INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH

SUPPORT VECTOR MACHINES

Laetitia Chapel

Lab-STICC, Universit

e Europ

eenne de Bretagne, Universit

e de Bretagne Sud, 56017 Vannes Cedex, France

Guillaume Deffuant

Laboratoire d’Ingnierie pour les Syst

emes Complexes, Cemagref, 63172 Aubi

ere Cedex, France

Keywords:

Viability theory, Capture basin, Optimal control, Support Vector Machines.

Abstract:

We propose a new approach to solve target hitting problems, that iteratively approximates capture basins at

successive times, using a machine learning algorithm trained on points of a grid with boolean labels. We

consider two variants of the approximation (from inside and from outside), and we state the conditions on the

machine learning procedure that guarantee the convergence of the approximations towards the actual capture

basin when the resolution of the grid decreases to 0. Moreover, we deﬁne a control procedure which uses the

set of capture basin approximations to drive a point into the target. When using the inner approximation, the

procedure guarantees to hit the target, and when the resolution of the grid tends to 0, the controller tends to

the optimal one (minimizing time to hit the target). We use Support Vector Machines as a particular learning

method, because they provide parsimonious approximations, from which one can derive fast and efﬁcient

controllers. We illustrate the method on two simple examples, Zermelo and car on the hill problems.

1 INTRODUCTION

We focus on the problem of deﬁning the control func-

tion for driving a dynamical system to reach a given

target compact set C ⊂ K in minimum time, without

going out from K, where K is a given set (called con-

straint set). This problem, often called the reacha-

bility problem, can be addressed by optimal control

methods, solving Hamilton-Jacobi-Bellman (HJB) or

Isaacs (HJI) equations. Several numerical techniques

are available; for example, (Mitchell et al., 2005) pro-

pose an algorithm that computes an approximation for

the backward reachable set of a system using a time

dependent HJI partial differential equation, (Lygeros,

2004) builds the value function of the problem which

can be then used to choose the best action at each time

step.

Reachability problem can also be addressed in the

viability theory framework (Aubin, 1991). To ap-

ply viability theory to target hitting problems, one

must add an auxiliary dimension to the system, rep-

resenting the time elapsed before reaching the tar-

get. The approach computes an approximation of the

envelopes of all t-capture basins, the sets of points

for which there exists a control function that allows

the system to reach the target in a time less than t.

(Frankowska, 1989) shows that the boundary of this

set is the value function in the dynamical program-

ming perspective. Hence, solving this extended via-

bility problem also provides the minimal time for a

state x to reach the target C while always staying in

K (minimal time function ϑ

(x) (Cardaliaguet et al.,

1998)). This function can then be used to deﬁne con-

trollers that drive the system into the target.

Several numerical algorithms (Saint-Pierre, 1994;

Cardaliaguet et al., 1998) provide an overapproxima-

tion of capture basins. (Bayen et al., 2002) imple-

ment an algorithm proposed by (Saint-Pierre, 2001)

that computes a discrete underapproximation of con-

tinuous minimal time functions (and thus an overap-

proximation of capture basins), without adding an ad-

ditional dimension. (Lhommeau et al., 2007) present

an algorithm, based on interval analysis, that provides

an inner and outer approximation of the capture basin.

In general, capture basin and minimal time function

approximation algorithms face the curse of dimen-

sionality, which limits their use to problems of low

dimension (in the state and control space).

This paper proposes a new method to solve tar-

get hitting problems, inspired by our work on via-

Chapel L. and Deffuant G..

INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH SUPPORT VECTOR MACHINES.

DOI: 10.5220/0003536000340040

In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 34-40

ISBN: 978-989-8425-74-4

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

bility kernel approximation (Deffuant et al., 2007).

The principle is to approximate iteratively the cap-

ture basins at successive times t. To compute time

t-capture basin approximation, we use a discrete grid

of points covering set K, and label +1 the points for

which there exists a control leading the point into the

t −δt-capture basin approximation, and -1 otherwise.

Then, we use a machine learning method to compute

a continuous boundary between +1 and -1 points of

the grid. We state the conditions the learning method

should fulﬁl (they are similar to the one established to

approximate viability kernels (Deffuant et al., 2007))

in order to prove the convergence toward the actual

capture basins.

We consider two variants of the algorithm: one

provides an approximation that converges from out-

side, and the other from inside. Although no conver-

gence rate is provided, the comparison of the two ap-

proximations gives an assessment of the approxima-

tion error for a given problem. Moreover, we deﬁne

a controller that guarantees to reach the target when

derived from the inner approximation.

We consider Support Vector Machines (SVMs

(Vapnik, 1995; Scholkopf and Smola, 2002)) as a rel-

evant machine learning technique in this context. In-

deed, SVMs provide parsimonious approximations of

capture basins, that allow the deﬁnition of compact

controllers. Moreover, they make possible to use op-

timisation techniques to ﬁnd the controls, hence prob-

lems with control spaces in more dimensions become

tractable. We can also more easily compute the con-

trol on several time steps, which improves the quality

of the solution for a given resolution of the grid.

We illustrate our approach with some experiments

on two simple examples. Finally, we draw some per-

spectives.

2 PROBLEM DEFINITION

We consider a controlled dynamical system in dis-

crete time (Euler approximation), described by the

evolution of its state variable x ∈ K ⊂ R

. We would

like to deﬁne the set of controls to apply to the sys-

tem starting from point x in order to reach the target

C ⊂ K in minimal time:

(

x(t + dt) = x(t) + ϕ(x(t),u(t)).dt, if x(t) /∈ C

x(t + dt) = x(t), if x(t) ∈ C

u(t) ∈ U,

(1)

where ϕ is a continuous and derivable function of x

and u. The control u must be chosen at each time step

in the set U of admissible controls.

The capture basin of the system is the set of states

for which there exists at least one series of controls

such that the system reaches the target in ﬁnite time,

without leaving K. Let G(x,(u

,..,u

)) be the point

reached when applying successively during n time

steps the controls (u

,..,u

), starting from point x. Let

the minimal time function (or hitting time function) be

the function that associates to a state x ∈ K the mini-

mum time to reach C:

(x) = inf

{

n|∃(u

,..,u

) ∈ U

such that G(x, (u

,..,u

)) ∈ C

and for 1 ≤ j ≤ n,G (x,(u

,..,u

)) ∈ K



(2)

This is the value function obtained when solving HJB

equations in a dynamic programming approach. It

takes values in N

∪ +∞, speciﬁcally ϑ

(x) = 0 if

x ∈ C and ϑ

(x) = +∞ if no trajectory included in K

can reach C. The capture basin of C viable in K is

then deﬁned as:

Capt(K, C) =

x ∈ K|ϑ

(x) < +∞

, (3)

and we can also deﬁne the capture basin in ﬁnite time

Capt(K, C,n) =

x ∈ K|ϑ

(x) ≤ n

. (4)

To solve a target hitting problem in the viability per-

spective, one must consider the following extended

dynamical system (x(t),y(t)) when x(t) /∈ C:



x(t + dt) = x(t) + ϕ(x(t), u(t)) .dt

y(t + dt) = y(t) − dt.

(5)

and (x(t + dt) = x(t),y(t + dt) = y(t)) when x(t) ∈ C.

(Cardaliaguet et al., 1998) prove that approximating

minimal time function comes down to a viability ker-

nel approximation problem of this extended dynam-

ical problem. In a viability problem, one must ﬁnd

the rule of controls for keeping a system indeﬁnitely

within a constraint set. (Bayen et al., 2002; Saint-

Pierre, 2001) give examples of such an application of

viability approach to solve a target hitting problem.

(Deffuant et al., 2007) proposed an algorithm,

based on (Saint-Pierre, 1994), that uses a machine

learning procedure to approximate viability kernels.

The main advantage of this algorithm is that it pro-

vides continuous approximations that enable to ﬁnd

the controls with standard optimization techniques,

and then to tackle problems with control in large di-

mensional space. The aim of this paper is to adapt

(Deffuant et al., 2007) to compute directly an approx-

imation of the capture basin limits, without adding the

auxiliary dimension, and then to use these approxima-

tions to deﬁne the sequence of controls.

INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH SUPPORT VECTOR MACHINES

3 MACHINE LEARNING

APPROXIMATION OF VALUE

FUNCTION CONTOURS AND

OPTIMAL CONTROL

For simplicity, we denote Capt(K, C,n.dt) = Capt

In all the following, continuous sets are denoted by

rounded letters and discrete sets in capital letters.

We consider function G:

G(x,u) =



x + ϕ(x,u).dt if x /∈ C

x if x ∈ C

(6)

We suppose that G is µ-lipschitz with respect to x:

∀(x,y) ∈ K

,∀u ∈ U|G(x,u) − G(y, u)| < µ|x − y|.

(7)

We deﬁne a grid K

as a discrete subset of K, such

that:

∀x ∈ K,∃x

∈ K

such that |x − x

| ≤ h. (8)

Moreover, we deﬁne an algebraic distance d

(x,∂E)of

a point x to the boundary ∂E of a continuous closed set

E, as the distance to the boundary when x is located

inside E, and this distance negated when x is located

outside E:

if x ∈ E, d

(x,∂E) = d(x,∂E), (9)

if x /∈ E, d

(x,∂E) = −d(x,∂E). (10)

3.1 Capture Basin Approximation

Algorithms

In this section, we describe two variants of an algo-

rithm that provides an approximation C

of the cap-

ture basin at time n.dt, one variant approximates the

capture basins from outside and the other one from

inside. At each step n of the algorithm, we ﬁrst build

a discrete approximation C

⊂ K

of the capture basin

Capt

, and then we use a learning procedure L (for in-

stance Support Vector Machines, as shown below) to

generalise this discrete set into a continuous one C

= L(C

) (11)

To simplifying the writing, we ﬁrst deﬁne:

{

∈ K

s.t. x

/∈ C

}

, (12)

{

x ∈ K s.t. x /∈ C

}

. (13)

The two variants differ on the conditions for deﬁning

the discrete set C

n+1

from C

, and on the conditions

the learning procedure L must fulﬁl. For both variant,

we construct an increasing sequence of approxima-

tions at time n.dt, by adding the points of the grid for

which there exists at least one control that drives the

system not too far away (in an algebraic sense - nega-

tive distance in the outer case and positive distance in

the inner one) from the boundary of the previous ap-

proximation. They also both require conditions on the

learning procedure, in order to guarantee the conver-

gence toward the actual capture basin when the step

of the grid h decreases to 0. In the inner approxima-

tion case, the condition is stricter on set C

and more

relaxed on set C

, while the outer case requires con-

verse conditions. We now describe in more details

both variants and conditions.

3.1.1 Outer Approximation

Given initial sets C

= C ∩ K

, C

= C, and supposing

that we deﬁne continuous approximations C

from C

(eq. (11)), we deﬁne discrete sets C

as follows:

n+1

= C



∈ C

s.t. ∃u ∈ U,

(G(x

,u),∂C

) ≥ −µh



(14)

Proposition. If the learning procedure L respects the

following conditions:

∀x ∈ C

,∃x

∈ C

s.t. |x − x

| ≤ h (15)

∃λ ≥ 1s.t.∀h,∀x ∈ C

,∃x

∈ C

s.t. |x − x

| ≤ λh (16)

then the convergence of the approximation from out-

side is guaranteed:

∀n,Capt

⊂ C

, (17)

→ Capt

when h → 0. (18)

Proof. The proof is given on appendix 5.

3.1.2 Inner Approximation

Given initial sets C

= C ∩ K

, C

= C, and supposing

that we deﬁne continuous approximations C

from C

we deﬁne discrete sets C

as follows:

n+1

= C



∈ C

s.t. ∃u ∈ U,

(G(x

,u),∂C

) ≥ µh



(19)

Proposition. If the learning procedure L respects the

following conditions:

∀x ∈ C

,∃x

∈ C

s.t. |x − x

| ≤ h (20)

∃λ ≥ 1s.t.∀h,∀x ∈ C

,∃x

∈ C

s.t. |x − x

| ≤ λh (21)

and that the dynamics are such that:

∀x ∈ K with d



x,∂Capt

n+1



> 0,

∃u ∈ U | d

(G(x,u), ∂Capt

) > 0.

(22)

then the convergence of the approximation from in-

side is guaranteed:

∀n,C

⊂ Capt

, (23)

→ Capt

when h → 0. (24)

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

Proof. Convergence proof of the algorithm from in-

side requires an additional condition on the dynamics

(eq. (22)): a point x of the interior of capture basin

at time (n + 1).dt, should be such that there exists

y ∈ G(x) belonging to the interior of capture basin at

time n.dt (and not on ∂Capt

). The proof of conver-

gence is on appendix 5.

3.2 Optimal Control

The aim of the optimal controller is to choose a

control function that reaches the target in minimal

time, without breaking the viability constraints. The

idea is to choose the controls which drive the system

to cross C

boundaries in a descending order.

Proposition. Consider x

, such that x

∈ C

n+1

and

/∈ C

. The procedure which, for i = 0 to n computes

∗

) deﬁned such that x

i+1

= G(x

∗

)) ∈ C

n−i

converges to the control policy minimizing the hitting

time, when h and dt tend to 0.

Proof. By construction, if x

∈ C

n+1−i

and x

/∈ C

n−i

there exists a control value u

∗

) such that x

i+1

G(x

∗

)) ∈ C

n−i

(see Appendix 5, part I.). There-

fore, the procedure leads to the target in n + 1 time

steps, i.e. in (n + 1).dt time. Moreover, by deﬁnition,

the optimum time for reaching the target from a point

x located on the boundary of capture basin Capt

n.dt. Hence, the optimum time for reaching the tar-

get from point x such that x ∈ Capt

n+1

and x /∈ Capt

with the dynamics deﬁned by ϕ, is between n.dt and

(n + 1).dt. Then, the fact that C

converges to Capt

when h tends to 0, ensures that the number of time

steps needed by the procedure applied to this point x

will tend to (n + 1).dt. When dt tends to 0, the differ-

ence with the optimum time to reach the target, which

is smaller than dt, tends to 0.

4 EXPERIMENTS

4.1 SVM as a Learning Procedure

We use Support Vector Machines (Vapnik, 1995;

Scholkopf and Smola, 2002) as the learning proce-

dure L to deﬁne capture basin approximations C

L(C

). At each iteration, we construct a learning set:

let S

be a set of couples (x

), where x

∈ K

and

= +1 if x

∈C

and −1 otherwise. Running a SVM

on learning sets S

provides a separating function f

between points of different labels and hence, allows

us to deﬁne a continuous approximation C

of C

follows:

{

x ∈ K such that f

(x) ≥ 0

}

. (25)

Points x of the boundary ∂C

are those such that

(x) = 0. The fulﬁlment of the conditions guaran-

teeing convergence is discussed in (Deffuant et al.,

2007) and the same arguments hold in both variants

of the algorithm.

In the following examples, we use libSVM (Chang

and Lin, 2001) to train the SVMs. As we did in (Def-

fuant et al., 2007), we use the SVM function as a

proxy for the distance to the boundary, in order to

simplify the computations.

4.2 Zermelo Problem

The state (x(t),y(t)) of the system represents the po-

sition of a boat in a river. There are two controls: the

thrust u and the direction θ of the boat. The system

in discrete time deﬁned by a time interval dt can be

written as follows:



x(t + dt) = x(t) + (1 − 0.1y(t)

+ u cosθ).dt

y(t + dt) = y(t) + (u sin θ).dt,

(26)

where u ∈ [0; 0.44] and θ ∈ [0; 2π]. The boat must

remain in a given interval K = [−6; 2] × [−2; 2], and

reach a round island C = B(0; 0.44). We suppose that

the boat must reach the island before time T = 7.

For this simple system, it is possible to derive ana-

lytically the capture basin, hence we can compare the

approximations given by the two algorithms with the

actual capture basin. Figure 1 compares some results

obtained with the outer and inner approximation. In

any cases, the quality of the approximation can be as-

sessed by comparing both approximations: by con-

struction, the contour of the actual capture basin is

surrounded by inner and outer approximations. A ex-

ample of a optimal trajectory deﬁned with the optimal

controller is also presented: with the inner approx-

imation, the trajectory will enable the boat to reach

the target, while it is not guaranteed in the outer case.

4.3 Car on the Hill

We consider the well-known car on the hill problem.

The state is two-dimensional (position and velocity)

and the system can be controlled with a continuous

one-dimensional action (thrust). For a description of

the dynamics and the state space constraints, one can

refers to (Moore and Atkeson, 1995). The aim of the

car on the hill system is to keep the car inside a given

set of constraints, and to reach a target (the top of the

hill) as fast as possible. Figure 2 shows the approx-

imation of the contours of the value function using

outer and inner variants of the algorithm, with an ex-

ample of optimal trajectories.

INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH SUPPORT VECTOR MACHINES

Figure 1: Approximation from inside (top) and from outside

(bottom) for Zermelo problem. The horizontal axis repre-

sents the position x and the vertical one position y. K is

the rectangle. The capture basin is represented in blue. The

black thick line limits the boundary of the actual capture

basin. The level lines represent approximation of the con-

tours of the capture basins for successive time steps. The

grid contains 41 points by dimension. The optimisation is

made on 4 time steps, with dt = 0.05. Each ﬁgure presents

an example of trajectory (in green) using the SVM optimal

controller.

5 DISCUSSION

We proposed an algorithm that approximates capture

basins and minimal time functions, using a classiﬁ-

cation procedure, in two variants (inner and outer).

The inner approximation can be used to deﬁne a opti-

mal controller that guarantees to ﬁnd a series of con-

trols that allows the system to reach the target. SVMs

Figure 2: Inner (top) and outer (bottom) approximation for

the car on the hill problem. The grid contains 51 points by

dimension. The optimisation is made on 2 time steps. An

example of an optimal trajectory is depicted in green.

appear as a particularly relevant classiﬁcation proce-

dure for this approach, because they provide parsimo-

nious representations of the capture basins and en-

able to use optimization techniques to compute the

controls. This latter point is particularly important to

deal with high dimensional control space. The parci-

monious property may allow to deﬁne compact and

fast controller, even for high dimensional state space.

However, although we generally manage to ﬁnd pa-

rameters in which the result respect the conditions of

convergence, this is not guaranteed. Therefore, con-

sidering other learning algorithms that would be even

more appropriate seems a relevant research direction.

A second direction of research is to investigate other

problems that could be solved by this approach. For

instance in the deﬁnition of resilience proposed by

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

(Martin, 2004), there is a problem of target hitting in

which a cost function is associated with the states. We

think that our approach could easily be adapted to this

problem.

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(1998). Set-Valued Numerical Analysis for Optimal

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APPENDIX

Proof of Proposition 3.1.1

Part I. First, let us prove by induction that ∀h >

0,Capt

⊂ C

By deﬁnition, Capt

= C = C

. Suppose that at step

n, Capt

⊂ C

. Consider x ∈ Capt

n+1

. Let us recall

that G(x,u) = x + ϕ(x, u).dt when x /∈ C.

Deﬁning B

(x,d) the set of points of K

such that

|x − x

| ≤ d, we can easily show that condition (16)

can be rewritten as:

(x,h) ⊂ C

⇒ x ∈ C

. (27)

By deﬁnition, we know that there exists u ∈ U such

that G(x, u) ∈ Capt

, which implies that for all

∈ B

(x,h), d (G(x

,u),Capt

) ≤ µh, because G is

µ-Lipschitz.

Moreover, for all x

∈ B

(x,h), d(G(x

,u),C

) ≤ µh,

because, by hypothesis, Capt

⊂ C

. Thus x

∈ C

n+1

Therefore, x ∈ C

n+1

(because of condition (27)).

We can thus conclude Capt

n+1

⊂ C

n+1

Part II. Now, we prove by induction that for

any n, C

→ Capt

when h → 0.

Suppose now that for a given value n, C

→ Capt

when h → 0.

Because Capt

⊂ C

, we have:

∀x ∈ K | x /∈ Capt

, ∃h > 0 | x /∈ C

Now, consider x ∈ K such that x /∈ Capt

n+1

This implies that for all u ∈ U such that

d (G(x,u),Capt

) > 0. One can choose h

> 0 and h

such that for all u ∈ U, d (G(x,u),Capt

) > h

+ µλh.

Condition (15) can be rewritten as:

(x,λh) ⊂

⇒ x ∈ C

. (28)

In this case, for all x

∈ B

(x,λh), all u ∈ U,

d (G(x

,u),Capt

) > h

, because G is µ-Lipschitz.

Since C

→ Capt

when h → 0, there exists h, such

that, for all x

∈ B

(x,λh), and all u ∈ U, G(x

,u) ∈

, hence x

∈ C

. Hence, there exists h such that

x /∈ C

(because of condition 27).

Therefore C

n+1

→ Capt

n+1

when h → 0.

Conclusion. Capt

⊂ C

and C

→ Capt

then C

an outer approximation of the capture basin at time

n.dt, which tends to the actual capture basin when the

resolution of the grid h tends to 0.

INNER AND OUTER CAPTURE BASIN APPROXIMATION WITH SUPPORT VECTOR MACHINES

Proof of Proposition 3.1.2

Part I. We begin to show by induction that C

⊂

Capt

Suppose that C

⊂ Capt

and consider x ∈ C

n+1

Because of condition (20):

∃x

∈ C

n+1

such that |x − x

| < h.

By deﬁnition of C

n+1

∃u ∈ U such that d

(G(x

,u),C

) > µh.

By hypothesis of induction C

⊂ Capt

, hence

: d

(G(x

,u),Capt

) > µh. By hypothesis

on G, |G(x

,u) − G(x,u)| < µ|x

− x|, hence

(G(x,u),Capt

) > 0. Therefore x ∈ Capt

n+1

. Thus

n+1

⊂ Capt

n+1

Part II. We prove by induction that, when h → 0,

→ Capt

Suppose that C

→ Capt

when h → 0.

Because C

⊂ Capt

, we have:

∀x ∈ Capt

| d

(x,∂Capt

) > 0,∃h > 0 | x ∈ C

We use the rewriting of condition (21):

(x,λh) ∩C

⇒ x ∈ C

. (29)

Consider x ∈ Capt

n+1

such that d

(x,∂Capt

n+1

) > 0

. One can choose h such that d

(x,∂Capt

n+1

) >

(µ + λ)h. With such a choice, for each x

∈ B

(x,λh),

,∂Capt

n+1

) > µh, hence, there exists u ∈ U

such that d

(G(x

,u),Capt

)) > 0 (because G is µ-

Lipschitz).

By induction hypothesis, there exists h such that



G(x

,u),C



> µh, hence x

∈ C

n+1

. Taking the

smallest value of h this is true for all x

∈ B

(x,λh).

Therefore x ∈ C

n+1

(because of condition (29)).

Therefore C

n+1

→ Capt

n+1

when h → 0.

Conclusion. C

⊂ Capt

and C

→ Capt

then C

is an inner approximation of the capture basin in ﬁ-

nite time n.dt, which tends to the actual capture basin

when the resolution of the grid tends to 0.

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics