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On the Decidability of theReachability Problemfor Planar Di�erential Inclusions

E. Asarin G. Schneider S. YovineVERIMAG, France

Overview of the presentation� Introduction� Simple Planar Di�erential Inclusion System(SPDI){ Reachability Problem� Di�culties� Our solution

{ Reachability Algorithm (Example)� Conclusions

Introduction� Reachability for Hybrid Systems{ Undecidable in most cases{ Approximation methods� Polyhedral approximations� Ellipsoidal approximations� Level-set approximations

{ Exact decision methods for sub-classes� Finite bisimulation (TA, InitializedRectangular Automata, etc.)� Quanti�er elimination� \Geometric" methods (only on the plane)

Introduction (Cont.)� \Geometric" methods - Ideas{ Based on: Topology + DynamicalSystems{ Require: Explicit solutions ofDi�erential Equations{ Take advantage of:� Topology of the plane� Explicit formula for Successors� Acceleration of cycles

SPDI: Simple Planar Di�erential InclusionSystem� SPDI:{ A partition of the plane into convexpolygonal regions{ A constant di�erential inclusion for eachregion

_x 2 \ba if x 2 Ri

b \baa

x0Ri

xab

SPDI: Simple Planar Di�erential InclusionSystem

Example: Swimmer trying to escape from awhirlpool

R3R7 R1R5

e3R4

R6

e4e5

e2 R2

e6 e7e8e1

R8

Reachability ProblemGiven x;x0 2 R 2, is there a trajectory � andt � 0 such that �(0) = x and �(t) = x0?

e4

e5 e6

e3

e7

e2e1

e8

Reachability Di�culties

In�nitely many trajectories (even locally)

R1e4e5R5

e1e8R7e6 e7R6

R4 e2

R8

R2R3e3

Reachability Di�cultiesIn�nitely many qualitative behaviors:

�1 = e1e2 e9e10e11e12 e9 � � � e12 � � ��2 = e1e2 � � � e8e1 � � �

R1e4e5R5 R7e6 e7R6

R4 e2

R8

R2R3e3e9e10 e11 e12

e8e1

Reachability Di�cultiesSelf-crossings:

�3 = e1 e2e9e10e11e1 e2e3 � � �

R1e4e5R5 R7e6 e7R6

R4 e2

R8

R2R3e3e9e10 e11 e12

e8e1

Reachability (Solution)

Our approach:� Simpli�cation of trajectories� Enumeration of qualitative behavior oftrajectories� Analysis of each qualitative behavior(computing successors)

Simpli�cation of Trajectories

Straightening:x0

Rix

ab

Simpli�cation of Trajectories (Cont.)

Removing self-crossings:

b a

b a

e2e02

x0

xf x0y

y0e01x e1

e02e01

e1

xfe2

y0

x0x

Qualitative Behavior(of Simpli�ed Trajectories)

� Classi�cation Theorem: Any edge signa-ture. � belongs to a typetype(�) = r1(s1)�r2(s2)� : : : rn(sn)�rn+1

� Example:

e10e11

e9e14

e2 e3e4

e8

e12 e13e15

e1 e5e6

e7

with � =e1e2e3 (e4e1e2e3)2e5e6 (e7 � � � e13e6)2 e14e15

Qualitative Behavior(of Simpli�ed Trajectories)

� Properties:{ ri is a seq. of pairwise di�erent edges;{ si is a simple cycle;{ ri and rj are disjoint{ si and sj are di�erent� Prop. The set of type of signatures is �nite

Preliminaries - for computing Successors

� A�ne function (AF):f(x) = ax+ b with a > 0� A�ne multi-valued function (AMVF):~F (x) = hf1(x); f2(x)i� Truncated a�ne multi-valued function(TAMVF): F (x) = ~F (x) \ hL;Ui

Lemma: AF, AMVF and TAMVF are closedunder composition.Lemma: Fixpoint equations F (I) = I can beexplicitely solved (without iterating)�.�Non trivial for TAMVF.

Successors (by �)

� One step (� = e2e3)

��

��e4

e5 e6 e7

e2e10 e11e12

e1e8

e13e9e3I 0 x

I 0 = Succe1e2(x) = [fb(x); fa(x)] = F (x)� Several steps (� = e1e2e3e4e5)

��

��e4

e5 e6 e7

e2e10 e11e12

e1e8

e13e9e3

Successors (by �)

� One cycle (� = s = e1e2 � � � e8e1)

��

��e4

e5 e6

e3

e7

e2e10 e11e12

e1e8

e13e9

� One cycle iterated: solution of �xpointequation (acceleration) (� = (s)�e13)

��

��

��

��e4

e5R5 e6

e3

e7

R4 e2e10 e11e12

e1e8

e13e9

Reachability Algorithm

� We have thatxf 2 Reach(x0)i�for some �; xf 2 Succ�(x0)

� Algorithm: Let x0 2 e0 and xf 2 ef .For each type type(�) = r1(s1)� � � � (sn)�rn+1,with e0 = first(r1) and ef = last(rn+1)check whetherxf 2 Succ�(x0)

Reachability Algorithm (Example)

R1e4

e5R5

R7e6

e3

e7

R3

R6

R4 e2

R8

R2e1

e8x0 = 12 xf = 34

� Type of signature: � = (e1 � � � e8)�

� Successor for the loop s = e1 : : : e8:Succe1:::e8(l; u) = [ l2 � 110; u2 + 13] \ (15;1)

Reachability Algorithm (Example)� Fixpoint equation: Succe1:::e8(I�) = I�� Solution: I� = [l�; u�] = [15; 23]� Hence: Succe1:::e8(x0) � [15; 23]

...e4

e5 e8e6 e7

e2 u� = 23x0 = 12

e1e3xf = 34l� = 15

Conclusion: xf 62 I� (34 62 [15; 23]).Hence, xf 62 Reach(x0)Note: the solution was found without iterating(acceleration).

Conclusions (Achievements)

� Reachability is decidable for SPDI� First application of \geometric" methodsto planar non-deterministic systems{ For deterministic planar systems:� PCD: Piece-wise Constant Derivatives(Maler & Pnueli - CAV'93)� Planar Piece-wise Hamiltonian Systems(�Cer�ans & J. V��ksna -HS III)

Conclusions (Future Works)

� Complexity analysis� Extensions:{ Hierarchical SPDI's{ 2-dim surfaces (ex.: torus){ Finding limit cycles

� Applications:{ Approximation of non-linear di�erentialequations

\Types" of cycles

1. STAY The cycle is not abandoned by anyof the two trajectories: L � l� � u� � U .2. DIE The right trajectory exits the cyclethrough the left (consequently the left onealso exits): u� < L _ l� > U .3. EXIT-BOTH Both trajectories exit thecycle (the left one through the left and theright one through the right):l� < L ^ u� > U .4. EXIT-LEFT The leftmost trajectory exitsthe cycle but not the other:l� < L � u� � U . Dual: EXIT-RIGHT.

\Types" of cycles

� Die

ul

(b)

ul

(c)

ul

(a)

e6 e6 e6

e4e4e4e7 e7e2 e2 e2 e3e3e3

e5 e5 e5e1 e1 e1

{ (a) Both trajectories leave the cycle(e1; e2; e3; e4)� through the left, whereasthe right one tends to the limit u�;{ (b) Reachable points on the cycle (inbold);{ (c) Possible continuation after leavingthe cycle (in bold).

\Types" of cycles

� Exit Both

(b)(a) (c)

ul

ul

ul

e6 e6 e6

e5 e5 e5e4 e4 e4

e3 e3 e3e2 e2 e2e7 e7 e7e1 e1 e1

{ (a) Both trajectories leave the cycle(e1; e2; e3; e4)�;{ (b) Reachable points on the cycle (inbold);{ (c) Possible continuation after leavingthe cycle (in bold).

\Types" of cycles

� Exit Left

(c)

lu

*

(a)

lu

(b)

lu

e6 e6

e5 e5 e5e4 e4 e4

e3 e3e2 e2 e2

u�1

u�3u�2u�3u�4e1

e6

e1 e1e3

u�1 u�1u�4

u�2u�4

u�2u�3

{ (a) The left trajectory leave the cycle(e1; e2; e3; e4)� through the left;{ (b) Reachable points on the cycle (inbold);{ (c) Possible continuation after leavingthe cycle (in bold).

Parametric Synthesis Problem

� Decidability of reachability for SPDI doesnot imply decidability for the parametriccase.

(a) (c)(b)

e2x0 xfe3

e1P0e1e2

x0 P0e0

e1e2

e3 x0 xfP0c0 b0 e3 xf e0e0

b0 b0 c0 c0

{ (a) xf is reachable from x0 following theleftmost (truncated) trajectory;{ (b) Parameter must be \at least" c0;{ (c) Taking c0 makes xf unreachable.


R1e4

e5R5

R7e6

e3

e7

R3

R6

R4 e2

R8

R2e1

e8x0 = 12 xf = 34

� Dynamics:{ a1 = b1 = (1;5),{ a2 = b2 = (�1; 12),{ a3 = (�1; 1160) and b3 = (�1;�14),{ a4 = b4 = (�1;�1),{ a5 = b5 = (0;�1),{ a6 = b6 = (1;�1),{ a7 = b7 = (1;0),{ a8 = b8 = (1;1).

� Type of signature: � = (e1 � � � e8)�


R1e4

e5R5

R7e6

e3

e7

R3

R6

R4 e2

R8

R2e1

e8x0 = 12 xf = 34

� Successor functions are:Succe1e2(x) = hx2; x2i \ (0;1)Succe2e3(x) = hx� 310; x+ 215i \ (0;1)Succeiei+1(x) = [x; x] \ (0;1), for all i 2 3::7Succe8e1(x) = hx+ 15; x+ 15i \ (0;1)

� Successor for the loop s = e1 : : : e8:Succe1:::e8(l; u) = [ l2 � 110; u2 + 13] \ (15;1)

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