PIECEWISE AFFINE SYSTEMS CONTROLLABILITY AND

HYBRID OPTIMAL CONTROL

Aude Rondepierre

Laboratoire de Mod

elisation et Calcul

50 av. des Math

ematiques - 38041 Grenoble, France

Keywords:

Piecewise afﬁne hybrid systems, polyhedral sets, controllability, optimal control synthesis, algorithms.

Abstract:

We consider a particular class of hybrid systems, deﬁned by a piecewise afﬁne dynamic over non-overlapping

regions of the state space. We want to control their behaviors so that it reaches a target by minimizing a given

cost. We provide a new numerical algorithm under-approximating the controllable domain under the given

hybrid dynamic. Given an optimal sequence of states of the hybrid automaton, we are then able to traverse the

automaton till the target, locally insuring optimality.

1 INTRODUCTION

Aerospace engineering, automatics and other indus-

tries provide a lot of optimization problems, which

can be described by optimal control formulations:

change of satellites orbits, ﬂight planning, motion co-

ordination (Fierro et al., 2001; Pesch, 1994). Since the

years 1950-1970, the optimal control theory has been

extensively developed and provides us with powerful

results like dynamic programming (Bellman, 1957)

or the maximum principle (Pontryagin et al., 1974).

However resolutions are mainly numerical.

Now, in “real-life”, optimal control problems are

fully nonlinear. There are today two main classes

of numerical methods: the ﬁrst one uses a discrete

version of the dynamical principle (Bertsekas, 1984;

Bardi and Capuzzo-Dolcetta, 1997). But those algo-

rithms are very expensive in high dimension. The

second is based on the Pontryagin Maximum Princi-

ple (Pontryagin et al., 1974), (Bryson and Ho, 1975),

which provides a pseudo-Hamiltonian formulation of

optimal control problems. However, the main difﬁ-

culty is actually the synthesis of optimal feedback,

even not solved for linear systems, except in some

very special cases as time-optimal problems (Bryson

and Ho, 1975; Pinch, 1993; Pesch, 1994).

In this paper, we consider a particular class of hy-

brid systems, deﬁned by a piecewise afﬁne dynamic

over non-overlapping regions of the state space:

X(t) = A

X(t)+B

u(t)+c

, for X(t) ∈ D

(1)

We present a hybrid algorithm controlling the system

(1) from an initial state X

at time t = 0 to a ﬁnal

state X

= 0 at an unspeciﬁed time t

. To reach

this state, we allow the admissible control functions

u to take values in a convex and compact polyhedral

set U

of R

, in such a way that: J(X, u(.)) =

l(X(t), u(t))dt is minimized.

Piecewise afﬁne models has become a relevant and

powerful tool in the approximation of general smooth

nonlinear systems (Johansson, 1999). They usually

manage to capture many features of general physical

systems, and enable a tractable mathematical analy-

sis. Where usual numerical methods suffers from

the curse of the dimension (and with the expansion

of aerospace, today algorithms in the control theory

have to deal with dimension 6 or 7), the analytical

approach by piecewise afﬁne models must allow to

improve approximations (Girard, 2004): the level of

details allows to reach a compromise between quanti-

tative quality of the approximation and the computa-

tional time. Such studies has already be done e.g. for

biological systems, where simpliﬁcations in relation

to real data and in regard of simulations are possible,

see (Dumas and Rondepierre, 2003).

Here, we provide a full implementation for the analy-

sis of polyhedral piecewise afﬁne control systems

in every dimension. In particular, we develop a

new efﬁcient numerical method to compute an under-

approximation of the controllable domain. We also

propose some promising directions towards generic

algorithms for solving piecewise afﬁne optimal con-

trol problems.

The paper is organized as follows. In section 2, we de-

ﬁne hybrid systems and formulate the hybrid optimal

control problem. In section 3, we provide a numeri-

cal controllability analysis and then, in section 4, an

294

Rondepierre A. (2005).

PIECEWISE AFFINE SYSTEMS CONTROLLABILITY AND HYBRID OPTIMAL CONTROL.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 294-299

DOI: 10.5220/0001185802940299

 SciTePress

algorithmic resolution of the hybrid optimal control

problem. Some examples are presented in section 5.

2 HYBRID OPTIMAL CONTROL

PROBLEM

Let us start deﬁning our hybrid problem. The con-

trol domain U

is a polytope in R

, deﬁned as the

convex hull of a ﬁnite number of points: U

Conv(s

, . . . , s

), such that: 0 ∈ U

. The points

are assumed to be the vertices of U

Let (D

)

be a polyhedral partition of the state

space R

. We thus introduce the hybrid automaton

H = (Q, D, E, F, G, R) deﬁned as follows:

1. Q the countable set of indices of the simplexes D

2. D = {D

/ q ∈ Q} the collection of domains over

the state space: ∀q ∈ Q, D

is a polyhedron of R

∀(q, q

′

) ∈ Q

, [int(D

)∩int(D

′

) 6= ∅ ⇒ D

= D

′

]

3. E = {(q, q

′

) ∈ Q × Q/ ∂D

∩ ∂D

′

6= ∅} the

transition set.

4. F = {f

/ q ∈ Q} the collection of afﬁne ﬁeld

vectors:

: D

× U

→ R

(X, u) → A

X + B

u + c

such that: [ 0 ∈ D

⇒ c

= 0 ].

5. G = {G

/ e ∈ E} the collection of the guards:

∀e = (q, q

′

) ∈ E, G

= ∂D

∩ ∂D

′

6. R = {R

/e ∈ E} the collection of Reset func-

tions: ∀e = (q, q

′

) ∈ E, ∀x ∈ G

, R

(x) = {x}

(Here, we do not need to reinitialize the continuous

variable x, since the D

are adjacent).

Remark 1 The assumption [ 0 ∈ D

⇒ c

= 0 ]

ensures that the target 0 is an equilibrium point of

our hybrid dynamic for u = 0.

From now on, the hybrid automaton H is assumed not

Zeno

In this paper, we focus on optimal control problems

) associated to the hybrid automaton H ; we con-

sider the hybrid dynamic induced by H:

X(t) = A

X(t)+B

u(t)+c

, for X(t) ∈ D

(2)

and want to control (2) from an initial state X

to a

target X

= 0 at an unspeciﬁed time t

. To reach this

state, we allow the admissible control functions u to

take values in the polytope U

, in such a way that:

J(X, u(.)) =

l(X(t), u(t))dt is minimized.

Zeno executions correspond to an inﬁnite number of

switch in a ﬁnite time. That often involves problems in the

simulation of hybrid system. Indeed the transition times

come closer and closer and in simulations, we can not dif-

ferentiate them any more (see (Girard, 2004; Zhang et al.,

2001)).

3 HYBRID SYSTEM

CONTROLLABILITY

In this section, we want to compute the set of con-

trollable points in R

, i.e. the set of initial points

for which the hybrid problem (P

) admits a solution.

The idea is, by time reversal, to come down to the

computation of the attainable set from 0 and to guar-

antee the controllability of given initial points.

In (Dumas and Rondepierre, 2005, §3.1), an algo-

rithm is proposed to compute an under-approximation

in time T of the controllable set for linear systems

without state constraints. In this paper, we propose

an extension of this algorithm to piecewise afﬁne sys-

tems. First we present our under-approximating al-

gorithm over one given cell of the space state. This

enables us then to build an under-approximation of

the controllable set over a path of cells.

3.1 Under-Approximation of the

Controllable set in a given cell

Let q be a discrete mode satisfying: 0 ∈ D

. We

want to compute an under-approximation of the con-

trollable set inside the cell D

⊂ R

under the control

constraints: u ∈ U

= Conv(s

, . . . , s

Deﬁnition 1 (Controllable set in D

when 0 ∈ D

)

X ∈ D

is controllable iff there exist T ≥ 0 and

u : [0, T ] → U

admissible, such that:

i. X = −

−A

u(ω) + c

]dω

ii. ∀t ∈ [0, T ], −

−A

u(ω) + c

]dω ∈ D

In the next, X

[0, s

](.) denotes the trajectory accord-

ing to u = s

that goes through 0 ; then, by time rever-

sal, X

q ,i

denotes the ﬁrst intersection of X

[0, s

](.)

with ∂D

. Let F

q ,i

be the encountered face:

q ,i

= X[0, s

](T

) ∈ F

q ,i

where: T

= sup{t < 0/ X[0, s

](t) ∈ ∂D

} as

shown on ﬁgure 1.

Notation 1 (X

q ,i

, F

q ,i

) := OutCell(q, 0, s

)

By convention, when X

[0, s

](.) goes out of D

, we

state: (X

q ,i

, F

q ,i

) = (∅, ∅).

By linearity of the hybrid dynamic in the mode q, the

controllable domain in D

is convex and (Dumas and

Rondepierre, 2005, Proposition 3) can be applied in

our context:

Proposition 1 Conv(X

q ,1

, . . . , X

q ,k

) is an under-

approximation of the controllable set in D

However the quality of the resulting under-

approximation is very poor, especially when

most of trajectories do not evolve inside D

PIECEWISE AFFINE SYSTEMS CONTROLLABILITY AND HYBRID OPTIMAL CONTROL

295

q,1

q,5

q,2

q,3

q,4

q,1

q,2

q,3

(a) (b)

Figure 1: Under-approximation in state q of the controllable

set when (a) O ∈ int(D

) (b) 0 ∈ ∂D

Example 1

X(t) =



−1 −2

−3 −1



u(t)

where u(t) ∈ Conv([0, 0], [1, 0], [0, 1]) and X(t) ∈

D = Conv([0, 0], [1, 0], [0, 1]).

As shown on ﬁgure 2-(a), the trajectory according

u = (1, 0) evolves outside D, so that there is no valid

intersection point with the boundary of D. Our ap-

proximation is actually insufﬁcient (see ﬁgure 2-(b)).

1,4

1,2

0,8

0,6

0,4

0,8

0,2

0,60,40,20

u=(0,1)

u=(1,0)

0,20 1

0,8

0,6

0,4

0,6

0,2

0,4

(a) (b)

Figure 2: (a) Exact controllable set in grey (b) Under-

approximation inside the cell D in ...

To improve our under-approximation, we so have to

compute more controllable points on the boundary of

. We then propose a new algorithm based on the

discretization of the edges of the control set and on

the following lemma:

Lemma 1 Let u be a constant control in ]s

, s

then: ∀t ≤ 0, X[0, u](t) ∈ ]X[0, s

](t), X[0, s

](t)[

Likewise, if u ∈ int(Conv(s

, . . . , s

)), then:

X[0, u](t) ∈ int(Conv(X[0, s

](t); i = 1 . . . k))

Let [s

, s

] be an edge of U

. The principle of the

algorithm 1 is the following: let us state:

q ,k

, F

q ,k

) := OutCell(q, 0, s

), k ∈ {i, j}

i. If F

q ,i

= F

q ,j

(6= ∅), any reﬁnement is required.

Indeed, if u ∈]s

, s

[, then the trajectory X[0, u](.)

evolves between X[0, s

](.) and X[0, s

](.), so that

its intersection with ∂D

already is in the under-

approximation (see e.g. X

q ,2

and X

q ,3

on ﬁgure 1).

ii. Otherwise, by dichotomy, we introduce the con-

trol u

i,j

and (X, F ) := OutCell(q, 0, u

i,j

)

to recursively complete the under-approximation.

The principle is illustrated on ﬁgure 3.

q,j

q,i

X[0,s ](.)

q,i

q,j

q,i

X[0,s ](.)

X[0,s ](.)0

X[0,s ](.)

(a)

(b)

(c)

Figure 3: Principle of the DiscreteEdge algorithm

(a) Recursive call for the controls [u

i,j

, s

]

(b) Recursive call for the controls [u

i,j

, s

]

, u

i,j

] and [u

i,j

, s

]

We so have a complete algorithm to under-

approximate the controllable set in a given state cell

in any dimension.

3.2 Controllability in a given cells

path

Let ∐ = (q

)

i=0...r

be a given sequence of discrete

modes of the hybrid automaton H, such that:



0 ∈ D

∀i ∈ [|0, r − 1|], G

i+1

)

6= ∅

Now, we want to build an under-approximation of the

controllable set over the sequence of adjacent states

. The principle is to start by computing the under-

approximation of the controllable set from 0 in the

cell q

as previously explained. Then, from its inter-

section with the guard G

)

, we pursue the under-

approximation, the same way. The only difference is

that the reverse starting point is not 0 any more, but

the extremal points of the intersection between the

guard and the current under-approximation. The al-

gorithm stops when this intersection is empty or when

the last state q

is reached.

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296

Algorithm 1 DiscreteEdge

Require: q, X

(target point), u

, u

, h > 0 the dis-

cretization step.{In §3.1 X

:= 0}

, F

) := OutCell(q, X

, u

);

, F

) := OutCell(q, X

, u

);

Ensure: Λ set of controllable points.

1: Λ := ∅;

2: if distance(u

, u

) ≥ h and

6= F

or F

= F

= ∅) {see i.} then

3: (X, F ) := OutCell(q, X

);

4: if X 6= ∅ then

5: Λ := {X};{X belongs to the under-

approximation}

6: end if

7: if X

6= ∅ and X

6= ∅ {Case (a) ﬁgure 3}

then

8: if F 6= F

then

9: Λ := Λ ∪ DiscreteEdge(q, X

, u

, h);

10: end if

11: if F 6= F

then

12: Λ := Λ ∪ DiscreteEdge(q, X

, u

, h);

13: end if

14: else

15: {Case (b) or (c) ﬁgure 3}

16: if F

= ∅ or X 6= ∅ then

17: Λ := Λ ∪ DiscreteEdge(q, X

, u

, h);

18: end if

19: if F

= ∅ or X 6= ∅ then

20: Λ := Λ ∪ DiscreteEdge(q, X

, u

, h);

21: end if

22: end if

23: end if

24: Return Λ;

4 SOLVING THE HYBRID

OPTIMAL CONTROL

PROBLEM

This section deals with the algorithmic solving of hy-

brid control problems: ﬁrst we focus on the control-

lability of given initial points. Then, a method is pro-

posed to solve local afﬁne optimal control problems

in each cell of the automaton H. Lastly, we detail a

generic algorithm for solving the whole hybrid prob-

lem.

4.1 Controllability of the initial point

Let X

be a given initial point in R

. Now we want

to deﬁne the controllability of X

. This leads us to in-

troduce the notion of solution of our hybrid problem:

Deﬁnition 2 (X(.), u(.)) is a solution of the hybrid

control problem (P

) (i.e. X

controllable) if there

exists a ﬁnite execution χ = (τ, ∐, X) satisfying:

i. (∐(τ

), X(τ

)) = (q

, X

) such that: X

∈ D

ii. ∀i, X(.) is continuously differentiable, ∐(t) = q

and X(t) ∈ D

over ]τ

, τ

i+1

[ (τ

< τ

i+1

iii. ∀i = 1, . . . , r, X(τ

) ∈ G

i−1

)

iv. (∐(τ

r+1

), X(τ

r+1

)) = (q

, 0).

where: τ = (τ

)

i=0...r+1

(τ

= 0) and ∐ = (q

)

0≤i≤r

Notation 2 For a given sequence of discrete modes

∐ = (q

)

i=0...r

, we deﬁne a successor function succ

as follows: succ

∐

(q) = q

i+1

if q = q

(i < r)

From this deﬁnition, the difﬁculty is to determine

the optimal sequence of modes. Some directions to

solve this problem include numerical pre-simulations

as done in (Bonnans and Maurin, 2000) or a variable

change ds = l(X(t), u(t))dt to come down to a time

optimal control problem. From now on, we then con-

sider the following assumption:

Hypothesis 1 let ∐ = (q

)

i=1...r

be a given admissi-

ble sequence of discrete modes i.e. there exists (τ, X),

such that χ = (τ, ∐, X) is a (non optimal) ﬁnite exe-

cution of the hybrid automaton H that steers the ini-

tial point X

to the target 0.

Under this hypothesis, the UnderApproximation

algorithm tests if the initial point X

is reachable by

time reversal from the 0 in the given path of cells.

4.2 Local Optimal Solutions

In this section, we analyze the dynamic behavior of

our hybrid system H in one given mode q. Let us

deﬁne our local afﬁne optimal control problem P

Minimize the cost function J(X

, u(.)) =

l(X(t), u(t))dt with respect to the control

u(.) under the dynamic:



X(t) = A

X(t) + B

u(t) + c

X(0) = X

and the constraints: ∀t ∈ [0, t

], X(t) ∈ D

, u(t) ∈

, where the ﬁnal time t

is unspeciﬁed.

So, in the mode q, we have to solve a state con-

strained optimal control problem P

. The main dif-

ﬁculty is then the choice of the target, when 0 is not

in the considered cell D

. Indeed, in this case, two

possible tactics could be considered:

X(t

) = 0. If 0 /∈ D

, P

is solved as an afﬁne

optimal control problem without state constraints. As

soon as the so computed optimal trajectory reaches

a guard G

(q,q

′

)

of the cell D

, the system switches

to the mode q

′

with a new problem P

′

. Methods

and algorithms have been developed in (Dumas and

Rondepierre, 2005) to solve afﬁne optimal control

problems via their Hamiltonian formulations. Unfor-

tunately, the convergence of trajectories towards the

origin is not guaranteed.

PIECEWISE AFFINE SYSTEMS CONTROLLABILITY AND HYBRID OPTIMAL CONTROL

297

X(t

) ∈ G

(q,q

succ

∐

(q)

)

. As deﬁned in hypothesis 1,

we are given a sequence ∐ = (q

)

i=0...r

of discrete

modes in our hybrid automaton, for which the initial

point X

is controllable. The strategy is then to reach

the guard between the current mode and its successor

towards ∐. We so compute local optimal trajectories

for the given path in the state space.

From now on, we choose these ﬁnal conditions.

Optimal control under state inequality constraints

is a hard and subtle problem. Indeed, for some spe-

cial conditions like bounded target curves, there can

be no generic solving methods, see e.g. (Pinch, 1993,

§5,Optimal Control to target curve). Let us show that

our problem can be solved via the Pontryagin maxi-

mum principle: we consider the afﬁne optimal con-

trol problem P

with the ﬁnal condition: X(t

) ∈

(q,q

succ

∐

(q)

)

. The state constraints induced by D

are afﬁne in the state variable, so that we can state:

X + M

≤ 0 State constraints in mode q

Under this constraints, the above ﬁnal condition is to

reach the hyperplan containing the face G

(q,q

succ

∐

(q)

)

We then introduce the Hamiltonian function:

(X, u, λ) = l(X, u)+(λ

+µ

)(A

X +B

u+ c

)

The Pontryagin principle (Bryson and Ho, 1975;

Clarke, 1990) provides then us the following opti-

mization problem: “Minimize the Hamiltonian func-

tion H with respect to the control variable u ∈ U

Conv(s

, . . . , s

) under the constraints:

•

X(t) =

∂H

∂λ

(X(t), u(t), λ(t), µ(t))

•

(t) =

∂H

∂X(t)

(X(t), u(t), λ(t), µ(t))

• H(X

⋆

(t), u

⋆

(t), λ

⋆

(t), µ

⋆

(t)) = 0 along the opti-

mal trajectory

• Transversality condition: < λ(t

), n

q,succ

∐

(q)

>= 0

where n

q ,suc c

∐

(q )

is the normal vector to the face

(q,q

succ

∐

(q)

)

”.

The parameter µ is a Lagrange multiplier verifying:

∀i, µ

(t)



= 0 if (L

X + M

)

< 0

> 0 if (L

X + M

)

= 0

4.3 Hybrid Solver

In regard of previous sections, we can now describe

the HybridSolving algorithm:

Let X

be the initial point and ∐ = (q

)

i=0...r

given sequence of discrete modes as expressed in

hypothesis 1. The principle is to replace the hybrid

problem P

by (r + 1) state constrained afﬁne

optimal control problems (P

)

i=0...r

as deﬁned in

section 4.2 and to compute cells by cells a local piece-

wise optimal solution of our initial hybrid problem

. For each problem P

, we respectively deﬁne

the initial condition: X(0) = X

(in mode q

) where:



= X

i+1

= X[X

, u

⋆

](.) ∩ G

i+1

)

, i = 0 . . . r − 1

Algorithm 2 HybridSolving

Require: X

, H, ∐ = (q

)

i=1..r

a sequence of dis-

crete modes s.t. X

∈ D

and 0 ∈ D

Ensure: (τ, X, u), V (X

)

where (τ, ∐, X) is a local optimal execution of

H, V (X

1: if X

/∈ UnderApproximation(H,q) then

2: Return “X

may not be controllable”.

3: end if

4: τ

:= 0; V := 0;

{Piecewise Afﬁne Resolution}

5: for all time step i (from 1 to r) do

6: Solve the afﬁne problem P

→

(X(.), u(.), t

, V

)

7: X

:= X(t

); τ

i+1

:= τ

+ t

; V := V + V

;

8: end for

9: Return (τ, X, u, V ).

5 UNDER-APPROXIMATION

EXAMPLES

5.1 In dimension 2

We consider the linear system:

X(t) =



1 1

0 1



X(t) +



−1 −2

−3 −1



u(t)

where u(t) ∈ U

= Conv([0, 0], [1, 0], [0, 1]). The

under-approximation algorithm is performed on the

simplex: D = Conv([0, 0], [1, 0], [0, 1]). On ﬁgure 4,

we show the successive steps to build a good under-

approximation of the controllable domain.

5.2 In dimension 3

We now consider the system for q ∈ N:

X(t) =

q 0 q

0 3q 0

0 q q

X(t)+

−1 0 0

0 −2 0

0 0 −1

u(t)

with u (t) ∈ Conv([0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]).

The under-approximation is performed on two adja-

cent cubes (see ﬁgure 5) in modes (q = 0, q = 6).

6 CONCLUSION

In this paper, we addressed the optimal control prob-

lem for piecewise afﬁne systems. We ﬁrst provided a

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298

0,20 1

0,8

0,6

0,4

0,6

0,2

0,4

0,20 1

0,8

0,6

0,4

0,6

0,2

0,4

0,20 1

0,8

0,6

0,4

0,6

0,2

0,4

0,20 1

0,8

0,6

0,4

0,6

0,2

0,4

0,20 0,60,4 1

0,8

0,6

0,8

0,4

0,2

0,20 0,6

0,2

0,4

0,8

0,6

0,8 1

0,4

0,20

0,8

0,6

0,60,4

0,2

0,4

Figure 4: Different steps in the under-approximation for re-

spectively h = 2, 1, 0.5, 0.25, 0.1, 0.05, 0.01

Figure 5: Under-Approximation in 3d

full algorithm to compute an under-approximation of

the controllable domain. Then the resolution of the

hybrid problem is reduced to several explicit resolu-

tions of state constraints afﬁne optimal control prob-

lems.

This algorithm however guarantees only a local op-

timization. Next step will be to give a way to ﬁnd

an optimal sequence of cells containing an optimal

trajectory. Several directions to solve this problem

include: exploration of different sequences of states,

partial numerical simulations to obtain some informa-

tions on the localization of optimal trajectories and

thus reduce the exploration. Another way could be to

replace l(X(t), u), the cost function with an admissi-

ble variable change ds = l, so that the problem comes

down to a time optimal control problem. Finding the

optimal sequence would then be to minimize the time

to reach the target.

Further developments are also a study of the approxi-

mation error and a rigorous proof of the convergence

of our under-approximation towards the real control-

lable set. Future works will include the analysis of

nonlinear dynamics:

X(t) = f(X(t), u(t)) by piece-

wise afﬁne models. The hybrid approximant is build

by linear interpolation of f at the vertices of a mesh of

× U

. In consequence, in each cell of the result-

ing automaton, the system is subject to mixed afﬁne

inequalities constraints in both state and control

REFERENCES

Bardi, M. and Capuzzo-Dolcetta, I. (1997). Optimal

Control and Viscosity Solutions of Hamilton-Jacobi-

Bellman Equations. Birkauser.

Bellman, R. (1957). Dynamic Programming. Princeton

University Press.

Bertsekas, R. (1984). Dynamic Programming and Optimal

Control. Athena Scientiﬁc.

Bonnans, J. and Maurin, S. (2000). An implementation

of the shooting algorithm for solving optimal control

problems. Technical Report RT-0240, INRIA.

Bryson, A. and Ho, Y. (1975). Applied Optimal Control.

Hemisphere.

Clarke, F. H. (1990). Optimization and Nonsmooth Analy-

sis. SIAM Classics in Applied Mathematics.

Dumas, J.-G. and Rondepierre, A. (2003). Modeling the

electrical activity of a neuron by a continuous and

piecewise afﬁne hybrid system. In Proceedings of the

2003 Hybrid Systems: Computation and Control.

Dumas, J.-G. and Rondepierre, A. (2005). Algorithms

for symbolic/numeric control of afﬁne dynamical sys-

tems. In Proceedings of the 2005 International Sym-

posium on Symbolic and Algebraic Computation.

Fierro, R., Das, A. K., V.Kumar, and Ostrowski, J. P. (2001).

Hybrid control of formations of robots.

Girard, A. (2004). Analyse Algorithmique des Syst

emes hy-

brides. PhD thesis, Institut National Polytechnique,

Grenoble.

Johansson, M. (1999). Piecewise Linear Control Systems.

PhD thesis, Lund Institute of Technology.

Pesch, H. (1994). A practical guide to the solutions of real-

life optimal control problems. Parametric Optimiza-

tion. Control Cybernet.

Pinch, E. (1993). Optimal Control and the Calculus of Vari-

ations. Oxford University Press.

Pontryagin, L., Boltiansky, V., Gamkrelidze, R., and

Michtchenko, E. (1974). Th

eorie math

ematique des

processus optimaux. Editions de Moscou.

Rondepierre, A. and Dumas, J.-G. (2005). Algorithms for

hybrid optimal control. Technical report, IMAG-ccsd-

00004191, arXiv math.OC/0502172.

Zhang, J., Johansson, K., Lygeros, J., and Sastry, S. (2001).

Zeno hybrid systems. International Journal of Robust

and Nonlinear Control.

Work in progress, see (Rondepierre and Dumas, 2005)

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