Speci ica ion and con ol syn hesis o ne wo ked
ehicle sys ems
Jo˜
ao Bo ges de Sousa
and Fe nando Lobo Pe ei a
Dep . Eng. Elec o ´
ecnica e de Compu ado es
Faculdade de Engenha ia da Uni e sidade do Po o
4200-465 Po o, Po ugal
Email: {j asso, lp}@ e.up.p
Ma ia Ben o Nunes
Labo a ´
o io de Ae onau ica
Academia da Fo c¸a A´
e ea Po uguesa
G anja do Ma quˆ
es
Sin a, Po ugal
E-mail: [email p o ec ed]
Abs ac —A amewo k o he ep esen a ion, o mal spec-
i ica ion, and con ol syn hesis o ne wo ked ehicle sys ems
is p esen ed. F om dynamic op imiza ion, his amewo k has
inhe i ed he concep s and heo ies o op imali y, each se
compu a ion and con ol, and he mo i a ion o imp o e he
pe o mance o inc easingly complex physical p ocesses. F om
se heo y, his amewo k bo owed he ep esen a ional powe
o he language o se s o cap u e he ela ions among ehicles
and con olle s in a way ha is consis en wi h con ol design.
The ANTEX-M p ojec is desc ibed o illus a e he challenges
posed by ne wo ked ehicle sys ems and o illus a e how he
amewo k add esses hese challenges.
I. INTRODUCTION
O e he las decade we ha e been designing and building
mul i- ehicle sys ems o unde wa e , sea, ai , and g ound
applica ions [5], [4], [3]. In his p ocess we de eloped a be e
unde s anding o he p oblem o b inging oge he echnologi-
cal de elopmen , heo e ical unde pinnings, and compu a ional
ools in he design and implemen a ion o ne wo ked semi-
au onomous and au onomous ehicles. This p oblem poses a
new challenge o con ol enginee ing. The challenge comes
om he dis ibu ed na u e o he p oblem and om he
na u e o in e ac ions. Fo example, in ne wo ked mul i- ehicle
sys ems, in o ma ion and commands a e exchanged among
mul iple ehicles, and he oles, ela i e posi ions, and de-
pendencies o hose ehicles change du ing ope a ions. This
challenge en ails a shi in he ocus o con ol heo y – om
p esc ibing and commanding he beha io o isola ed sys ems
o p esc ibing and commanding he beha io o in e ac ing
sys ems.
In his pape we ou line a amewo k o he o mal ep-
esen a ion, speci ica ion, and con ol syn hesis o ne wo ked
ehicle sys ems.
Fo ou pu poses, he wo ld consis s basically o egions,
physical objec s, eams o physical objec s, and ne wo ks o
eams. While some objec s ha e a physical exis ence, o he s
a e b ough in o exis ence as so wa e agen s. Examples o
so wa e agen s a e con olle s ha may be c ea ed, modi ied,
and des uc ed in eal- ime. Examples o physical objec s
include ehicles and o he de ices. Vehicles ha e a ibu es
and a e capable o deli e ing a omic se ices (e.g. sensing),
o execu ing asks (e.g. ly a ce ain pa h), and o pe o ming
ac ions (e.g. launch a missile). A complex se ice is a se ice
ha canno be deli e ed by a single physical objec : i equi es
he composi ion o a omic se ices deli e ed by mul iple
physical objec s. This is done wi h a omic links. An a omic
link is a ela ion on he posi ions, mo ions and a omic se ices
p o ided by wo physical objec s. An a omic con igu a ion is
a lis o a omic links connec ing a g oup o ehicles. Vehicles
a e eamed o deli e se ices, and o pe o m asks ha canno
be deli e ed by a single physical objec .
In wha conce ns o mal ep esen a ion we ep esen all o
he objec s, hei dynamic beha io , and he ela ions among
hemsel es wi h simple concep s om se heo y and om
dynamic op imiza ion. Rela ions o in e es o us a e: 1)
se ices and hei composi ion; 2) se ices and hei physi-
cal implemen a ion; 3) se ices and hei o de ela ions; 4)
objec s and modes o coo dina ion; 5) objec s and p ope ies
o hei composi ion; 6) se ices and se ice p o ide s; 7)
objec s and hei con ol s uc u e; 8) con ol s uc u es and
se ices. Some o he ela ions a e s a ic, o he s conce n he
dynamic beha io o ehicles unde coo dina ion cons ain s
ha change wi h ime. In o de o ep esen he las ones
we use a se - alued desc ip ion o he dynamic beha io o
ehicles and eams. We use each se s o desc ibe he e olu ion
o a dynamic sys em, in a ian se s o desc ibe he loca ions
whe e he pe manence o an objec wi hin a ce ain se is
ensu ed, and sol abili y se s o desc ibe he loca ions om
which a sys em can e ol e o each a gi en se .
We speci y ope a ions on objec s, and exp ess he speci i-
ca ion in a o mal language. The key obse a ion is ha we
can ep esen he objec s, he ela ions hey sa is y, and hei
ope a ions in he language o se s. We will show ha se - alued
cons ain s exp ess all o he ela ions o in e es o sys ems
o ne wo ked objec s. This way we a e able o ep esen he
wo ld o sys ems o ne wo ked objec s wi h simple concep s
om se heo y. This is why we will be able o o mally ela e
design and speci ica ion. In ac , his is he key idea behind
ou speci ica ion and con ol amewo k.
We w i e pa ial plan speci ica ions and de ine a planning
p ocedu e ha esul s in a da a s uc u e de ining all o he
con olle speci ica ions ha p ecede con olle design, and
whe e all logical ela ions a e al eady sa is ied.
Finally, we use echniques om dynamic op imiza ion o
syn hesize con olle s ha implemen he plan, o ha p o e
ha he plan is no easible.
The pape is o ganized as ollows. In sec ion II we discuss
he ANTEX-M p ojec o desc ibe pa o he mo i a ion
o ou de elopmen s. In sec ion III we in oduce a mission
example ha we will use in he eminde o he pape o
illus a e ou amewo k. In sec ion IV we discuss he issues
o o mal ep esen a ion and speci ica ion. In sec ion V we
o mula e he con ol p oblem o he mission example and
in sec ion VI we discuss he solu ion me hodology. In sec ion
VII we d aw some conclusions.
II. ANTEX-M
The ANTEX-M p ojec conce ns he design and he con-
s uc ion o a low cos unmanned ai ehicle (UAV) pla o m
o he Po uguese Ai Fo ce. The objec i es o he p ojec
a e:
•To design and build a low cos UAV pla o m o expe -
imen a ion, de elopmen , and in eg a ion o sensing and
communica ion echnologies.
•To de elop he echnological and expe imen al expe ise
equi ed o in eg a e UAV echnology in he Po uguese
A med Fo ces.
•To demons a e he ope a ional capabili ies o UAVs,
ei he in isola ed ope a ion o in eg a ed in a sys em.
The p ima y mission o he ANTEX-M UAV conce ns
su eillance. The speci ic applica ions a e:
•Sea ch and escue ope a ions in he Po uguese coas al
wa e s.
•Moni o mili a y ac i i ies in ac ical ope a ions.
•An i- e o is ope a ions.
•De ec ion and acking o ma i ime pollu ion.
•Fi e de ec ion.
The ANTEX-M UAV pla o m is an e olu ion o a Re-
mo ely Pilo ed Vehicle (RPV) de eloped by he Po uguese Ai
Fo ce Academy o conduc esea ch on adap i e ae o-elas ic
s uc u es [11]. The UAV will se e as a pla o m o moun
senso s de eloped by he Po uguese A med Fo ces and by he
Di ec o a e o A mamen and De ense Equipmen (Di ecc¸˜
ao
Ge al do A mamen o e Equipamen o e Equipamen os de
De esa). These senso s include in a ed senso s wi h image
p ocessing, o de ec ion and au oma ic acking on boa d;
lase emission de ec o s; and ada lase sys ems o h ee-
dimensional image gene a ion.
The p elimina y design speci ica ions o he UAV a e: 1)
emp y weigh – 5kg; 2) maximum ake o weigh – 8kg; 3)
wing span – 2.4 m; 4) max le el speed – 151 km/h ; 5) c uise
speed – 139 km/h .
III. MISSION EXAMPLE
We illus a e ou amewo k wi h one o he concei able
missions o he ANTEX-M ype Unmanned Ai Vehicle.
Fig. 1. ANTEX-M UAV
A concei able mission o he ANTEX-M UAVs consis s in
he su eillance and mapping o selec ed egions. One such
example consis s in moni o ing he e olu ion o oil spills1.
Conside he ollowing mission in ol ing wo ehicles, A
and B, ha coo dina e hei mo ions o execu e a “mapping”
ask. The “mapping” ask consis s o ha ing ehicle A ollow-
ing a p esc ibed pa h in he geog aphic (x,y) plane and aking
measu emen s along ha pa h wi hou colliding wi h obs acles.
The e a e no cons ain s on he z geog aphic coo dina e excep
o hose a ising om unknown obs acles. Vehicle A has a
mapping senso and does no ha e any senso o obs acle
a oidance. Vehicle B su eys he a ea in on o ehicle A
o iden i y he p esence o po en ial obs acles. B is as e
han A, and communica es he p esence o obs acles o A. To
do his B, ca ies an obs acle de ec ion senso . The p oblem
is o coo dina e he mo ions o he wo ehicles so ha ,
unde mild assump ions on he opog aphy o he wo ld, he
ehicles a e able o execu e he mapping ask success ully, i.e.
ehicle A does no collide wi h an obs acle be o e eaching
i s des ina ion.
He ea e , and unless s a ed o he wise, we will e e o his
mission as ou “example”.
IV. FORMAL REPRESENTATION AND SPECIFICATION
The wo ld consis s basically o egions,physical objec s,
eams o physical objec s, and ne wo ks o eams. While
some objec s ha e a physical exis ence, o he s a e b ough
in o exis ence as so wa e agen s.
Regions a e subse s o <n. Physical objec s a e ehicles,
and de ices. Each ehicle has a Type and each physical objec
is loca ed wi hin a leas one egion.
A ehicle/de ice has a ibu es (e.g. ange), i is capable
o deli e ing a omic se ices (e.g. sensing), o execu ing
asks (e.g. ly a ce ain pa h), and o pe o ming ac ions (e.g.
1This ype o mission is pa icula ly impo an o Po ugal. The in ense
ma i ime a ic o and om Eu ope p esen s a conside able en i onmen al
h ea , as demons a ed ecen ly by he oil spill om he P es ige anke .
launch a missile). A ehicle is con olled o mo e, and o
deli e a omic se ices while mo ing. Physical objec s ha e
he po en ial o es ablish in e ac ions among hemsel es. This
is done wi h a omic links. An a omic link is a ela ion on
he posi ions, mo ions and a omic se ices p o ided by wo
physical objec s. An a omic con igu a ion is a lis o a omic
links connec ing a g oup o ehicles.
We use physical objec s as he building blocks o eams
and o ne wo ks o eams. Teams and ne wo ks o eams
a e b ough in o exis ence o deli e complex se ices, and
o pe o m asks ha canno be deli e ed by a single physical
objec .
Acomplex se ice is a se ice ha canno be deli e ed by
a single physical objec : i equi es he composi ion o a omic
se ices deli e ed by mul iple physical objec s, in pa icula
ehicles. In o de o do his, hese ehicles ha e o be in a
pa icula a omic con igu a ion. In p ac ice, complex se ices
eme ge om modes o coope a ion among mul iple objec s,
o example physical objec s and so wa e agen s.
A eam is a se o ehicles ha is able o pe o m eam
missions. A eam mission consis s o eam asks and o ask
swi ching logic (also called a eam play). A eam ask consis s
o he deli e y o se ices and mo ions.
Aplan is a da a s uc u e consis ing o eam asks, con olle
speci ica ions o each ask, o de ing cons ain s, a iable
binding cons ain s, and causal links. The plan is e ined in o
eam asks. The e inemen p ocess in ol es eam composi ion
and asking, esou ce alloca ion, and pa h planning.
Nex we illus a e hese concep s wi h he ep esen a ion o
he p oblem domain o ou mission example.
In ou example he se Vehicles is:
Vehicles={A,B,C,D}
The e a e wo ypes o ehicles Mappe and Scou :
Type(A)=Mappe , Type(B)=Scou , Type(C)=Scou , Type(D)=Mappe
The unc ion P o ideA omicSe ice e u ns he lis o a omic
se ices p o ided by each ehicle ype. The unc ion A -
ibu eA omicSe ice e u ns he lis o a ibu es o an a omic
se ice and he unc ion ValueA ibu e(a,c) e u ns he alue o
a ibu e ao he ype ca omic se ice.
P o ideA omicSe ice(Mappe )={Coms,MapSenso ,Mo ion}
P o ideA omicSe ice(Scou )=
{Coms, Obs acleDe ec ionSenso ,Mo ion}
A ibu eA omicSe ice(Coms)=Range
ValueA ibu e(Range,Coms)=RComs
The equa ions o mo ion o all ehicles a e gi en by:
˙xi( ) = i( , xi( ), ui( )) ui( )∈ Ui, i =A, B, C, D
The unc ion Posi ion( ,Z) e u ns he geog aphic posi ion
(x,y,z) o ehicle Za ime Posi ion( ,Z) =Π(xZ( )).Πgi es
he p ojec ion o he s a e o ehicle Z on o he geog aphical
posi ion o he ehicle.
A omic se ices a e he building blocks o complex se -
ices. This is because some o he a omic se ices ha e he
po en ial o es ablish in e ac ions among he espec i e se ice
p o ide s. We call he a oms o hese in e ac ions a omic
links: an a omic link is a ela ion on he ela i e mo ions,
posi ions and a omic se ices p o ided by wo di e en se ice
p o ide s. The p edica e A omicLink(l, 1, 2) ep esen s he
ac ha ehicles 1and 2a e linked wi h a link o ype l.
The ype de ines he ole – he a omic se ices and he lis o
commands accep ed and issued – o each o he pa icipan s in
he link and he glue – he way he wo pa icipan s in e ac .
The glue is a ela ion on he ela i e posi ions and mo ions o
bo h se ice p o ide s, and on he commands hey exchange.
The glue is de e mined om he a ibu es o he co esponding
a omic se ices.
We ep esen he ac ha any wo ehicles in Vehicles a e
able o communica e unde well-de ined condi ions wi h he
a omic link o ype Coms:
A omicLink(Coms, 1, 2)⇔
Coms ∈P o ideA omicSe ice( 1)∧
Coms ∈P o ideA omicSe ice( 2)∧
φComs(Posi ion( , 1),Posi ion( , 2)) ≤1
whe e
φComs(a, b) : <3× <3→ <,s. .
φComs(a, b) = d2(a, b)
R2
Coms
, d(a, b) = ka−bk2
I is con enien o exp ess he unc ion φComs in e ms o
he ull s a e o bo h ehicles 1and 2.
φComs(x 1, x 2) = φComs(Π(x 1),Π(x 2))
We use he ollowing p edica es and unc ions o ep esen
he complex se ice o ype s:
•Requi edVehicleType(s) e u ns a lis wi h he ypes o
ehicles equi ed o implemen he se ice.
•Requi edVehicles(s,c,V) e u ns all he subse s o Vcapable
o deli e ing he complex se ice o ype swi h he alue
o a ibu es speci ied in c.
•Requi edCon igu a ionS yle(c,a) e u ns he con igu a ion
s yle ha each se o ehicles in Requi edVehicles(c,a) mus
sa is y o deli e he se ice cwi h he alue o a ibu es
as speci ied a.
Fo example, we ep esen he in e ac ions be ween ehicles
A and B in ou mission example as he Scou edMapping
complex se ice. To do his we conside wo gene ic Mappe
and Scou ehicles, 1and 2 espec i ely.
The ehicle o ype Scou , 2, e ol es in a icini y P
o he cu en geog aphic posi ion o 1and in o ms 1o
he exis ence o obs acles so ha 1can pe o m obs acle
a oidance success ully. P(P osi ion( , 1)) is gi en as a se -
alued map om he cu en geog aphic posi ion o 1 o a
subse o <3.
P(a) : a∈ <3,→P(a)⊂ <3
We ep esen his ype o in e ac ions be ween 1and 2
wi h he a omic link o ype Inside.
A omicLink(Inside, 1, 2)⇔
(Type( 1)=Mappe )∧(Type( 2)=Scou )∧
(φInside(Posi ion( , 1),Posi ion( , 2)) ≤1)
whe e
φInside :<3× <3→ < s. .
φInside(a, b) = d2
c(b, P (a)) + 1, dc(b, P ) = min
s∈Pd(s, b)
As be o e we de ine φInside(x 1, x 2)as ollows:
φInside(x 1, x 2) = φInside(Π(x 1),Π(x 2))
The implemen a ion o he se ice Scou edMapping also
equi es bo h ehicles o communica e. This means ha hey
ha e o sa is y a con igu a ion, i.e. a lis o a omic links. We
use an a omic con igu a ion s yle as a compac ep esen a ion
o a se o a omic con igu a ions sha ing a common p ope y.
We ep esen he con igu a ion s yle ywi h a p edica e Con ig-
u a ionS yle(y,c), whe e cis a eam o ehicles.
Con igu a ionS yle(Scou Mappe ,V) ⇔
∃X, Y ∈V:Type(X)=Mappe ∧Type(Y)=Scou ∧
A omicLink(Coms,X,Y) ∧A omicLink(Inside,X,Y)
Finally we a e able o ep esen he Scou edMapping complex
se ice:
Requi edVehicleType(Scou edMapping)={Scou ,Mappe }
Requi edVehicles(Scou edMapping,nil,Vehicles)=
{{A, B},{A, C},{D, B},{D, C}}
Requi edCon igu a ionS yle(Scou edMapping,Vehicles)=
Scou Mappe
Single ehicles and eams o ehicles execu e asks. A ask
has a ype. Conside , o example, he Mapping ask. This ask
is de ined as ollows.
Task(Mapping, {X, Y }, Scou edMapping(X,Y),
Pa h(x0, x , p, X), φ0(Posi ion( 0,X),Posi ion( 0,Y)))
whe e Mapping is he ype o he ask, {X, Y }is he eam
o ehicles execu ing he ask while deli e ing he se ice
Scou edMapping,p={(x, y)∈ <2: (x, y) = p( ), ∈[ 0, ]},
and P a h(x0, x , p, X)and φ0(Posi ion( 0,X),Posi ion( 0,Y)) a e
de ined as ollows (X, Y a e ehicle a iables):
∀ ∈[ 0, F] : φpa h( ,Posi ion( ,X),p( )) ≤1
whe e
φpa h( , a, b) : < × <3× <3s. .
φpa h( , a, b) = d2(a, b)−δ+ 1
φ0(Posi ion( 0, X),Posi ion( 0, Y )) ≤1
δis he pa h- acking ole ance and he las equa ion de ines
he se o ini ial posi ions o ehicles X and Y. We de ine
φpa h in he manne desc ibed abo e:
φpa h( , x i, b) = φpa h( , Π(x i), b)
The plan speci ica ion is a da a s uc u e consis ing o asks
and a pa ial o de on hese asks. In ou example he plan
speci ica ion consis s only o he mapping ask:
Plan = {Task(Mapping, {X, Y }, Scou edMapping(X,Y),
Pa h((0,10),(100,10),p, X), φ0(Posi ion( 0,X),Posi ion( 0,Y)))}
We need o ans o m his plan speci ica ion on o an im-
plemen able plan, i.e. we need a planne . He e we a e no
conce ned wi h planning p ocedu es and we assume ha he
planne p oduced he ollowing plan.
Plan = {Task(Mapping, {A, B}, Scou edMapping(A,B),
Pa h((0,10),(100,10),p,X)), φ0(Posi ion( 0,A),Posi ion( 0,B))}
A his poin he plan consis s o con ol speci ica ions om
which we de i e a easible s uc u e o con olle s in case i
exis s.
V. FORMULATION
The con ol p oblem o mula ion a ises na u ally om he
p e ious speci ica ion and is exp essed as ollows.
∀ ∈[0,1] : φpa h( , xA( ), p( )) ≤1∧(1)
φInside(xA( ), xB( )) ≤1∧
φComs(xA( ), xB( )) ≤1∧
φ0(xA( 0), xB( 0)) ≤1
We ob ain his o mula ion om he ins an ia ed plan, whe e
he a iables X and Y a e bound o ehicles A and B.
Rema k 1: We ep esen all o he s a e cons ain s as
inequali ies o he o m φ(x)≤1. We use his ep esen a ion
o simpli y he no a ion. In ac , all s a e cons ain s can be
ep esen ed in his o m.
In wha ollows we conside he ollowing hypo heses:
H1. The se - alued map Pis closed, con ex, and bounded.
H2. The pa h pis con inuous in .
H3. φ0(x, y)is con inuous in bo h a iables.
Lemma 1: Unde hypo heses H1−2 he unc ions
φpa h( , x, y),φInside(x, y), and φComs(x, y)a e con inuous.
De ine φ( , x, y),uand Uas ollows:
φ( , x, y) = max{φpa h( , x, p( )), φInside(x, y), φComs(x, y)}
u={uA, uB},U=UA× UB
( , xA, xB, u) = col( A( , xA, uA), B( , xB, uB))
We use he app oach om [10] o o mula e his con ol
p oblem as an in a iance p oblem (see [8], [9], [2], [1]). To
do his, we in oduce he ollowing alue unc ion.
V( , xA, xB) = min
u(.)max{{ max
τ∈[ 0, ]{φ(τ, xA[τ], xB[τ])},(2)
φ0(xA( 0), xB( 0))}, xA[ ] = xA, xB[ ] = xB}
whe e u(.) is a easible con ol unc ion (u(τ)∈ U, τ ∈[ 0, ]).
Now conside he sub-le el se o his alue unc ion gi en
by he ollowing equa ion:
R( , xA, xB) = {(xA, xB) : V( , xA, xB)≤1}(3)
A ime , R ep esen s he se o all loca ions o A and B
ha sa is y equa ion 1.
The ques ion now is how o calcula e he alue unc ion.
This is no a i ial ma e . The idea is o ans o m his global
p oblem in o a local one. We do his by ans o ming he global
p oblem on o a pa ial di e en ial equa ion.
VI. SOLUTION
In gene al he alue unc ion V can be calcula ed h ough
he gene alized Hamil on-Jacobi-Bellman (HJB) equa ion. We
can only do his i he alue unc ion sa is ies he p inciple o
op imali y.
Theo em 1: The alue unc ion V sa is ies he p inciple o
op imali y.
Basically he p inciple o op imali y s a es ha he alue
unc ion sa is ies a semi-g oup p ope y. The alue unc ion
inhe i s his p ope y om he semi-g oup p ope y o he
each se .
Using he echniques om [10] we can de i e he HJB
equa ion o his p oblem. Fi s we in oduce some no a ion:
H( , x, y, V, u) = V ( , x, y) + hVx( , x, y)· ( , x, y, u)i(4)
An in ini esimal e sion o he p inciple o op imali y leads
o Hamil on-Jacobi-Bellman equa ion:
V ( , x, y) + max
u∈U hVx( , x, y)· ( , x, y, u)i= 0 (5)
when V( , x, y)6=φ( , x, y)
max
u∈U {min{H( , x, y, V, u),H( , x, y, φ, u)}}
when V( , x, y)6=φ( , x, y)
V( 0, x, y) = max{φ( 0, x, y), φ0( 0, x, y)}
whe e V , Vx ep esen he co esponding sub-di e en ials.
Since V is non-di e en iable he usual no ion o solu ion o
a pa ial di e en ial equa ion does no apply. We conside
gene alized “ iscosi y”, o equi alen concep s, o solu ions
o his equa ion (see [6], [7]).
Gi en a solu ion V o he Hamil on-Jacobi-Bellman equa-
ion we a e able o ind he in a ian se R om equa ion
3. Now we ha e all o he ing edien s equi ed o syn hesize
he con olle o ou p oblem (see [7]). De ine U( , xA, xB)
as he se o con ol alues whe e he maximum o equa ion
5 is a ained when xAand xBa e he alues o he s a e o
ehicles A and B a ime . In he in e io o R we can use any
easible con ol. On he bounda y o R he con ol selec ion
is es ic ed o he se - alued map U( , xA, xB).
VII. CONCLUSIONS
In his pape we p opose a speci ica ion, planning, and
con ol syn hesis amewo k o ne wo ked ehicles sys ems.
We use he language o se heo y o uni o mly ep esen
ehicles, pa e ns o in e ac ions among hese ehicles, and
he beha io o o dina y di e en ial equa ions – such as he
ones desc ibing he mo ions o a ehicle – and echniques om
dynamic op imiza ion o he se - alued ep esen a ion o his
beha io and o con ol syn hesis unde se - alued cons ain s.
The calcula ion o he alue unc ion is no a i ial ma e .
We a e in es iga ing compu a ional me hods o do his.
ACKNOWLEDGMENT
The au ho s hank P o esso s P a in Va aiya and Alexan-
de Ku zhanski o s imula ing discussions on dynamic op i-
miza ion and each se compu a ion, D . Raja Sengup a o
he mo i a ion o he UAV example, and Tenen e Co onel
An ´
onio Cos a and Capi ˜
ao Del im Do es om he Po uguese
Ai Fo ce o he discussions on he ANTEX-M p ojec .
Jo˜
ao Bo ges de Sousa and Fe nando Pe ei a ha e been
suppo ed by Fundac¸˜
ao da Ciˆ
encia e Tecnologia unde p ojec
Co dyal.
REFERENCES
[1] Jean-Pie e Aubin. Viabili y heo y. Bi khause , 1991.
[2] Jean-Pie e Aubin and Helene F ankowska. Se - alued analysis.
Bi khause , 1990.
[3] J. Bo ges de Sousa, An´
ıbal C. Ma os, and F. Lobo Pe ei a. Dynamic
op imiza ion in he coo dina ion and con ol o au onomous unde wa e
ehicles. In P oceedings o Decision and Con ol Con e ence. IEEE,
2002.
[4] J. Bo ges de Sousa and F. Lobo Pe ei a. Speci ica ion and design
o coo dina ed mo ions o au onomous ehicles. In P oceedings o
Decision and Con ol Con e ence. IEEE, 2002.
[5] J. Bo ges de Sousa and Raja Sengup a. Tu o ial on au onomous and
semi-au onomous ne wo ked mul i- ehicle sys ems, decision and con ol
con e ence. Decembe 2001.
[6] L. C. E ans. Pa ial Di e en ial Equa ions. G adua e S udies in
Ma hema ics. Ame ican Ma hema ical Socie y, 1998.
[7] N.N. K aso skii and A.I. Subbo in. Game- heo e ical con ol p oblems.
Sp inge -Ve lag, 1988.
[8] A. B. Ku zhanskii. Ad ances in nonlinea dynamics and con ol : a
epo om Russia. Bi khause , 1993.
[9] A. B. Ku zhanskii. Ellipsoidal calculus o es ima ion and con ol.
Bi khause , 1997.
[10] A. B. Ku zhanskii and P. Va aiya. Op imiza ion me hods o a ge
p oblems o con ol. In P oceedings o Ma hema ical Theo y o
Ne wo ks and Sys ems Con e ence, 2002.
[11] A. Suleman, P. A. Moniz, and A. P. Cos a. Design and es ing o adap i e
p ae olas ic demons a o . In P oceedings o he AIAA, pages 1361–71,
2001.