion p : Rn ^H R m (see [20, CLARKE]) is defined by:

Theorelm 4.3 ([31, H.F.]) Let ( r , l ) be a trajectory-control pair of ( I ) , (10). Then at least one of the following two statentents holds true:

i ) 3 X E R k and an absolutely continuous p : [O,1]

-

R n not both equal to zero, satisfying the rnazimum principle

where N K ( x ) denotes the Clarke normal cone to I< at x and a,f the gen- eralized Jacobzan with respect to x .

ii) 3 L

>

0 , ^E

>

0 such that for all ( a , b, c ) E R k x Rn x R n satisfying

there exisls a trajectory-control pair ( x u , u ) such that

and ~ ( { t

I

^{u ( t )}

#

~ ( t ) ) )

i

L(lla - s ( x a ( l ) ) l l

+

llbll

+

IIcII).

In particular, if g ( t ( 1 ) ) is a boundary point of g ( R ( l ) ) , then the state- ment i) holds true.

T h e above results from the set-valued inverse mapping theorem on metric spaces. Denote by U the set of all measurable functions u : [ O ,

11

⁺ U . Let x ( . ; U , x O ) be the solution of (8) corresponding to the control u and define the set-valued map G : Rn x U

-

^{R x Rn x Rn by}

G ( x o , u ) = ( g ( x ( 1 ; u , x o ) ) , xo, x ( 1 ; u , x o ) ) - ( 0 ) x ICo x I\ll T h e "strategy" of tlie proof is the following one:

1. Approximate G via "smootli" maps by regularizing f and g 2. Use tlie inverse mapping theorem on approximations.

3 . Go t o the limit.

Regularization technics implying nonsmooth maximum principle go back [39, WARGA]. In [26, FRANKOWSKA] it was shown t h a t Warga's leme may be refined to get smaller objects than the derivatives contain- ers. T h e inverse mapping theorem used on approximations is Theorenr 4 . 4 below. Finally Stability Theorem 4.5 is applied to take limits.

Consider G : S I. Y , where X is a complete separable nretric space and Y is a Barlach space with the norm Gateaux differelltiable away from zero. Let yo E G ( x 0 ) . T h e graph of G is defined by

T h e first order "contingent" variation is defined by

G ( ' ) ( x ~ , yo) = L i m s ~ p ~ + ~ + G ( B h ( x 0 ) ) - YO h

Theorern 4.4 ([31, H.F.]) If for some ^E

>

0 , p

>

0, M

>

0

(11) P B

c r‘l

= ( G ( ~ ) ( X , ~ )

n

M B )

( 2 , Y ) E Graph(G) ( x I Y ) E B E ( X O , Y O )

then for all ( x i , y l , y2) E G r a p h ( G ) x Y near (xo, yo, yo) 1

dist ( x i , C i ( y z ) )

5

- IJyi - ~ 2 1 1 , where G - ' ( ~ ) = {x

1

y E G ( x ) ) P

Theorern 4.5 ([31, H . F . ] ) Consider set-valued maps { G ; ) i > o from a com- plete metric space X to a Banach space Y having closed graphs. Let yo E G o ( x o ) . W e assume that for some 6

>

0 and for every X

>

0 there

exists a n integer I A such that for all i

>

I A and all x E B a ( x o )

If G, have "a Lipschitt inverse" on a neighborhood of ( x o , yo) with the s a m e Lipschitt constant, t h e n so does G .

References

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[2] AUBIN J.-P. (1991) VIABILITY THEORY, Birkhauser, Boston, Basel, Berlin

[3] AUBIN J.-P. & FRANKOWSKA H . (1990) SET-VALUED ANALYSIS, Birkhauser, Boston, Basel, Berlin

[4] AUBIN J.-P. & FRANKOWSKA H. (to appear) Set-valued solutions t o the C a u c h y problem for hyperbolic systents of partial differential inclusions, NODEA

[5] AUBIN J.-P. 8.: FRANKOWSICA H. ( t o appear) T l t f viability kernel algorithm for computing value functions of i n j i n i t f h o r i z o n optintal control problems, JMAA

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