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Introduction Reachability Parameter Synthesis Case Studies Conclusion Reachability Computation and Parameter Synthesis for Polynomial Dynamical Systems Tommaso Dreossi April 4, 2016 1 / 32

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

Reachability Computation andParameter Synthesis for Polynomial

Dynamical Systems

Tommaso Dreossi

April 4, 2016

1 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Overview

What’s this thesis about:

• Formal analysis of dynamical systems

• Dynamical system: mathematical model used to describe theevolution of a system

Why dynamical systems?

• Help to model, understand, make predictions

• Dynamical systems are ubiquitous

2 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewFormal Verification

• Dynamical systems are fundamental in the design of complexsystems (e.g., cyber-physical systems)

• Find application in safety-critical situations

• It is important to formally verify a system

• Important questions are:• Does the system reach an unsafe state? (Reachability)• Can we correctly tune the system? (Parameter synthesis)

3 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Overview

Two important problems:

• Reachability: Compute all the reachable states from a set ofinitial conditions

• Parameter Synthesis: Find a set of parameters such that thesystem satisfies a given a property

t

x

X0,P X1X2 X3 X4

Reachability

Compute X0,X1,X2,X3,X4, . . .

t

x

X0,P

ϕ

Parameter Synthesis

Find Pϕ ⊆ P

Xi and P can be infinite – bad for nonlinear dynamics

4 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Overview

Two important problems:

• Reachability: Compute all the reachable states from a set ofinitial conditions

• Parameter Synthesis: Find a set of parameters such that thesystem satisfies a given a property

t

x

X0,P X1X2 X3 X4

Reachability

Compute X0,X1,X2,X3,X4, . . .

t

x

X0,P

ϕ

Parameter Synthesis

Find Pϕ ⊆ P

Xi and P can be infinite – bad for nonlinear dynamics

4 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Overview

Two important problems:

• Reachability: Compute all the reachable states from a set ofinitial conditions

• Parameter Synthesis: Find a set of parameters such that thesystem satisfies a given a property

t

x

X0,P X1X2 X3 X4

Reachability

Compute X0,X1,X2,X3,X4, . . .

t

x

X0,P

ϕ

Parameter Synthesis

Find Pϕ ⊆ P

Xi and P can be infinite – bad for nonlinear dynamics

4 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewState of the Art

Reachability computation:

• Lot results on linear systems

• Hundreds of variables [FLGD+11, KV00, Fre05]

• No efficient solutions for nonlinear systems

• Low dimensions (≈ 10) [CAS13, KGCC15]

Parameter synthesis:

• Analytic/optimization techniques (scalability issues)

• Simulation based approaches (not formal/exhaustive)[Don10, MMB03, HWT96]

• No formal approaches dealing with infinite sets

5 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewOur contributions

We consider discrete-time polynomial dynamical systems (nonlinear) andinfinite compact sets (for both states and parameters)

Reachability analysis:

• Image computation based on boxes, parallelotopes, andparallelotope bundles

• Bernstein coefficients (new efficient algorithm, symbolicapproach)[DD14, DDP14, DDP16]

Parameter synthesis:

• Formalization using Signal Temporal Logic (STL)

• Definition of synthesis semantics for STL

• Algorithm to synthesize parameter sets using STL[DDP15]

Implementation:

• Sapo1: C++ tool that gathers all the developed methods1https://github.com/tommasodreossi/Sapo

6 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewOur contributions

We consider discrete-time polynomial dynamical systems (nonlinear) andinfinite compact sets (for both states and parameters)

Reachability analysis:

• Image computation based on boxes, parallelotopes, andparallelotope bundles

• Bernstein coefficients (new efficient algorithm, symbolicapproach)[DD14, DDP14, DDP16]

Parameter synthesis:

• Formalization using Signal Temporal Logic (STL)

• Definition of synthesis semantics for STL

• Algorithm to synthesize parameter sets using STL[DDP15]

Implementation:

• Sapo1: C++ tool that gathers all the developed methods1https://github.com/tommasodreossi/Sapo

6 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewOur contributions

We consider discrete-time polynomial dynamical systems (nonlinear) andinfinite compact sets (for both states and parameters)

Reachability analysis:

• Image computation based on boxes, parallelotopes, andparallelotope bundles

• Bernstein coefficients (new efficient algorithm, symbolicapproach)[DD14, DDP14, DDP16]

Parameter synthesis:

• Formalization using Signal Temporal Logic (STL)

• Definition of synthesis semantics for STL

• Algorithm to synthesize parameter sets using STL[DDP15]

Implementation:

• Sapo1: C++ tool that gathers all the developed methods1https://github.com/tommasodreossi/Sapo

6 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewOur contributions

We consider discrete-time polynomial dynamical systems (nonlinear) andinfinite compact sets (for both states and parameters)

Reachability analysis:

• Image computation based on boxes, parallelotopes, andparallelotope bundles

• Bernstein coefficients (new efficient algorithm, symbolicapproach)[DD14, DDP14, DDP16]

Parameter synthesis:

• Formalization using Signal Temporal Logic (STL)

• Definition of synthesis semantics for STL

• Algorithm to synthesize parameter sets using STL[DDP15]

Implementation:

• Sapo1: C++ tool that gathers all the developed methods1https://github.com/tommasodreossi/Sapo

6 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewRoadmap

1 Reachability Analysis

1 How to transform/approximate a set?2 Bernstein coefficients for polynomials3 Box-based reachability4 Parallelotope-based reachability5 Parallelotope bundle-based reachability

2 Parameter Synthesis

1 Problem formalization via STL2 Synthesis semantics3 Synthesis algorithm

3 Application

1 Tool overview2 Case studies

4 Conclusion

7 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewRoadmap

1 Reachability Analysis

1 How to transform/approximate a set?2 Bernstein coefficients for polynomials3 Box-based reachability4 Parallelotope-based reachability5 Parallelotope bundle-based reachability

2 Parameter Synthesis

1 Problem formalization via STL2 Synthesis semantics3 Synthesis algorithm

3 Application

1 Tool overview2 Case studies

4 Conclusion

7 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ReachabilityComputation

Problem

Given a dynamical system f : Rn → Rn and a set of initialconditions X0 ⊂ Rn, compute the reachable sets up to time T ∈ N

• How to compute/represent nonlinear set transformations?(nonconvexity)

• Idea: Over-approximate sets with simpler objects (polytopes)

8 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ReachabilityComputation

Problem

Given a dynamical system f : Rn → Rn and a set of initialconditions X0 ⊂ Rn, compute the reachable sets up to time T ∈ N

• How to compute/represent nonlinear set transformations?(nonconvexity)

• Idea: Over-approximate sets with simpler objects (polytopes)

8 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ReachabilityComputation

Problem

Given a dynamical system f : Rn → Rn and a set of initialconditions X0 ⊂ Rn, compute the reachable sets up to time T ∈ N

• How to compute/represent nonlinear set transformations?(nonconvexity)

• Idea: Over-approximate sets with simpler objects (polytopes)

8 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ReachabilitySingle Step

Polytopes as solution of linear systems

X0 ≡ Dx ≤ c (D, c : template and offset)

How to find c ′j?

c ′j ≥ maxx∈Xi

Dj f (x)

Nonlinear optimization problemHow to bound a polynomial?

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

ReachabilitySingle Step

Polytopes as solution of linear systems

X0 ≡ Dx ≤ c (D, c : template and offset)

How to find c ′j?

c ′j ≥ maxx∈Xi

Dj f (x)

Nonlinear optimization problemHow to bound a polynomial?

9 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ReachabilitySingle Step

Polytopes as solution of linear systems

X0 ≡ Dx ≤ c (D, c : template and offset)

How to find c ′j?

c ′j ≥ maxx∈Xi

Dj f (x)

Nonlinear optimization problemHow to bound a polynomial?

9 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ReachabilitySingle Step

Polytopes as solution of linear systems

X0 ≡ Dx ≤ c (D, c : template and offset)

How to find c ′j?

c ′j ≥ maxx∈Xi

Dj f (x)

Nonlinear optimization problemHow to bound a polynomial?

9 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bernstein Polynomials

Power basis

π(x) =∑i≤d

aixi

Bernstein basis

π(x) =∑i≤d

biBd ,i (x)

bi =∑ (i

j

)(dj

)ajRange enclosure property

For all x ∈ [0, 1]n : m ≤ π(x) ≤ Mwith m and M minimum and maximum Bernstein coefficients

π(x) can be bounded over [0, 1]n using the Bernstein coefficients

How to generalize to other domains?

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bernstein Polynomials

Power basis

π(x) =∑i≤d

aixi

Bernstein basis

π(x) =∑i≤d

biBd ,i (x)

bi =∑ (i

j

)(dj

)ajRange enclosure property

For all x ∈ [0, 1]n : m ≤ π(x) ≤ Mwith m and M minimum and maximum Bernstein coefficients

π(x) can be bounded over [0, 1]n using the Bernstein coefficients

How to generalize to other domains?

10 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bernstein Polynomials

Power basis

π(x) =∑i≤d

aixi

Bernstein basis

π(x) =∑i≤d

biBd ,i (x)

bi =∑ (i

j

)(dj

)ajRange enclosure property

For all x ∈ [0, 1]n : m ≤ π(x) ≤ Mwith m and M minimum and maximum Bernstein coefficients

π(x) can be bounded over [0, 1]n using the Bernstein coefficients

How to generalize to other domains?

10 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Box-basedApproximation

Idea: Map [0, 1]n to other sets (v : [0, 1]n → Xi )

c ′j ≥ maxx∈[0,1]n

Dj f (v(x))

c ′j ← maximum Bernstein coefficient of Dj f (v(x))

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

Parallelotope-basedApproximation

Idea: Map [0, 1]n to parallelotopes

• More generic set

• More flexibility

• Precision improvement

What about combing different sets?(Boxes + Parallelotopes)

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

Parallelotope-basedApproximation

Idea: Map [0, 1]n to parallelotopes

• More generic set

• More flexibility

• Precision improvement

What about combing different sets?(Boxes + Parallelotopes)

12 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bundle-BasedApproximation

Idea: Represent polytopes as intersection of parallelotopes

Definition (Parallelotope bundle)

A bundle B = {P1, . . . ,Pb} is a finite set of parallelotopes s.t.Q = I(B) ∩n

i=1 Pb

Theorem (Polytope decomposition)

For any polytope Q there exists a bundle B = {P1, . . . ,Pb} s.t.Q = I(B)

13 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bundle-BasedApproximation

Idea: Represent polytopes as intersection of parallelotopes

Definition (Parallelotope bundle)

A bundle B = {P1, . . . ,Pb} is a finite set of parallelotopes s.t.Q = I(B) ∩n

i=1 Pb

Theorem (Polytope decomposition)

For any polytope Q there exists a bundle B = {P1, . . . ,Pb} s.t.Q = I(B)

13 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bundle-basedApproximation

Basic steps:

• Bound direction Dj over each image f (Pi )• Keep the tightest offset c ′j• Repeat for all the directions

• Sensible precision improvement• Complexity O(kb) (k directions, b number of parallelotopes)

14 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bundle-basedApproximation

Basic steps:

• Bound direction Dj over each image f (Pi )

• Keep the tightest offset c ′j• Repeat for all the directions

• Sensible precision improvement

• Complexity O(kb) (k directions, b number of parallelotopes)

14 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bundle-basedApproximation

Basic steps:

• Bound direction Dj over each image f (Pi )

• Keep the tightest offset c ′j• Repeat for all the directions

• Sensible precision improvement

• Complexity O(kb) (k directions, b number of parallelotopes)

14 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bundle-basedApproximation

Basic steps:

• Bound direction Dj over each image f (Pi )

• Keep the tightest offset c ′j• Repeat for all the directions

• Sensible precision improvement

• Complexity O(kb) (k directions, b number of parallelotopes)

14 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Bundle-basedApproximation

Basic steps:

• Bound direction Dj over each image f (Pi )

• Keep the tightest offset c ′j• Repeat for all the directions

• Sensible precision improvement

• Complexity O(kb) (k directions, b number of parallelotopes)

14 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewRoadmap

1 Reachability Analysis

1 How to transform/approximate a set?2 Bernstein coefficients for polynomials3 Box-based reachability4 Parallelotope-based reachability5 Parallelotope bundle-based reachability

2 Parameter Synthesis

1 Problem formalization via STL2 Synthesis semantics3 Synthesis algorithm

3 Application

1 Tool overview2 Case studies

4 Conclusion

15 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Parameter SynthesisProblem

g0(x) ≥ 0g1(x) ≥ 0 g2(x) ≥ 0

X0

X1 X2

P

f (X0,P)f (X1,P)

f (X0,Ps)f (X1,Ps)

1) How to express time-dependent

properties over traces?

2) How to synthesize Ps?

16 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Parameter SynthesisProblem

g0(x) ≥ 0g1(x) ≥ 0 g2(x) ≥ 0

X0

X1 X2

P

f (X0,P)f (X1,P)

f (X0,Ps)f (X1,Ps)

1) How to express time-dependent

properties over traces?

2) How to synthesize Ps?

16 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Parameter SynthesisProblem

g0(x) ≥ 0g1(x) ≥ 0 g2(x) ≥ 0

X0

X1 X2

P

f (X0,P)f (X1,P)

f (X0,Ps)f (X1,Ps)

1) How to express time-dependent

properties over traces?

2) How to synthesize Ps?

16 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Signal Temporal LogicFrom LTL to STL

Extension of LTL with real-time and real-valued constraints

• LTL G( r ⇒ F g)Boolean predicates, discrete-time

• MTL G( r ⇒ F[0,.5s] g )Boolean predicates, real-time

• STL G( x [t] > 0 ⇒ F[0,.5s]y [t] > 0 )Predicates over real values , real-time

Definition (STL Syntax (DNF))

ϕ := > | f (x1[t], . . . , xn[t]) > 0 | ϕ ∧ ψ | ϕ ∨ ψ |ϕ U[a,b] ψ

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

Parameter Synthesis

Given a dynamical system f (x , p), initial state set X0, initialparameter set P, and an STL specification ϕ,

find Ps ⊆ P such that f (X0,Ps) |= ϕ

• STL is defined on single signals

• We extended its semantics to sets of signals (flows)

18 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Parameter Synthesis

Given a dynamical system f (x , p), initial state set X0, initialparameter set P, and an STL specification ϕ,

find Ps ⊆ P such that f (X0,Ps) |= ϕ

• STL is defined on single signals

• We extended its semantics to sets of signals (flows)

18 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Atomic PredicateParameter Synthesis Algorithm

Rewrite safe condition:

Xi+1 = f (Xi ,Ps)

Valid if g(Xi+1) < 0

g(f (Xi ,Ps)) < 0

Xi

Xi+1

g(x) ≥ 0

f (Xi ,Ps)

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

Conjunction and DisjunctionParameter Synthesis Algorithm

Conjunction ϕ1 ∧ ϕ2:

1 Solve the problem for ϕ1 → P1

2 Solve the problem for ϕ2 → P2

3 Return the intersection → P1 ∩ P2

Disjunction ϕ1 ∨ ϕ2:

1 Solve the problem for ϕ1 → P1

2 Solve the problem for ϕ2 → P2

3 Return the union → P1 ∪ P2

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

UntilParameter Synthesis Algorithm

Cases on temporal interval [a, b]

Until ϕ1U[a,b]ϕ2:

1 ϕ1U[0,0]ϕ2

2 ϕ1U[0,b]ϕ2 (b > 0)

3 ϕ1U[a,b]ϕ2 (a, b > 0)

0

a, b

0

a b

0

a b

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Introduction Reachability Parameter Synthesis Case Studies Conclusion

OverviewRoadmap

1 Reachability Analysis

1 How to transform/approximate a set?2 Bernstein coefficients for polynomials3 Box-based reachability4 Parallelotope-based reachability5 Parallelotope bundle-based reachability

2 Parameter Synthesis

1 Problem formalization via STL2 Synthesis semantics3 Synthesis algorithm

3 Application

1 Tool overview2 Case studies

4 Conclusion

22 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Case Studies

Implementation:

• Sapo2: C++ tool for reachability analysis and parametersynthesis of polynomial dynamical systems

• Symbolic computation of Bernstein coefficients with improvematrix method

• Boxes / Parallelotopes / Parallelotope bundles (system states)

• STL specifications + Polytopes (parameter sets)

Applications:

• System biology: SIR, SARS, Influenza, Ebola (3-7d)

• Population growth: Honeybees nest choice (5d)

• Robotics: Quadcopter drone (17d)

2https://github.com/tommasodreossi/Sapo23 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

Case Studies

SIR epidemic model (3d)

1.1 Reachability (parallelotope, bundle) 1.2 Param. Synth (G( i ≤ 0.5))

Quadcopter (17d)

2.1 Drone height 2.2 Vertical speed

24 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ConclusionSummary:

• Formal verification is important

• Nonlinear dynamical systems are difficult to threat

Contributions:

• Reachability and parameter synthesis for polynomial dynamicalsystems

• Reachability analysis using boxes, parallelotopes, and parallelotopebundles

• Parameter synthesis using STL and polytopes

• Techniques based on Bernstein coefficients

Future works:

• Parallelization (bundles easy to parallelize)

• From parameter to input synthesis (controller)

• Hybrid automata verification

25 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ConclusionSummary:

• Formal verification is important

• Nonlinear dynamical systems are difficult to threat

Contributions:

• Reachability and parameter synthesis for polynomial dynamicalsystems

• Reachability analysis using boxes, parallelotopes, and parallelotopebundles

• Parameter synthesis using STL and polytopes

• Techniques based on Bernstein coefficients

Future works:

• Parallelization (bundles easy to parallelize)

• From parameter to input synthesis (controller)

• Hybrid automata verification

25 / 32

Introduction Reachability Parameter Synthesis Case Studies Conclusion

ConclusionSummary:

• Formal verification is important

• Nonlinear dynamical systems are difficult to threat

Contributions:

• Reachability and parameter synthesis for polynomial dynamicalsystems

• Reachability analysis using boxes, parallelotopes, and parallelotopebundles

• Parameter synthesis using STL and polytopes

• Techniques based on Bernstein coefficients

Future works:

• Parallelization (bundles easy to parallelize)

• From parameter to input synthesis (controller)

• Hybrid automata verification

25 / 32

References I

Xin Chen, Erika Abraham, and Sriram Sankaranarayanan, Flow*: Ananalyzer for non-linear hybrid systems, Computer Aided Verification,CAV, 2013, pp. 258–263.

Tommaso Dreossi and Thao Dang, Parameter synthesis forpolynomial biological models, Hybrid Systems: Computation andControl, HSCC, 2014, pp. 233–242.

Thao Dang, Tommaso Dreossi, and Carla Piazza, Parametersynthesis using parallelotopic enclosure and applications to epidemicmodels, Hybrid Systems and Biology, HSB, 2014, pp. 67–82.

, Parameter synthesis through temporal logic specifications,FM 2015: Formal Methods - 20th International Symposium, Oslo,Norway, June 24-26, 2015, Proceedings, 2015, pp. 213–230.

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References II

Tommaso Dreossi, Thao Dang, and Carla Piazza, Parallelotopebundles for polynomial reachability, Hybrid Systems: Computationand Control, HSCC, 2016.

Alexandre Donze, Breach, a toolbox for verification and parametersynthesis of hybrid systems, Computer Aided Verification, CAV,Springer, 2010, pp. 167–170.

Thao Dang and Romain Testylier, Reachability analysis forpolynomial dynamical systems using the Bernstein expansion,Reliable Computing 17 (2012), no. 2, 128–152.

Goran Frehse, Colas Le Guernic, Alexandre Donze, Scott Cotton,Rajarshi Ray, Olivier Lebeltel, Rodolfo Ripado, Antoine Girard, ThaoDang, and Oded Maler, Spaceex: Scalable verification of hybridsystems, Computer Aided Verification, CAV, Springer, 2011,pp. 379–395.

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References III

Goran Frehse, Phaver: Algorithmic verification of hybrid systemspast hytech, Hybrid Systems: Computation and Control, HSCC,Springer, 2005, pp. 258–273.

Thomas A. Henzinger and Howard Wong-Toi, Using hytech tosynthesize control parameters for a steam boiler, Formal Methods forIndustrial Applications (London, UK, UK), Springer-Verlag, 1996,pp. 265–282.

Soonho Kong, Sicun Gao, Wei Chen, and Edmund Clarke, dreach:δ-reachability analysis for hybrid systems, Tools and Algorithms forthe Construction and Analysis of Systems, TACAS, Springer, 2015,pp. 200–205.

Alexander B Kurzhanski and Pravin Varaiya, Ellipsoidal techniquesfor reachability analysis: internal approximation, Systems & controlletters 41 (2000), no. 3, 201–211.

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References IV

Carmen G Moles, Pedro Mendes, and Julio R Banga, Parameterestimation in biochemical pathways: a comparison of globaloptimization methods, Genome research 13 (2003), no. 11,2467–2474.

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Bernstein ExpansionBasis

Power basis

π(x,p) =∑i≤d

ai(p)xi

Bernstein basis

π(x,p) =∑i∈I

bi(p)Bd,i(x)

Basis

Bd,i(x) = Bd1,i1(x1) . . .Bdn,in(xn)

Bdi ,ii (xi ) =

(diii

)x iii (1− xi )

di−ii

30 / 32

Bernstein ExpansionApproximation

Lemma ([DT12])

Let Cπ : Rn → R be the piecewise linear function defined by theBernstein control points of the polynomial π : Rn → R, with respect tothe box [0, 1]n. For all x ∈ [0, 1]n

| π(x)− Cπ(x) |≤ maxx∈[0,1]n;i,j∈{1,...,n}

| ∂i∂jπ(x) | (1)

where | · | is the infinity norm on Rn.

31 / 32