jou nal o op imiza ion heo y and applica ions: Vol. 124, No. 1, pp. 55–77, Janua y 2005 (© 2005)
Necessa y Condi ions o Impulsi e Nonlinea
Op imal Con ol P oblems wi hou a p io i
No mali y Assump ions1
A. A u yuno ,2V. Dykh a,3and F. Lobo Pe ei a4
Communica ed by B. Polyak
Abs ac . Fi s -o de and second-o de necessa y condi ions o op i-
mali y o an impulsi e con ol p oblem ha emain in o ma i e
o abno mal con ol p ocesses a e p esen ed and de i ed. One o
he main ea u es o hese condi ions is ha no a p io i no mal-
i y assump ions a e equi ed. This ea u e ollows om he ac ha
hese condi ions ely on an ex emal p inciple which is p o ed o
an abs ac minimiza ion p oblem wi h equali y cons ain s, inequali y
cons ain s, and cons ain s gi en by an inclusion in a con ex cone.
Two simple examples illus a e he powe o he main esul .
Key Wo ds. Op imal impulsi e con ol, ex emal p inciple, second-
o de op imali y condi ions, abno mali y.
1. In oduc ion
Le us conside he ollowing fixed- ime op imal con ol p oblem:
(A) min J(x
0,u,w)=L0(a), (1)
s. . dx( ) = ( , x( ), u( ))d +G( , x( ))dw( ), ∈[ 0,
1],(2)
1The fi s au ho was pa ially suppo ed by he Russian Founda ion o Basic Resea ch
G an 02-01-00334. The second au ho was pa ially suppo ed by he Russian Founda-
ion o Basic Resea ch G an 00-01-00869. The hi d au ho was pa ially suppo ed by
Fundacao pa a a Ciencia e Tecnologia and by INVOTAN G an .
2P o esso , Depa men o Di e en ial Equa ions and Func ional Analysis, Peoples F iend-
ship Uni e si y o Russia, Moscow, Russia.
3P o esso , Depa men o Ma hema ics, Baikal S a e Uni e si y o Economy and Law,
I ku sk, Russia.
4Associa e P o esso , Depa men o Elec o echnical and Compu e Enginee ing, Facul y
o Enginee ing, Uni e si y o Po o, Po o, Po ugal.
55
0022-3239/05/0100-0055/0 © 2005 Sp inge Science+Business Media, Inc.
56 JOTA: VOL. 124, NO. 1, JANUARY 2005
L1(a) ≤0,L
2(a) =0,(3)
dw∈K.(4)
He e,
a=(x( 0), x( 1)), x( 0)=x( −
0)=x0,x(
1)=x1,
0<
1
a e gi en. The mappings
:[ 0,
1]×Rn×Rm→Rn,G:[ 0,
1]×Rn→Rn×k,
Li:Rn×Rn→Rd(Li),i=0,1,2,
a e gi en, wi h d(Li) he dimension o he ec o unc ion Li,d(L
0)=1,
and dw is a k-dimensional Bo el measu e associa ed wi h he unc ion o
bounded a ia ion w( ), igh con inuous on ( 0,
1].The cone Kis defined
by
K={dw∈C∗([ 0,
1];Rk):∀con inuous φsuch ha
φ( )∈K0∀ ,
B
φ( )dw≥0,∀Bo el B⊂[ 0,
1]},
whe e Kis a gi en con ex, closed, poin ed cone om Rkand K0is i s
dual. In ano he wo ds, he measu e dw sa isfies
B
dw( ) ∈K, o all Bo el subse s B.
The pai (u, w) is called an admissible con ol i u∈Lm
∞and w∈BVkis
such ha dw∈K.
Le us desc ibe ou assump ions o p oblem A:
(H1) The unc ions L0,L
1,L
2a e C2.
(H2) The unc ion is wice di e en iable w. . . xand u o almos
all ∈[ 0,
1]; he unc ion plus he fi s -o de and second-
o de de i a i es a e measu able w. . . and bounded on any
bounded subse .
(H3) The ma ix unc ion G∈C2.
(H4) The ma ix Gsa isfies he F obenius condi ion, i.e.,
Gi
x( , x)Gj( , x) −Gj
x( , x)Gi( , x) ≡0,(5)
whe e Giis he i h column o G.
JOTA: VOL. 124, NO. 1, JANUARY 2005 57
No ice ha , unde (H4), he dynamic sys em (2) is obus w. . . app ox-
ima ions o he gene alized con ol dw by con en ional con ols (·)∈
Lk
∞([ 0,
1];K); see Re s. 1–4. I he F obenius condi ion holds, hen o
any gi en admissible con ol (u, w) and ini ial condi ion x0, he co e-
sponding ajec o y (whose exis ence is assumed) is he unique igh -con-
inuous unc ion o bounded a ia ion on ( 0,
1],wi h x( 0)=x0such ha
x( )=x0+
0
(θ, x(θ), u(θ))dθ
+[ 0, ]
G(θ, x(θ))dwc(θ) +
Si≤
(z(1;si,ci)−x(s−
i)). (6)
He e, dwc ep esen s he con inuous pa o dw,
dwa( ) :=ciδSi
is he a omic pa , si∈[ 0,
1] a e he jump imes o dw ( imes o impulses),
δSis he Di ac measu e a ime s,ci∈Ka e he jumps o dw, and he
unc ion zi(τ) =z(τ;si,c
i)is he solu ion o he limi ing sys em
dzi/dτ =G(si,z
i)ci,z
i(0)=x(S−
i);(7)
hence,
zi(1)=x(s+
i).
The obus ness o he sys em (2), due o (H4), implies ha he solu ion
(6) belongs o he closu e o he se o absolu ely con inuous solu ions o
equa ion (2) co esponding o (u, w) ∈L∞×AC.
An admissible con ol p ocess is a iple (x0,u,w), whe e (u, w) is an
admissible con ol and he co esponding s a e ajec o y sa isfies he gi en
endpoin cons ain s. The p oblem unde conside a ion is o minimize J
o e he se o admissible con ol p ocesses.
By (x∗
0,u
∗,w∗)and x∗, we deno e espec i ely an admissible con ol
p ocess and he co esponding s a e ajec o y in es iga ed o a minimum
o p oblem (A). I is assumed ha his con ol p ocess sa isfies he ollow-
ing addi ional assump ion:
(H5) dw∗( ) = ∗( )d +
s∈S∗
csδs( ), (8)
58 JOTA: VOL. 124, NO. 1, JANUARY 2005
whe e ν∗( ) =˙w∗( ) a.e. wi h espec o he Lebesgue measu e on [ 0,
1]5,
S∗⊂[ 0,
1] is he se o jump imes o w∗(·), assumed o be fini e, and
cs=[w∗(s)]:=w∗(s+)−w∗(s−),
i.e., he unc ion w∗(·)has no singula con inuous pa and has a fini e
numbe o jump imes.
Mo eo e , since (x∗
0,u
∗,w∗)is in es iga ed o a local minimum only
(in he sense o Defini ion 1.1 below), hen wi hou loss o gene ali y
we can assume ha all endpoin inequali y cons ain s a e ac i e a he
op imal ajec o y x∗,i.e.,
L1(a∗)=0,whe e a∗=(x∗( 0), x∗( 1)). (9)
Dynamic op imiza ion p oblems a ising in a a ie y o applica ion a eas
such as finance, mechanics, esou ces managemen , and space na iga ion
(see Re s. 4–10, jus o men ion a small bu ep esen a i e sample o e e -
ences), whose solu ions migh in ol e discon inuous ajec o ies, ha e been
conside ed o e he yea s, mo i a ing a significan esea ch e o on he
impulsi e con ol p oblem.
In o de o no obscu e he aim o his a icle, we selec ed he sim-
ples con ol p oblem pa adigm enabling us o deal wi h he issues ele an
o fi s -o de and second-o de condi ions o impulsi e con ol p oblems
ha emain in o ma i e, e en o abno mal con ol p ocesses. I is no
di ficul o see ha his esul can be de i ed o a numbe o di e en
and mo e complex con ol o mula ions. In pa icula , by s anda d s a e-
a iable manipula ions, one can con e Bolza and Lag ange ypes o cos
unc ionals in o he one s a ed he e.
The app oach o his a icle can be used o de i e hese op i-
mali y condi ions o p oblems wi h egula con ol cons ian s o he
ype R(u, ) =0. Unde egula i y assump ions (see Re . 11), he implici
unc ion heo em can be used o sol e ( o each ) his equa ion in u, hus
con e ing he con ol p oblem in o he one conside ed he e.
Defini ion 1.1. We say ha he admissible p ocess (x∗
0,u
∗,w∗)is a
local minimize o he p oblem (A) i ∃ε>0 and, o any fini e-dimen-
sional subspace R⊂Lm
∞[ 0,
1],∃εR>0 such ha p ocess (x∗
0,u
∗,w∗)yields
he minimum o p oblem (1)–(4) wi h he addi ional cons ain s
a−a∗<ε, dw−dw∗C∗([ 0, 1];Rk)<ε,
u−u∗Lm
∞[ 0, 1]<ε
R,u(·)∈R.
5He e o o e, L-a.e. deno es a.e. w. . . he Lebesgue measu e.
JOTA: VOL. 124, NO. 1, JANUARY 2005 59
The defined ype o local minimum is fini e dimensional in uand
weak in dw.
In his a icle, we ob ain fi s -o de and second-o de necessa y condi-
ions o op imali y o he p oblem unde conside a ion. The main ea u es
o he esul s is ha no a p io i no mali y assump ions a e equi ed and
ha hey a e in o ma i e o abno mal con ol p ocesses as well. Ano he
issue conce ns he ac ha , in he p oblem conside ed, he unc ion G
depends also on x. The p oo o hese condi ions is based on a nonlin-
ea ans o ma ion o he ini ial p oblem A (Re . 4) in o ano he one o
which Gdoes no depend on xand fi s -o de and second-o de necessa y
condi ions o op imali y we e de i ed in Re . 12.
In spi e o he well-de eloped heo y o highe -o de necessa y con-
di ions o op imali y o con en ional op imal con ol p oblems (see o
example, Re s. 11, 12, 14), i is somewha su p ising ha , om he as
amoun o li e a u e add essing op imal impulsi e con ol p oblems (Re s.
1–3, 15–22), only a ew publica ions a e a ailable (Re s. 23–25, 27).
We no ice ha , while he condi ions in Re s. 23, 24 become i ial
(i.e., degene a e, o abno mal p oblems), ou s emain in o ma i e. Also,
ou esul s di e subs an ially om hese condi ions as i can be seen
om he ac ha hese ollow di ec ly om he maximum p inciple in
he case he op imal ajec o y is absolu ely con inuous, i.e., wi h no
impulses.
In Re . 3, second-o de necessa y condi ions o op imali y o he
Legend e-Jacobi-Mo se ype o ime-op imal con ol a e de i ed by using
in an essen ial way an ex emal p inciple and he no ion o index o
quasiex emali y p o ided in Re . 26.
Howe e , he app oach ollowed he e di e s subs an ially om all he
ones in he e e ences ci ed abo e as we ega d his p oblem as a specific
ins ance o a gene al abs ac p oblem o which powe ul second-o de
op imali y condi ions a e de i ed.
This a icle is o ganized as ollows. In Sec ion 2, we in oduce key
defini ions and s a e fi s -o de and second-o de necessa y condi ions o
op imali y o he dynamic op imiza ion p oblem desc ibed in Sec ion 1.
Issues conce ning abno mali y, geome ic in e p e a ion, and compu a ion
a e also discussed. In Sec ion 3, we p esen he p oo , which is o ga-
nized in h ee pa s: ans o ma ion o he gi en p oblem in o ano he
one o which he e a e fi s -o de and second-o de necessa y condi ions
o op imali y a ailable; s a emen o he men ioned op imali y condi ions
o he p oblem conside ed; and decoding o he hus ob ained fi s -o de
and o second-o de condi ions in e ms o he da a o he o iginal p ob-
lem. Finally, in Sec ion 4, wo examples illus a e he applica ion o hese
condi ions.

60 JOTA: VOL. 124, NO. 1, JANUARY 2005
2. Second-O de Necessa y Condi ions o Op imali y
Le us s a e he necessa y condi ions o op imali y o p oblem A.
Be o e p esen ing he main esul , we discuss some auxilia y concep s
which a e undamen al o he s a emen o ou main esul : local maxi-
mum p inciple, c i ical cone, and quad a ic o m.
2.1. Local Maximum P inciple. Le
F( ,x,u, )= ( ,x,u)+G( , x)
and le
ψ∈Rn,λ=(λ0,λ
1,λ
2)∈R1×Rd(L1)×Rd(L2).
Define he Pon yagin unc ion H=H0+H1and he endpoin Lag angian
lλby
H0( ,x,ψ,u)=ψ, ( , x, u),
H1( ,x,ψ, )=ψ, G( , x) ,
lλ(a) =λ0L0(a) +λ1,L
1(a)+λ2,L
2(a).
Defini ion 2.1. We say ha a p ocess (x∗
0,u
∗,w∗)sa isfies he Eule -
Lag ange condi ions o he local maximum p inciple i he e exis s λ=0
such ha
λ0≥0,λ
1≥0,λ1,L
1(a∗)=0 (10)
and he ec o unc ion ψ, solu ion o he adjoin sys em
−dψ( ) =H0x( )d +Hx ( )dw∗( ), −ψ( 1)=lλ
x1(a∗), (11)
which sa is y he ollowing condi ions:
ψ( 0)=lλ
x0(a∗), (12)
Hu( ) =0,L-a.e.,(13)
H ( ), ≤0,∀( , ) ∈[ 0,
1]×K, (14)
H ( ), ¯ω∗( )=0,dw
∗-a.e.,(15)
whe e
¯ω∗( ) =dw∗( )/d|w∗( )|
JOTA: VOL. 124, NO. 1, JANUARY 2005 61
is he Radon-Nicodym de i a i e o he measu e dw∗wi h espec o i s
o al a ia ion measu e.
No ice ha he solu ion o he adjoin sys em (11) is in he same
sense as he one o (6), i.e.,
ψ( 1)=−lλ
x1(a∗)
and ha
ψ( )=−lλ
x1(a∗)+ 1
H0x(θ)dθ + 1
Hx (θ)dw∗
c(θ)
+
si>
(ψ(si)−q(0;si,ci)), ∈[ 0,
1). (16)
He e, he unc ions qi(τ)=qi(τ ;si,ci)a e solu ions o he adjoin limi ing
sys em
−dqi/dτ =Hx (si,z
i(τ), qi(τ))ci,q
i(1)=ψ(si), (17)
wi h he co esponding solu ion zi(τ) o he sys em (7) when x(s−
i)=
x∗(s−
i). The no a ion
Hx ( ) =∂2H
∂ ∂x( )
e e s o he e alua ion o he unc ion Hx along he p ocess examined
( his no a ion is adop ed also o o he unc ions in simila con ex s). We
ema k ha any adjoin ajec o y ψ( ) and he unc ion H( ) depend on
λdue o he ans e sali y condi ion (12).
Deno e by
=(x∗
0,u
∗,w∗)
he se o all no malized Lag ange mul iplie s λ,λ=1, sa is ying he
local maximum p inciple. I is well known ha =∅ is a fi s -o de nec-
essa y condi ion o a weak local minimum o p oblem A. Howe e , we
shall p o e he e ha i is also necessa y o he local minimum in he sense
o Defini ion 1.1. No e ha he local maximum p inciple holds wi hou
(H5).
62 JOTA: VOL. 124, NO. 1, JANUARY 2005
2.2. C i ical Cone. In o de o ensu e a compac s a emen o he
second-o de condi ions, we shall use he o al de i a i e w. . . ime along
he solu ion o he ollowing o dina y di e en ial sys em:
˙x=F ( , x, u, ), (18a)
−˙
ψ=Hx( ,x,ψ,u, ), (18b)
˙w= , ( )∈K. (18c)
Fo example,
(˙
H )x=(∂/∂x)[(d/d )(∂H/∂ )]| ,x∗( ),u∗( ),w∗( )).
Unde he F obenius condi ion, his de i a i e does no depend on ,bu
in any o he case, we pu always ∗( )=˙w∗( ); see (8). Deno e by BVn(S∗)
he se o n-dimensional ec o unc ions o bounded a ia ion whose
jump imes a e suppo ed on S∗. Clea ly, each e m in (x∗(·), ψ(·), w∗(·))
isinaBV(S∗)space o he co esponding dimension.
Defini ion 2.2. A a ia ion (δx0, δu, δw) ∈Rn×Lm
∞×BVk(S∗)is
called c i ical i he co esponding s a e ajec o y a ia ion δx ∈BVn(S∗)
sa isfies he ollowing condi ions:
Lia(a∗), δa+Lix1(a∗), G( 1)δw1≤0,i=0,1,
=0,i=2,(19)
δa =(δx( 0), δx( 1)), δw1=δw( 1), (20)
d(δx)/d =Fx( )δx +Fu( )δu −(˙
H )T
ψ( )δw, /∈S∗,(21)
d(δw)∈K+Lin {dw∗},δw(
0)=0,(22)
δx(s) =δq(1;s,c), ∀s∈S∗.(23)
He e,
G( 1)=G( 1,x∗( 1)),
δq(τ;s,c):=δqs(τ ) is he solu ion o he sys em
d(δqs)/dτ =H1ψx(s, zs(τ), c)δqs,(24a)
δq 0(0)=δx0,(24b)
δqs(0)=δx(s−), s > 0,(24c)
and he unc ion zs(τ) is solu ion o (7) when si=s,x(s−)=x∗(s−); ecall
ha c=[w∗(s)].
Deno e by Kc he cone o all c i ical a ia ions.
JOTA: VOL. 124, NO. 1, JANUARY 2005 63
2.3. Quad a ic Fo m. Fo any λ∈, define he quad a ic o m
λ(δx0, δu, δw) =δaTlλ
aa(a∗)δa +Qλ
1(δa, δw1)
− 1
0
Qλ(δx, δu, δw)( )d , (25)
whe e Qλand Qλ
1a e he ollowing quad a ic o ms:
Qλ(δx, δu, δw) =δuTHλ
uuδu+2δxTHλ
xuδu−2δwT(˙
Hλ
)uδu
−δwT(¨
Hλ
) δw−2δwT(˙
Hλ
)xδx +δxTHλ
xxδx, (26)
Qλ
1(δx(·), δw1)=2[δx( 0)Tlλ
x0x1(a∗)G( 1)−δx( 1)THλ
x ( 1)]δw1
+δwT
1GT( 1)[lλ
x1x1(a∗)G( 1)−Hλ
x ( 1)]δw1
−
s∈S∗
[δxT(s)λ(s)δx(s) −δxT(s−)λ(s−)δx(s−)].
(27)
He e, dependence on op Qλis omi ed, Hλ e e s o he Pon yagin
unc ion e alua ed along ( , x∗,ψ,u
∗, ∗), ψ sa isfied (11) o a ce ain λ,
and δx(·)is he co esponding solu ion o (24) wi h (22), (23), δx( −
0)=
δx0,w( −
0)=0. Only λ( ) ∈BVn×n(S∗) emains o be defined in o mula
(27). Fo his, le z∗(τ; ) and q∗(τ; ) be solu ions o
dz∗/dτ =G( , z∗)w∗( ), z∗(1; )=x∗( ),
−dq∗/dτ =H1x( , z∗,q∗,w∗( )), q∗(1; )=ψ( ),
and le he n×nma ix Z(τ; ) sa is y
−dZ/dτ =ZH1ψx( , z∗(τ, ), w∗( )), Z(0; )=I.
Then,
λ( ) =−ZT(1; )1
0
Z−1T(τ; )H1xx(τ; )Z−1(τ ; )dτZ(1; ), (28)
whe e he exp ession H1xx(τ; ) is a sho no a ion o H1xx e alu-
a ed along ( , z∗(τ; ),q∗(τ; ),w∗( )). We ema k ha ( −
0)=0 because
w( −
0)=0.
70 JOTA: VOL. 124, NO. 1, JANUARY 2005
whe e δy(·)is he co esponding solu ion o (47). Pu
˜
Kπ=Ke ˜
A.
P oposi ion 3.2. We ha e ha ˜
d=codim (Im ˜
A)and also µ∈Mais
equi alen o he ac ha he index o he o m ˜
µ
aconside ed on he lin-
ea subspace ˜
Kπis no g ea e han ˜
d.
This p oposi ion ollows om he ac ha BVkis dense in Lk
∞wi h
espec o he Lk
2me ic and i s p oo is based on s anda d Lebesgue in e-
g a ion a gumen s.
S ep 3. Decoding o he Local Maximum P inciple. F om he defini-
ion o ξand η, i ollows ha ηsa isfies he pa ial di e en ial sys em
ηw( ,x,w)+ηx( , x, w)G( , x) =0,(52)
wi h bounda y condi ion η( ,x,0)=x. Mo eo e , he ollowing equali ies
hold:
η( ,ξ( ,y,w),w)=y, ξ( ,η( ,x,w),w)=x. (53)
We shall use hese ela ions and hei consequences ob ained by di e en-
ia ion.
Le us conside he ollowing p oposi ion.
P oposi ion 3.3. The unc ion σ( ,y,p,w) =ηT
x( , ξ( , y, w), w)p is a
solu ion o he ollowing comple ely in eg able sys em:
σw=−Hx ( , ξ( , y, w), σ, u, ), σ |w=0=p. (54)
P oo . Ob iously, σsa isfies he ini ial condi ion. Deno e he le -
hand side o (52) by F( ,x,w). By di e en ia ing σwi h espec o w, and
using Fx≡0, we ob ain
σw( ,y,p,w)=ηT
xx( , ξ( , y, w), w)GT( , ξ( , y, w))p
+ηwx( , ξ( , y, w), w)p
=Fx( , ξ( , y, w), w)p
−GT
x( , ξ( , y, w))ηT
x( , ξ( , y, w), w)p
=−Hx ( , ξ( , y, w), σ ( , y, p, w)).
This p o es P oposi ion 3.3.

JOTA: VOL. 124, NO. 1, JANUARY 2005 71
P oposi ion 3.4. Le
s( ,x,ψ,w)=ξT
y( , ηx( ,x,w),w)ψ.
Then, he unc ion
p( )=s( ,x∗( ), ψ( ), w∗( )) (55)
is a solu ion o (36) i and only i he unc ion
ψ( )=σ( ,y∗( ), p( ), w∗( )) (56)
is solu ion o he sys em (11).
P oo . Any solu ion (x(·), ψ(·)) o he sys ems (2), (11) can be
app oxima ed in he weak s a opology o he space o unc ions o
bounded a ia ion by solu ions o he sys em (18). Hence, i is su ficien
o p o e ha he ela ions (55) and (56) hold o any absolu ely con in-
uous ajec o y o he sys em (18). Fo his, i su fices o p o e ha he
o mula
(d/d )σ ( , y, p, w) =−Hx( ,ξ( ,y,w),σ( ,y,w),u, ) (57)
holds, wi h
(˙y, ˙p, ˙w)=(g, −˜
Hy, ).
We ha e
(d/d )σ ( , y, p, w) =−ηT
x( ,ξ( ,y,w),w) ˜
Hy( ,y,p,u,w)
+(d/d )ηT
x( ,ξ( ,y,w),w)p. (58)
No e ha he unc ion ˜
Hmay be ep esen ed in he o m
˜
H( ,y,p,u,w)=H0( ,ξ( ,y,w),σ( ,y,p,w),u)
+p,η ( ,ξ( ,y,w),w).
By di e en ia ing w. . . y, we ob ain
˜
Hy( ,y,p,u,w)=ξT
y( ,y,w)[H0x( ,ξ( ,y,w),σ( ,y,p,w),u)
+[ηT
x( , ξ( , y, w), w)p]x ( ,ξ( ,y,w),u)
+η x( , ξ( , y, w), w)p].
72 JOTA: VOL. 124, NO. 1, JANUARY 2005
By subs i u ing his equali y in (58), we ob ain
(d/d )σ ( , y, p, w) =−H0x( ,ξ( ,y,w),σ( ,y,p,w),u)
−[ηT
x( , ξ( , y, w), w)p]x ( ,ξ( ,y,w),u)
−ηT
x( , ξ( , y, w), w)p
+(d/d )ηT
x( ,ξ( ,y,w),w)p. (59)
By conside ing he iden i y Fx( ,x,w)≡0 and using i in he las e m o
(59), we ob ain
(d/d )[ηT
x( , ξ( , y, w), w)p]x
=[ηT
x( , ξ( , y, w), w)p] +[ηT
x( , ξ( , y, w), w)p]x ( ,ξ( ,y,w),u)
−Hx ( ,ξ( ,y,w),ζ( ,y,p,w),u, ) .
By subs i u ing in (59), we ob ain (57). This p o es P oposi ion 3.4.
In a simila way (see he de ails in Re s. 4, 18, 23), we can ob ain he
impo an equali y
−˙pw=˜
Hw( ,y,p,u,w)=−(d/d )H ( , ξ( , y, w), σ ( , y, p, w)). (60)
Le us conside he ans e sali y condi ions o he adjoin sys ems. We
ha e
p( 0)=˜
Ly0(b∗)=ξT
y( 0,y∗( 0), 0)Lx0(a∗)=Lx0(a∗),
−p( 1)=˜
Ly1(b∗)=ξT
y( 1,y∗( 1), w∗)Lx1(a∗),
−pw( 1)=˜
Lw1(b∗)=GT( 1,x∗( 1))Lx1(a∗)=H ( 1,x∗
1,ψ(
1)).
F om hese ela ions and P oposi ion 3.4, i ollows ha he se o
adjoin ajec o ies p(·)and ψ(·)can be ob ained om each o he by
coo dina es ans o ma ion. Mo eo e , bea ing in mind he equali y (60),
we ob ain
pw( ) =−H ( ), ∀ ∈[ 0,
1],
since unde (H4) he unc ion →H ( ) ∈AC. Due o his and he ob i-
ous equali y ˜
Hu( ) =Hu( ), we conclude ha he se s and Ma e dis-
inguished only by he no a ion. F om now on, we shall use he common
no a ion λand ins ead o µand M. This conclusion is s a ed as ollows.
JOTA: VOL. 124, NO. 1, JANUARY 2005 73
P oposi ion 3.5. =∅⇔M=∅. Tha is, he condi ions o he local
maximum p inciple in p oblems A and B a e equi alen .
S ep 4. Decoding o he Second-O de Condi ions. We shall p o e
ha he cone ˜
Kc and he o m ˜
λa e ans o med in o Kc and λ
espec i ely by he simple linea mapping o he y- a ia ions
:δx( ) =ξy( )δy( ). (61)
Since his ans o ma ion is in e ible [de ξy( )=0on[ 0,
1] due o (33)],
hen in o de o ob ain he second-o de necessa y condi ions o op imal-
i y o p oblem A om hose o p oblem B, we need o show he ollow-
ing p oposi ion.
P oposi ion 3.6. The ollowing equali ies hold:
◦˜
Kc =Kc ,
◦˜
λ=λ,∀λ∈. (62)
P oo . The linea ela ions (44), (45) speci ying he c i ical cone can
be ans o med easily in o (19). Le us p o e ha , o /∈S∗, he ollowing
o mulas hold:
gy( ) =(ηx( ) x( ) +˙ηx( ))ξy( ), (63)
gw( ) =ηx( )( x( )G( ) −˙
G( )) =−ηx( )( ˙
H )ψ( ), (64)
gu( ) =ηx( ) u( ). (65)
In ac , by using he equali y Fx≡0, we ob ain
gy( ) =[ηxx( ) ( ) + x( )ηx( ) +η x( )]ξy( )
={[Fx( )ηx( ) +(d/d )ηx( )]−Fx( ) ∗( )}ξy( ).
Analogously, by using he addi ional equali y F ≡0, we ob ain (64) and
(65).
F om (63)–(65) and (61), we see ha (43a) is ew i en in he o m
(21). Fo any s∈Sd(w∗), le
z(τ;s)=ξ(s,y∗(s), w∗(s−)+τ[w∗(s)]), τ ∈[0,1],
−q(τ,s)=ξy(s, y∗(s), w∗(s−)+τ[w∗(s)])δy(s), τ ∈[0,1].
I is easy o check ha hese unc ions sa is y equa ions (7), (24) and
desc ibe he jump condi ions o he a ia ion δx in (23) and (24). The
p oo o he second equali y in (64) is analogous o he co esponding one
in Re . 23; he e o e, i is omi ed.
74 JOTA: VOL. 124, NO. 1, JANUARY 2005
F om P oposi ions 3.2 and 3.6, i ollows ha condi ion (31) is equi -
alen o condi ion (50) o p oblem (B); hence, (31) is ob ained. Theo em
3.1 is p o ed.
4. Examples
Example 4.1. Take n≥5,k=n−1,x=col(x1,... ,x
n)∈Rn, and le Q
be a symme ic k×kma ix such ha he index o each o he ma ices Q
and −Qis no less han 2. The case k=4 and Q=diag(1,1,−1,−1)is a
good example. Conside he p oblem
min J=ζ,(x1(1),... ,x
k(1)),
s. . dxi= i(x, )d +dwi,i=1,k, w=col(w1,... ,w
k),
dxn= n(x, )d +Qcol(x1,... ,x
k), dw,
∈[0,1],x(0)=0,x
n(1)=0,K=Rk,
whe e ζ∈Rkis a gi en nonze o ec o ; o i=l,... ,n, le he ibe a bi-
a ily gi en smoo h unc ions such ha
i(0, )≡0,
ix(0, )≡0,
nxx(0, )≡0.
Because o he symme y o Q, i can be shown easily ha he F obenius
condi ion holds. We in es iga e he admissible con ol p ocess (0,0,0) and
p o e ha i is no a locally op imal con ol p ocess.
Fix any λ∈. F om (14), o ψ(·)=ψλ(·)=(ψ1(·),...ψ
n(·)),we
ob ain ψi( ) ≡0,i=l,... ,k, and om (11), we ha e ψn( ) ≡ψn,0=cons .
Hence, by using (11), (12), and ζ=0, we ob ain
={λ:λ0=0,λ
2,i =0,i=1,n−1,λ
2,n =−λ2,n+1};
consequen ly, consis s o only wo ec o s,
¯
¯
λ=−¯
λ, ¯
λ=(1/√2)(0,... ,0,1,−1)and ψn,0=±1/√2.
I can be shown easily ha
d=1,
λ
a(δw) =ψn,01
0Qcol(δx1,... ,δx
k), δwd .
Hence,
λ
a(δw) =(1/2)ψn,0Qcol(δx1(1),... ,δx
k(1)), col(δx1(1),... ,δx
k(1)).
JOTA: VOL. 124, NO. 1, JANUARY 2005 75
This implies ha , o any ψn,0=±1, he index o he unc ion λ
ais no
less han 2. So, a=∅; consequen ly, he p ocess (0,0,0) is no op imal.
Also no ice ha his p ocess is abno mal, ha
max
λ∈λ(δw) ≥0,∀δw,
because o ¯
λ, ¯
¯
λ∈, and ha he las inequali y is no use ul.
Example 4.2. Conside he ollowing op imal con ol p oblem wi h
pa ame e s α1,α
2.
min x3(1),
s. . dx1=x2d , x1(0)=0,
dx2=dw, x2(0)=x20 <0,
dx3=(α1x1+α2x2)dw, x3(0)=0,
dw≥0.
The con ol unc ion
w∗( ) =−x20,∀ ∈(0,1],
wi h w∗(0)=0 sa isfies he maximum p inciple o any pa ame e alues.
The co esponding ajec o ies and adjoin a iables a e, espec i ely,
(x∗
1( ), x∗
2( )) =(0,0), ∀ ∈(0,1],
wi h
(x∗
1(0), x∗
2(0)) =(0,x
20),
and
(ψ1( ), ψ2( ) =(0,0), ∀ ∈(0,1],
wi h
(ψ1(0), ψ2(0)) =x20(α1,α
2)and ψ3≡−1.
The c i ical cone o he con ol is desc ibed by he condi ions
δ˙x1=δw, δx1(0)=δx1(0+)=0,
δ˙x2=0,δx
2(0)=0,
d(δw)∈C∗([0,1],R+)+γδ
0,γ∈R.

76 JOTA: VOL. 124, NO. 1, JANUARY 2005
Since H1ψx ≡0,δx
1(·), and δx2(·)a e con inuous; hence, δx2≡0. The
ac ha H1xx ≡0 implies ha ψ≡0. The e o e, he o m is gi en by
(δw)=2α1δx1(1)δw1+α2δw2
1−2α11
0
δw(δw +δx1)d .
The necessa y condi ions o Theo em 2.3 amoun o he inequal-
i y ≥0onkc ; consequen ly, he unc ion δw∗≡0 has o minimize 
on Kc . F om he maximum p inciple, we ha e ha α1≤0. No ice ha
i su fices o conside a needle-shaped a ia ion δw concen a ed a a le
semineighbo hood o poin =1.
I α1<0 and α2>0, hen he con ol w∗is globally op imal.
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