Thomas Le Mézo

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Thomas Le Mézo

^∗

, Luc Jaulin, Benoît Zerr

ENSTA-Bretagne, LabSTICC, 2 rue François Verny, Brest 29806, France

a r t i c l e i n f o

Article history:

Available online xxx Keywords:

Abstract interpretation ODE

Inﬁnity

Interval computation Dynamical systems

a b s t r a c t

Inthispaper,wepresentanewmethodforbracketing(i.e.,characterizingfrominsideand fromoutside)allsolutionsofanordinarydifferentialequationinthecasewheretheinitial timeisinsideanintervalandtheinitialstateisinsideabox.Theprincipleoftheapproach istocasttheproblemintobracketingthelargestpositiveinvariantsetwhichisincluded insideagivenset X.Althoughthereexists an eﬃcientalgorithmto solve thisproblem whenXisbounded,weneedtoadaptittodealwithcaseswhereXisunbounded.

1. Introduction

Inthispaper,wedealwithadynamicalsystemSdeﬁnedbythefollowingstateequation:

˙

x

(

^t

)

=f

(

^x

(

^t

))

⁽¹⁾

wherex(^t)∈Rⁿisthestatevectorandf:Rⁿ→RⁿistheevolutionfunctionofS.Denoteby

ϕ

ftheﬂowmapofthesystem.

Thismeansthatifattimet₀,theinitialstatevectorisx₀,thenthesolutionofthestateequationis

x

(

^t

)

⁼

ϕ

f

(

^t⁻^t0,x₀

)

^. ⁽²⁾

Inthispaper,weconsiderthattheinitialstate x₀ isnotknownexactly.Moreprecisely,x₀ belongstoabox[x₀]ofRⁿ. Twoproblemswillbetreated.

Problem1(Forwardreachableset). Wedeﬁnetheforwardreachableset[2,16,21]as

F⁺_[_x₀_]=

{

^x^a

| ∃

^x0∈[x0],

∃

^t≥0,xa=

ϕ

f

(

t,x0

) }

. (3) The problem that we will consider is to bracket the set F⁺_[_x

0] which means that we want to characterize this set from insideandfromoutside.Thissetcan beinterpreted asan approximationofall solutionsof (1),except thatwe loosethe dependencywithrespecttot.

Problem2(Positivegraph). Foraﬁxed,t₀andx₀,thepositivegraphofthesolutionof(1)correspondstotheset[3]

G⁺t0,x0=

{ (

t,xa

) |

^t≥t0,xa=

ϕ

f

(

^t−t0,x0

) }

. (4)

∗ Corresponding author.

E-mail addresses: [email protected] (T. Le Mézo),[email protected] (L. Jaulin),[email protected] (B. Zerr).

Please citethisarticleas:T.LeMézo etal., Bracketingthesolutions ofan ordinarydifferential equation withuncertain

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Fig. 1. Illustration of the set F ⁺_[_x₀_]. The orange backward trajectory is outside F ⁺_[_x₀_]since it never reaches [ x 0]. The black trajectory is included in F ⁺_[_x₀_]. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

Westillassumethatx₀∈[x₀]butalso,weconsiderthattheinitialtimet₀ isuncertainandisonlyknowntobelongtothe interval[t₀].Inthiscontext,wedeﬁnethepositivegraphastheset

G⁺_[_t₀_]_,_[_x₀_]=

{ (

t,xa

) |∃

^t0∈[t0],

∃

^x0∈[x0]

|

t≥t0,xa=

ϕ

f

(

^t−t0,x0

) }

, (5)

whichcanbeinterpretedasthesolutionofthestateequationwithuncertaininitialstateandtime.SimilarlytoProblem1, wewanttobracketthesetG⁺_[_t

0],[x₀] frominsideandoutside.

OurobjectiveistofindauniquealgorithmabletofindaguaranteedinnerandouterapproximationofthesetsF⁺_[_x₀_]and G⁺_[_t₀_]_,_[_x₀_][9].Someexisting approachesuseguaranteed integration[6,20,23]tobracketthosesets[7].Forefficiencyreasons wewillproposeinthispaper,aguaranteedapproachbasedonintervalcomputation[9,13]andconstraintnetworking[14]that donotuseguaranteedintegration.Themaindifferencewithexistingapproachesisthatbisectionswilltakeplacebothinthe timespaceandthestatespace,whichmakesthemethodbothEulerianandLagrangian [15].Thisincreasesthecomplexity ofthemethodbutallowsustohaveabettercontrolontheaccuracyoftheresults.

2. Mainresults

ThissectionshowsthatbothproblemsproposedinSection1canbeexpressedasthecomputationofthelargestpositive invariantset[10]whichisincludedinsideagivensetX.

AsetAispositiveinvariantforthesystem(1)ifforanytrajectoryx(·),wehave

x

(

⁰

)

^∈^A^,^t^≥⁰ ^⇒^x

(

^t

)

^∈^A^. ⁽⁶⁾

GivenasetX,wedenotebyIn

v

⁺(^f,X),the largestsubset ofX(withrespecttotheinclusion)whichispositiveinvariant.

Thelargestsetexistsandisunique,duetothefactthatthesetofpositiveinvariantsetsisacompletelatticewithrespect totheinclusion (e.g.,theunionortheintersectionbetweentwopositiveinvariant setsispositiveinvariant).From [3],we knowthat

xa∈In

v

⁺

₍

f,X

)

^⇔

ϕ

f

(

^[0^,^∞^]^,^x^a

)

^⊂^X^. ⁽⁷⁾

Asaconsequence,In

v

⁺(^f,X)^can^be^deﬁnedⁱⁿ^two^different^manners In

v

⁺

(

^f,X

)

=

{

A∈P

(

X

) |

Ais positiveinvariant

}

=

{

^x^a^∈^Rⁿ

| ϕ

f

(

^[0,∞],xa

)

⊂X

}

⁽⁸⁾

whereP(X)^is^the^power^set^ofX.

ThetwofollowingtheoremsshowthatourtwosetsF⁺_[_x

0]andG⁺_[_t

0],[x₀]canbedeﬁnedintermsofpositiveinvariantsets.

Theorem1. Wehave

F⁺_[_x₀_]=Rⁿ

\

^In

v

⁺

₍

−f,Rⁿ

\

^[^x0]

)

⁽⁹⁾

where\isthesettheoreticdifferenceoperator(i.e.,A

\

B=

{

^x∈A

|

^x∈/B

}

⁾^and−fistheoppositeoff(i.e.,

∀

^x,−f(^x)+f(^x)=0). Proof. Takeanelementx_aofIn

v

⁺₍₋f,Rⁿ

\

^[^x0]),asillustratedbyFig.1,wehave

xa∈In

v

⁺

(

−f,Rⁿ

\

^[^x⁰^]

)

⇔

ϕ

−f

(

^[0,∞],xa

)

⊂Rⁿ

\

^[^x⁰^]

(3)

(For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

⇔

∀

^t^≥⁰^,

ϕ

₋f

(

^t^,^x^a

)

^∈^Rⁿ

\

^[^x⁰^]

⇔

∀

^t≥0,

ϕ

₋f

(

^t,xa

)

∈/[x₀]

⇔ ¬

( ∃

^t≥0,

ϕ

₋f

(

t,xa

)

∈[x0]

)

⇔ ¬

( ∃

^x0∈[x0],

∃

^t≥0,

ϕ

₋f

(

t,xa

)

=x0

)

⇔ ¬

( ∃

^x⁰^∈^[^x⁰^]^,

∃

^t^≥⁰^,

ϕ

f

(

t,x0

)

=xa

)

⇔ ¬

(

^xa∈F⁺_[_x₀_]

)

.

Asaconsequence,thetwosetsF⁺_[_x₀_] andIn

v

⁺(−f,Rⁿ

\

^[^x0])^arecomplementary.

Theorem2. Wehave G⁺_[_t

0],[x0]=Rⁿ

\

^In

v

⁺

₍

g,Z

)

⁽¹⁰⁾

where

g

(

^t^,^x

)

⁼

−1

−f

(

t,x

)

, Z=Rⁿ⁺¹

\

^[^t⁰^]^×^[^x⁰^]^.

Proof.Takeanelement(ta,xa)ofIn

v

⁺(^g,Z),asillustratedbyFig.2,wehave

(

^t^a^,^x^a

)

^∈^In

v

⁺

₍

g,Z

)

⇔

ϕ

g

(

^[0,∞],

(

^t^a,xa

))

⊂Z

⇔

∀

^t1≥0,

ϕ

g

(

^t1,

(

^ta,xa

))

∈Z

⇔

∀

^t¹^≥⁰^,

ϕ

g

(

^t1,

(

^ta,xa

))

∈/[t0]×[x0]

⇔

∀

^t1≥0,

−t1+ta

ϕ

₋f

(

^t1,xa

)

/

∈[t0]×[x0]

⇔

∀

^t¹^≥⁰^,^−t¹⁺^t^a^∈^/^[^t⁰^]^∨

ϕ

−f

(

^t1,xa

)

∈/[x0]. Takingthecontraposite,weget

(

^t^a,xa

)

∈/In

v

⁺

₍

g,Z

)

⇔

∃

^t1≥0,−t1+ta∈[t0]

∧

ϕ

₋f

(

^t1,xa

)

^∈^[^x0]

⇔

∃

^t¹^,^t¹^≥⁰^,^−t¹⁺^t^a^∈^[^t⁰^]

∧

∃

^x0∈[x0],

ϕ

₋f

(

^t1,xa

)

=x0. Ifdeﬁnet₀=ta−t₁,weget

(

^ta,xa

)

∈/In

v

⁺

(

^g,Z

)

⇔

∃

^t⁰^,^t^a⁻^t⁰^≥⁰^,^t⁰^∈^[^t⁰^]

∧

∃

^x0∈[x0],

ϕ

₋f

(

^ta−t0,xa

)

=x0

⇔

∃

^t0∈[t₀],ta−t₀≥0,

∃

^x0∈[x₀]

ϕ

−f

(

^t^a⁻^t⁰^,^x^a

)

⁼^x⁰

(4)

Fig. 3. Illustration of a part of a maze where a possible trajectory is shown in orange. The {[ x i] }i∈{1,...,6}form a paving of X . (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

⇔

∃

^t⁰^∈^[^t⁰^]^,^t^a⁻^t⁰^≥⁰^,

∃

^x⁰^∈^[^x⁰^]^,

ϕ

f

(

^ta−t0,x0

)

=xa

⇔

(

^t^a,xa

)

∈G⁺_[_t₀_]_,_[_x₀_].

WehavethusprovedthatIn

v

⁺₍g,Z)^andG⁺_[_t

0],[x₀]aretwocomplementarysets.

3. Bracketingthelargestinvariantset

GivenasetX⊂Rⁿ,thesearch set,whichisbounded,we willshowinthispartthatthesetIn

v

⁺(^f,X)^can^be^bracketed frominsideandoutsideusingthealgebraofmazes.Bycombiningtheinner[19]andtheouter [18]approach,whichwere used separatelybefore andusedmazesalgebra,we providean eﬃcientmethod tobracketthesetIn

v

⁺₍f,X)^.^Indeed,^less bisectionsofthestatespacewillberequiredbecauseofthesimultaneouscharacterizationfrominsideandoutside.

Maze. Wewill now recallbriefly themain ideasbehindmazes.The difficulty to work withtrajectories isthat the di- mensionofthetrajectory spaceisinfinite.Thereforeweneed amathematical objecttograspandworkwithtrajectories:

themazes. Mazeswasintroducedusing theframeworkof ConstraintNetworktheoryto addressthisproblem. Aconstraint network[14] iscomposed ofa setof variables,constraints anddomains. Here,the variablesare the paths ofRⁿ that are consistentwith(1)andtheunique constraintis“thepathshould startfromtheinitialcondition[x₀]”.Mazesareatype of domainsthatallowtoenclosepaths,theyarecomposedof:

• ApavingPwhichcoversX,

• Apolygonineachbox[p]ofP thatenclosesallthepossibletrajectoriesinsidethebox,

• Doorsontheboundaryofeachbox[p] thatallowtrajectoriestoleaveorenterinsidepolygons.Inourimplementation, doorsareunionpolygons(oneperfaceof[p]).Wehavetwotypesofdoors:outputdoorsfromwhichthetrajectoriesare allowedtoleavethebox[p],andtheinputdoorsfromwhichtheyareallowedtoenterinside[p].

Fig.3showsa representationofapartofamazewhereleavingdoorsarepaintedblueandenteringdoorsarepainted redforbox[x₄].

Fromtheuniqueconstraint,wecandeducetwocontractorsthatwillbeappliedondoorsandpolygons[1,4,5]:

• Aﬂowcontractorthatcontractsthepolygonsanddoorsconsistentlywithf,

• Acontinuitycontractorthatcontractsthepolygonsanddoorsofaboxpaccordingtoitsneighbors.

Fig.4illustratestheprincipleofthecontinuityandtheﬂowcontractors.Theblueoutputdoorof[a]iscontractedaccord- ingtotheredinputdoorof[b]usingthecontinuitycontractors.Thepolygonsassociatedto[a]and[b]arethencontracted usingtheﬂowcontractor.Thislastcontractioncouldhavethencontractthedoorsofeachboxesbutthisisnotthecasein theexampleofFig.4.

Algorithm.Tocombinetheouterandtheinnerapproachwewillusetwomazes,onefortheinnerandonefortheouter approximation.Theinnerapproximation isverysimilartotheoutermethodasitworksonacomplementary approachas describedin[17].Weproposethefollowingalgorithmtobracketthelargestinvariantset:

1. Initialize the two mazes, one for the inner and one for the outer approximation, with the initial conditionwhich meansthatthedoorsandpolygonswhichcorrespondtotheinitialconditionaresetconsistentlywithit;

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Fig. 4. Illustration of the ﬂow and the continuity contractors for a maze . The maze at the top is before the contraction and the bottom maze is after applying the contractors. The gray cones represent the union of all possible trajectories consistent with f in the box. This cone can be computed using interval computation. The orange curve is a possible path enclosed by the maze . (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

Fig. 5. Bracketing F ⁺_[_x₀_] for the Van der Pol system. The magenta set corresponds to the inner approximation of F ⁺_[_x₀_] and the union of the yellow and magenta set to the outer approximation of F ⁺_[_x₀_]. The initial box [ x 0] is painted in red and the blue area corresponds to the set where there is no solution.

Gray cones correspond to the union of all directions of the vector ﬁeld in each boxes. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

2. Applythecontractorsoneachmazeuntilaﬁxpointisreachedfortheinnerandtheoutermaze;

3. ShowtheresultORbisectallboxeswheretheinnerandtheouterpolygonsarenotempty,andgotoStep1.

Test-cases

We will now test our algorithm on two test-cases. All computations were conducted on a single processor i5- [email protected].

Test-case1.VanderPolsystem.Considerthesystemdescribedbythestateequation:

x˙1=x2

x˙2=

1−x²₁

·x2−x1 (11)

Theinitialvectorsatisﬁesx0∈[x0]=[−4,−3]×[3,4].Ouralgorithmisabletocharacterizetheinnerandouterapproxima- tionofthe forwardreachablesetF⁺_[_x₀_] asshowninFig. 5.Thealgorithm wasstoppedafter17steps whichtakes8s. The searchset,whichalsocorrespondstotheframeboxoftheﬁgure,correspondstoX=[−6,6]×[−6,6].

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Fig. 6. Bracketing F ⁺_[_x₀_]for the Car on the hill system. The color code is the same as Fig. 5 . Now, since the boundary of the approximations intersets the boundary of the initial box [ x 0], we cannot guarantee anymore that the blue area is outside F ⁺_[_x₀_]. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

WehavetestedthisexampleonthesolverCAPD[23]whichisthestateoftheartofintervalintegration.Itworkswith a guaranteed Lagrangian approach. We start the simulation att=0, the solver rapidlydiverges andstops computing at t=0.57 withtheanswerx∈[−722.795,716.825]×[−3.24089e+8,3.24089e+8].Thisisduetothefact thattheinitialbox islarge.

Test-case2.Caronthehillsystem[12].Considerthesystemdescribedbythestateequation

⎧ ⎨

⎩

˙ x1=x2

˙

x2=−9.81sin

₁_.₁

1.2sin

(

^x1

)

−1.2sin

(

¹.1x₁

)

2

−0.7x2+2.0. (12)

andtheinitialbox[x₀]=[−1,1]×[−1,1]andasearchsetX=[−2,13]×[−6,6].Asin[18]weareabletocharacterizethe outerapproximationoftheset.Themainimprovementistobeabletocharacterizeaninsideandoutsideapproximationin thesametime:theneedtobisectboxesisthenreduced.Indeed,thebisectioneffortofthealgorithmisnowfocusedonthe boundarybetweentheinnerandtheouterapproximationwhereasonlargeouterapproximation zonesthatwerebisected inpreviousworks[18].TheresultisshownonFig.6.Thealgorithmwasstoppedafter18stepswhichtakes30s.

OneofthedrawbackofouralgorithmisthatwhentrajectoriesleavethesearchsetX,wearenotabletoconcludeifthat theywillnotcomebacklater.ThisisthecaseonFig.6wheretrajectoriescanleavethesearchsetXonitsrighttoppart.

Thenextsectionshowshowthisproblemcanbeaddressed, atleastforsomespeciﬁccases,byconsideringunbounded searchboxes.

4. Dealingwithinﬁnity

The previous section explainedhow to compute inner and outer approximationsof In

v

⁺(^f,X) ^whereX is a bounded box.Now,tocharacterizeapositivegraph,thesetXisnotboundedanymoreandtheabstractinterpretationapproachwill need tohandle unboundedintervals. Solving equations onan inﬁnite intervalscan still beperformed combininginterval constraintpropagationwiththegeneralizedinterval arithmetic[8].Inthisarithmetic aninterval isa connectedsubsetof theextendedrealnumbers,thesetofallrealnumbersaugmentedwith−∞and+∞.Forinstance,withthisarithmetic,

[−∞,∞]²∩

(

^[−∞,2]³−[−∞,∞]²

)

=[0,∞]∩

(

^[−∞,8]−[0,∞]

)

=[0,∞]∩[−∞,8]=[0,8].

Similarresultscouldhavebeenobtainedandprobablymademoreaccurate,usingthearithmeticofinﬁnity[22]. Example.Toillustratethepropagationprocess,considerthethreefollowingequations:

₍

_C

1

)

^: ^y⁼^x²

(

^C²

)

^: ^xy⁼¹

(

^C³

)

^: ^y⁼⁻²^x⁺¹^.

Using interval propagation, we want to prove that this system has no solution.To each of the variables, we assign the domain[−∞,∞].Then,we contractthe domainswithrespecttothe constraintsinthefollowingorder:C₁,C₂,C₃,C₁,C₂ andwegetemptyintervalsforxandy.AgeometricinterpretationofthepropagationisgivenonFig.7.

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Fig. 8. Illustration of the maze with unbounded boxes. The orange path corresponds to a trajectory that reaches the bottom inﬁnity box but stays inside it according to the green doors. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

Theresultingintervalcomputationisasfollows:

(

^C1

)

⇒y∈[−∞,∞]² = [0,∞]

(

^C2

)

⇒x∈1/[0,∞] =[0,∞]

(

^C3

)

⇒y∈[0,∞] ∩

( (

⁻²

)

^.^[0^,^∞^]⁺¹

)

=[0,∞] ∩

(

^[^−∞^,^1]

)

⁼^[0^,^1]

x∈[0,∞] ∩

(

^−[0^,^1]^/²⁺¹^/²

)

⁼

0,1 2

(

^C1

)

⇒y∈[0,1]∩[0,1/2]²=[0,1/4]

(

^C2

)

⇒x∈[0,1/2] ∩ 1/[0,1/4]=∅ y∈[0,1/4]∩1/∅=∅.

untilasteadybox(alsocalledtheﬁxedpoint)isreached.

Bisection.Toapplythemazemethod,we needtobisectintervalsthatcanbeunbounded.Now,intheintervalliterature, onlyboundedintervalsofRarebisectedandthecut pointisthecenter.Forinstance,theinterval[−1,3]isbisectedinto [−1,1] and [1, 3].Here, to bisect unbounded intervals we propose to ﬁx a support interval[a, b] andto never allow any bisectionsoutsidethissupportinterval.Forinstancetheinterval[−∞,∞]isbisectedinto[−∞,a]and[a,∞];theinterval [a, ∞] is bisected into [a, b] and [b, ∞]. But the intervals [−∞,a] and [b, ∞] are never bisected. They are considered as’toosmall’.

5. Bracketingthelargestinvariantsetwithunboundedsearchset

Method.Wearenowabletobuildmazeswithinﬁniteboxes.Wewilluse2·nextraunboundedboxesaroundasearchset YtopaveRⁿ.Fig.8showsanexampleofanunboundedsetXcomposedoftheboundedsearchsetYandextraunbounded

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Fig. 9. Bracketing the positive graph of the sinusoidal system with an unbounded search set. The color code is the same as Fig. 5 . The surrounding boxes of the blue frame are the inﬁnite boxes where their polygon is colored in orange. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

Fig. 10. Bracketing the forward reachable set of the Car on the hill system with an initial condition and an infinite search space. Infinite boxes are orange colored and gray circles mean that all directions might be possible inside the box. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

boxes

{

^[^b1],...,[b_k]

}

^such ^asX∈

{

Y∪[b₁]∪,...,∪[b_k]

}

^.^We ^canûse^the ^sameâlgorithm ^from^Section³ âs^we âreâble ^to

applycontractorsonunboundeddomains.Indeed,weareabletocomputetheconeofalldirectionvectorsinsideunbounded boxestoapplytheﬂowcontractorandabletoapplythecontinuitycontractor.

Inthispart,we willillustrate themethodofinﬁniteextension to computeF⁺_[_x₀_] orG⁺_[_t₀_]_,_[_x₀_] whenthesearch boxXis unbounded.

Test-case3.Sinusoidalsystem.Considerthesystemdescribedbythestateequation

˙

x=−sinx+e (13)

with the initial conditiont₀∈[0.3, 0.5] and x0∈[−0.5,0.5] and where e∈[−0.1,0.1]. We choose X=[0,5]×[−2,2]. As illustrated byFig.9trajectoriesleavesthesearch setontherightpartofthefigureonly andthat theycannot comeback on the left part of the figure. This result is consistent with the state equation where time always increases andforces trajectoriestonotcomebackontherightofthefigure.Ifwewanttostudymoregloballythissystem, wenowknowthat itisonlynecessarytoextendthesearchsetontherightpartofthefigure.Thealgorithmwasstoppedafter10stepswhich takes0.7s.

Test-case4.Caron thehillsystem. Wenow takebacktest-case 2withtheexampleoftheCaronthehillsystemwhere theboundedsearchsetwasX=[−2,13]×[−6,6].Fig.10showstheresultofthealgorithmwithinﬁniteboxes.Unlikethe resultoftest-case2,itwasnotpossibleheretocontracttheouterapproximation(nobluezones).Infact,wehaveseenthat trajectorieswereabletoleave theboundedsearchsetontherightpartofFig.6inSection3.Accordingtothecones ofall possibletrajectoriesoftheunboundedboxes,thesestrajectoriescanre-enterinsidethesearch-set(seetherightcone).This iswhywecannothaveabetterresultoftheouterapproximation.Alargerzoneshouldbestudied:anextendofthesearch spaceontherightpartoftheﬁgureisrequiredtoimprovetheresult.Thealgorithmwasstoppedafter18stepswhichtakes 190s.

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Fig. 11. Bracketing the forward reachable set of the Car on the hill system with an initial condition and an inﬁnite search space. The search space was set larger than Fig. 10 .

Test-case4-bis.Fig.11showstheresultofouralgorithmontheCaronthehillsystemwithalargerboundedsearchset X∈[−2,17]×[−6,6].Wecanseethatnotrajectoriesleavetheboundedsearchset:thisiswhythereisanemptypolygon inalloftheinﬁniteboxesandwhywewereabletocontracttheouterapproximation.AllthetrajectoriesstayinsideX.The algorithmwasstoppedafter16stepswhichtakes14s.

6. Conclusion

Inthispaper,we haveproposedanewapproachtobracketthesolutionsofan ordinarydifferentialequation withun- certaininitialconditions.Wehaveshownthatthisproblemcanbeseenasanenclosureofthepositiveinvariantsetofthe system.Byusingmazestoworkwithtrajectoriesandbyusingtoolsfromconstraintprogrammingandabstractinterpretation theory,wehavebeenabletoproposeanalgorithmtocomputesomeinnerandtheouterapproximationsofthesolutionof ourproblem.Finallywe haveextendedthe useofmazestounboundedset whichincreasesthenumberofproblemsthat ourmethodcandealwith.Themethodwasalsotestedonseveraltest-cases.

Ourmethodneedstobisectthestatespace,contrarytomostintervalintegrationmethodsmakingourapproachEulerian.

Ontheonehand,thisimpliesanexponentialincreasesofthecomputationalcomplexitywithrespecttothedimensionof thestatespace.Thismakesourmethodimpracticablewithhighdimensionalsystems.

Ina near future,we plan to extendour approach todealing withhybrid systems[11] when jumps could occur with respecttosomeguardconditions.

Acknowledgment

ThisworkhasbeensupportedbytheFrenchGovernmentDefenseprocurementandtechnologyagency(DGA).

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