Singular solutions in optimal control: second order conditions and a shooting algorithm

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by  M. Soledad Aronna

SINGULAR SOLUTIONS IN OPTIMAL CONTROL:
SECOND ORDER CONDITIONS AND
A SHOOTING ALGORITHM1,2
2
1
0
M. SOLEDADARONNA
2

t
c
O
8 Abstract. In this article we study optimal control problems for sys-
2 tems that are affine in one part of the control variable. Finitely many
equalityandinequalityconstraintsontheinitialandfinalstatearecon-

] sidered. We investigate singular solutions for this class of problems.
C
First,weobtainsecondordernecessaryandsufficientconditionsforweak
O optimality. Afterwards,weproposeashootingalgorithm andshowthat
. the sufficient condition above-mentioned is also sufficient for the local
h
quadratic convergence of the algorithm.
t
a
m
1. Introduction
[
The purpose of this paper is to investigate optimal control problems gov-

1
erned by ordinary differential equations that are affine in one part of the
v
5 control variable. This class of system includes both the totally affine and
2 the nonlinear cases.
4
Many models that correspond to this framework can be found in practice
7
and, in particular, in the existing literature. Among these we can mention:
.
0
the Goddard’s problem [25] in 3 dimensions analyzed in Martinon et al.
1
2 [11], other models concerning the motion of a rocket in Lawden [38], Bell
1 and Jacobson [9], Goh [27, 31], Oberle [46], Azimov [8] and Hull [33]; an
:
v hydrothermalelectricity productionproblemstudied inBortolossi etal. [13]
i andAronnaetal. [6],theproblemofatmosphericflightconsideredbyOberle
X
in [48], and an optimal production process in Cho et al. [17] and Maurer at
r
a al. [41].
Thesubjectofsecondorderoptimality conditionsforthesepartiallyaffine
problems have been studied by Goh in [28, 29, 27, 31], Dmitruk in [21],
Dmitruk and Shishov in [22], Bernstein and Zeidan [10], and Maurer and
Osmolovskii [42]. The first works were by Goh, who introduced a change of
variables in [28] and used it to obtain optimality conditions in [28, 26, 27],
always assuming uniqueness of the multiplier. The necessary conditions we
presentimply those by Goh [26] whenthereis only onemultiplier. Recently,
Keywordsandphrases. optimalcontrol,singularcontrol,secondorderoptimalitycon-
dition, weak optimality, shooting algorithm, Gauss-Newton method.
∗
This article was published as theINRIAResearch Report Nr. 7764.
†ThisworkissupportedbytheEuropeanUnionunderthe7thFrameworkProgramme
FP7-PEOPLE-2010-ITN Grant agreement number264735-SADCO.
1
2 M.S.ARONNA
Dmitruk and Shishov [22] analysed the quadratic functional associated with
the second variation of the Lagrangian function and provided a set of nec-
essary conditions for the nonnegativity of this quadratic functional. Their
results are consequence of a second order necessary condition we present. In
[21]Dmitrukproposed,withoutproof,necessaryandsufficientconditionsfor
a problem having a particular structure: the affine control variable applies
to a term depending only on the state variable, i.e. the affine and nonlinear
controls are ‘uncoupled’. This hypothesis is not used in our work. The con-
ditions established here coincide with those suggested in Dmitruk [21] when
the latter are applicable. In [10], Bernstein and Zeidan derived a Riccati
equation for the singular linear-quadratic regulator, which is a modification
of the classical linear-quadratic regulator where only some components of
thecontrol enter quadratically in the cost function. All of these four articles
use Goh’s Transformation to derive their conditions; we use this transfor-
mation as well. On the other hand, in [42] Maurer and Osmolovskii gave
a sufficient condition for a class of problems having one affine control sub-
ject to bounds and such that it is bang-bang at the optimal solution. This
structure is not studied here since no control constraints are considered, i.e.
our optimal control is suppose to be totally singular.
Regarding second order optimality conditions, we provide a pair of nec-
essary and sufficient conditions for weak optimality of totally singular solu-
tions. Theseconditions are‘nogap’inthesensethatthesufficientcondition
is obtained from the necessary one by strengthening an inequality. We do
not assume uniqueness of multiplier.
Among the applications of the shooting method to the numerical solution
of partially affine problems we can mention the articles Oberle [45, 48] and
Oberle-Taubert [49]. In these articles the authors use a generalization of
the algorithm that Maurer [39] suggested for totally affine systems. These
workspresentinterestingimplementations ofashooting-likemethodtosolve
partiallyaffinecontrolproblemshavingbang-singularorbang-bangsolutions
and, in some cases, running-state constraints are considered. No result on
convergence is given in these articles.
In this paper we propose a shooting algorithm which can be also used to
solve problems with bound on the controls. Our algorithm is an extension
of the method for totally affine problems in Aronna et al. [7]. We give
a theoretical support to this method, by showing that the second order
sufficientconditionabove-mentionedensuresthelocalquadraticconvergence
of the algorithm.
The article is organised as follows. In Section 2 we present the problem,
thebasicdefinitionsandfirstorderconditions. InSection3wegive thetools
for second order analysis and establish a second order necessary condition.
We introduce Goh’s Transformation in Section 4. In Section 5 we show
a new necessary condition, and in Section 6 we give a sufficient one. A
shooting algorithm is proposed in Section 7, and in Section 8 we prove
SINGULAR SOLUTIONS IN OPTIMAL CONTROL 3
thatthesufficientcondition above-mentioned guarantees thelocal quadratic
convergence of the algorithm.
Notations. We denote by h the value of function h at time t if h is a
t
functionthatdependsonlyont,andbyh theithcomponentofhevaluated
i,t
at t.Partial derivatives of afunction hof (t,x) are referred as D h and D h.
t x
When dealing with derivatives of higher order we may use the notation of
type h since it is not ambiguous. By Rk we denote the k dimensional
xx
−
real space, i.e. the space of column real vectors of dimension k; and by
Rk,∗ its corresponding dual space, which consists of k dimensional row real
vectors. By Lp(0,T;Rk) we mean the Lebesgue space−with domain equal to
the interval [0,T] R and with values in Rk. The notation Wq,s(0,T;Rk)
⊂
refers to the Sobolev spaces (see Adams [1] for further details on Sobolev
spaces).
2. Statement of the problem and assumptions
2.1. Statement of the problem. We study the optimal control problem
(P) given by
(1) J := ϕ (x ,x ) min,
0 0 T
→
m
(2) x˙ = v f (x ,u ), a.e. on [0,T],
t i,t i t t
i=0
X
(3) η (x ,x )= 0, for j = 1...,d ,
j 0 T η
(4) ϕ (x ,x ) 0, for i = 1,...,d .
i 0 T ϕ
≤
Here f : Rn+l Rn for i = 0,...,m, ϕ : R2n R for i = 0,...,d ,
i i ϕ
η :R2n R for→j = 1,...,d and we put, in sake o→f simplicity of notation,
j η
→
v 1 which is not a variable. The nonlinear control u belongs to :=
0
L∞≡(0,T;Rl), while by := L∞(0,T;Rm) we denote the space of aUffine
controls v,and := W1V,∞(0,T;Rn)referstothestatespace. Whenneeded,
X
we write w = (x,u,v) for a point in := . The hypothesis below
W X ×U×V
is considered along all the article.
Assumption 2.1. AlldatafunctionshaveLipschitz-continuoussecondderiva-
tives.
A trajectory is an element w that satisfies the state equation (2). If
∈ W
inaddition,constraints(3)and(4)hold,wesaythatwisafeasible trajectory
of problem (P).
Definition 2.2. A feasible trajectory wˆ = (xˆ,uˆ,vˆ) is a weak minimum
∈ W
of (P) if there exists ε > 0 such that the cost function attains at wˆ its
minimum in the set of feasible trajectories w =(x,u,v) satisfying
x xˆ < ε, u uˆ < ε, v vˆ < ε.
∞ ∞ ∞
k − k k − k k − k
4 M.S.ARONNA
In the sequel, we study a nominal feasible trajectory wˆ = (xˆ,uˆ,vˆ) .
∈ W
An element δw is termed feasible variation for wˆ if wˆ+δw is feasible
for (P). Take λ =∈ W(α,β,p) in Rdϕ+1,∗ Rdη,∗ W1,∞(0,T;Rn,∗). Define the
× ×
pre-Hamiltonian function
m
H[λ](x,u,v,t) := p v f (x,u),
t i i
i=0
X
the terminal Lagrangian function ℓ :R2n R by
→
dϕ dη
ℓ[λ](q) := α ϕ (q)+ β η (q),
i i j j
i=0 j=1
X X
and the Lagrangian function
T m
(5) L [λ](w) := ℓ[λ](x ,x )+ p v f (x ,u ) x˙ dt.
0 T t i,t i t t t
Z0 i=0 − !
X
We assume, in sake of simplicity, that whenever some argument of f , H, ℓ,
i
L or their derivatives is omitted, they are evaluated at wˆ. Without loss of
generality we suppose that
(6) ϕ (xˆ ,xˆ ) = 0, for all i= 1,...,d .
i 0 T ϕ
2.2. Lagrange multipliers. We introduce here the concept of multiplier.
The second order conditions that we prove in this article are expressed in
termsofthesecondvariationoftheLagrangianin(5)andthesetofLagrange
multipliers associated with wˆ that we define below.
Definition 2.3. Anelementλ = (α,β,p) Rdϕ+1,∗ Rdη,∗ W1,∞(0,T;Rn,∗)
∈ × ×
is a Lagrange multiplier associated with wˆ if it satisfies the following condi-
tions,
(7) α + β =1,
| | | |
(8) α = (α ,α ,...,α ) 0,
0 1 dϕ ≥
the function p is solution of the costate equation
(9) p˙ = D H[λ](xˆ ,uˆ ,vˆ,t),
t x t t t
−
and it satisfies the transversality conditions
p = D ℓ[λ](xˆ ,xˆ ),
(10) 0 − x0 0 T
p = D ℓ[λ](xˆ ,xˆ ),
T xT 0 T
and the stationarity conditions
D H[λ](xˆ(t),uˆ(t),vˆ(t),t) = 0,
u
(11) a.e. on [0,T],
D H[λ](xˆ(t),uˆ(t),vˆ(t),t) = 0,
v
hold true. Denote by Λ the set of Lagrange multipliers associated with wˆ.
Recall the®following well-known result.
SINGULAR SOLUTIONS IN OPTIMAL CONTROL 5
Theorem 2.4. If wˆ is a weak minimum, the set Λ is non empty and com-
pact.
Proof. RegardingtheexistenceofaLagrangemultiplierthereaderisreferred
to [3, 37], [44, Thm. 2.1]. In order to prove the compactness, observe that p
may be expressed as a linear continuous mapping of (α,β). Thus, since the
normalization (7) holds, Λ is a finite-dimensional compact set. (cid:3)
In view of previous result, note that Λ can be identified with a compact
subset of Rs, where s := d +d +1.
ϕ η
Given (x¯ ,u¯,v¯) Rn , consider the linearized state equation
0
∈ ×U ×V
(12) x¯˙ = A x¯ +E u¯ +B v¯, a.e. on [0,T],
t t t t t t t
(13) x¯(0) = x¯ ,
0
where
m m
(14) A := vˆD f (xˆ,uˆ), E := vˆD f (xˆ,uˆ),
t i x i t i u i
i=0 i=0
X X
and B : [0,T] (R) such that for every v Rm,
n×m
→ M ∈
m
(15) B v := v f (xˆ ,uˆ ).
t i i t t
i=1
X
Here (R) refers to the space of n m real matrices. Hence, the ith.
n×m
M × −
column of B is f (xˆ,uˆ). The solution x¯ of (12)-(13) is called linearized state
i
variable.
2.3. Critical cones. Wedefinenowthesetsofcriticaldirectionsassociated
with wˆ, both in the L∞ and the L2 norm. Even if we are working with
control variables in L∞−and hence th−e control perturbations are naturally
taken in L∞, the second order analysis involves quadratic mappings and it
is useful to extend them continuously to L2.
Set := W1,2(0,T;Rn), := L2(0,T;Rl) and := L2(0,T;Rm). Put
2 2 2
X U V
:= for the corresponding product space. Given w¯
2 2 2 2 2
W X ×U ×V ∈ W
satisfying (12)-(13), consider the linearization of the endpoint constraints
and cost function,
(16) Dη (xˆ ,xˆ )(x¯ ,x¯ ) = 0, for j = 1,...,d ,
j 0 T 0 T η
(17) Dϕ (xˆ ,xˆ )(x¯ ,x¯ ) 0, for i = 0,...,d .
i 0 T 0 T ϕ
≤
Define the critical cones in and by
2
W W
(18) := w¯ :(12)-(13),(16)-(17) hold ,
C { ∈ W }
(19) := w¯ :(12)-(13),(16)-(17) hold .
2 2
C { ∈ W }
Lemma 2.5. The critical cone is a dense subset of .
2
C C
Inordertoprovepreviouslemma, recall thefollowingtechnical result(see
e.g. Dmitruk [20, Lemma 1] for a proof).
6 M.S.ARONNA
Lemma 2.6 (on density of cones). Consider a locally convex topological
space X, a finite-faced cone Z X, and a linear manifold Y dense in X.
⊂
Then the cone Z Y is dense in Z.
∩
Proof. [of Lemma 2.5]
Set X := w¯ : (12)-(13)hold , Y := w¯ : (12)-(13)hold , and
2
Z := and{app∈lyWLemma 2.6. } { ∈ W } (cid:3)
2
C
3. Second order analysis
We begin this section by giving an expression of the second derivative
of the Lagrangian function L , in terms of the derivatives of ℓ and H. We
denote it by Ω. All the second order conditions we present are established
in terms of either Ω or some transformed form of Ω. The main result of the
current section is the necessary condition in Theorem 3.9, which is applied
in Section 5 to get Theorem 5.3.
3.1. Second variation. Let us consider the quadratic mapping
(20)
Ω[λ](x¯,u¯,v¯) := 1D2ℓ[λ](xˆ ,xˆ )(x¯ ,x¯ )2
2 0 T 0 T
T
+ [1x¯⊤Q[λ]x¯+u¯⊤F[λ]x¯+v¯⊤C[λ]x¯+ 1u¯⊤R [λ]u¯+v¯⊤K[λ]u¯]dt,
2 2 0
Z0
where the involved matrices are, omitting arguments,
(21) Q := H , F := H , C := H , R := H , K := H .
xx ux vx 0 uu vu
Recall the following notation: given two functions h : RN RM and k :
RN RL, we say the h is a big-O of k around 0 and denote→it by
→
h(x) = (k(x)),
O
if there exists positive constants δ and M such that h(x) M k(x) for
| | ≤ | |
x < δ. It is a small-o if M goes to 0 as x goes to 0. Denote this by
| | | |
h(x) = o(k(x)).
Lemma 3.1 (Lagrangian expansion). Let w = (x,u,v) be a solution
∈ W
of (2), and set δw = (δx,δu,δv) := w wˆ. Then for every multiplier λ Λ,
− ∈
(22) L [λ](w) = L [λ](wˆ)+Ω[λ](δx,δu,δv)+τ[λ](δx,δu,δv)+ (δx,δu,δv),
R
where the time variable is omitted in the sake of simplicity, τ is a cubic
mapping given by
τ[λ](δx,δu,δv) :=
T
[H [λ](δx,δx,δv)+2H [λ](δx,δu,δv)+H [λ](δu,δu,δv)]dt,
vxx vux vuu
Z0
and satisfies the estimate
R
(δx,δu,δv) = ((δx ,δx )3)+(1+ v ) (δx,δu) ( (δx,δu) 2).
R O | 0 T | k k1 k k∞O k k2
SINGULAR SOLUTIONS IN OPTIMAL CONTROL 7
Proof. Omit the dependence on λ for the sake of simplicity. In order to
achieve the expression (22) consider the second order Taylor representations
below, written in a compact form,
(23) ℓ(x ,x )= ℓ+Dℓ(δx ,δx )+ 1D2ℓ(δx ,δx )2+ ((δx ,δx )3),
0 T 0 T 2 0 T O | 0 T |
(24) f (x ,u ) = f +Df (δx ,δu )+ 1D2f (δx ,δu )2+ ( (δx,δu) 3),
i t t i,t i t t 2 i t t O k k3
where, whenever the argument is missing, the corresponding function is
evaluated on the reference trajectory wˆ. Observe that the transversality
conditions (10) and the costate equation (9) yield
T m
(25) Dℓ(δx ,δx ) = p δx +p δx = p vˆD f δx+δ˙x dt.
0 T 0 0 T T i x i
− Z0 "−i=0 #
X
RecalltheexpressionoftheLagrangiangivenin(5). Replacingℓ(x ,x )and
0 T
f (x,u) in (5) by their Taylor expansions (23)-(24) and using the identity
i
(25) we get
T
L (w) = L (wˆ)+ [H δu+H δv]dt+Ω(δx,δu,δv)
u v
Z0
T
+ [H (δx,δx,δv)+2H (δx,δu,δv)+H (δu,δu,δv)]dt
vxx vux vuu
Z0
T m
+ ((δx ,δx )3)+ (δx,δu) p v ( (δx,δu) 2)dt.
O | 0 T | k k∞ iO k k2
Z0 i=0
X
Finally, to obtain (22) use stationarity condition (11) and the compactness
of Λ. (cid:3)
Remark 3.2. The last lemma yields the equality
(26) Ω[λ](w¯)= 1D2L [λ](wˆ)w¯2.
2
3.2. Second order necessary condition. Recall the second order condi-
tion below.
Theorem 3.3 (Classical second order necessary condition). If wˆ is a weak
minimum of problem (P), then
(27) maxΩ[λ](x¯,u¯,v¯) 0, on .
λ∈Λ ≥ C
A proof of Theorem 3.3 can be found in Osmolovskii [50]. Nevertheless,
for the sake of completeness, we give a proof here.
Weshallwriteproblem(P)inanabstractformand,therefore,weconsider
the functions
(28) η¯ :Rn R, η¯ (x ,u,v) := η (x ,x ),
j j 0 j 0 T
×U ×V →
(29) ϕ¯ :Rn R, ϕ¯ (x ,u,v) := ϕ (x ,x ),
i i 0 i 0 T
×U ×V →
8 M.S.ARONNA
where x is the solution of (2) associated with (x ,u,v). Hence, (P) can
0
be writte∈nWas the following problem in the space Rn ,
×U ×V
min ϕ¯ (x ,u,v);
0 0
(AP) s.t. η¯ (x ,u,v) = 0, for j = 1,...,d ,
j 0 η
ϕ¯ (x ,u,v) 0, for j = 1,...,d .
i 0 ϕ
≤
Notice that if wˆ is a weak solution of (P) then (xˆ ,uˆ,vˆ) is a local solution
0
of (AP).
Definition 3.4. We say that the endpoint equality constraints are qualified
if
(30) Dη¯(xˆ ,uˆ,vˆ) is onto from Rn to Rdη.
0
×U ×V
When (30) does not hold, the constraints are not qualified.
The proof of Theorem 3.3 is divided in two cases: qualified and not qual-
ified endpoint equality constraints. In the latter case the condition (27)
follows easily and it is shown in Lemma 3.5 below. The proof for the qual-
ified case is done by means of an auxiliary problem written in an abstract
form and its dual.
Lemma 3.5. If the equality constraints are not qualified then (27) holds.
Proof. Observe that since Dη¯(xˆ ,uˆ,vˆ) is not onto there exists β Rdη,∗
0
∈
with β = 1 such that dη β Dη¯ (xˆ ,uˆ,vˆ) = 0 and consequently,
| | j=1 j j 0
P dη
β Dη (xˆ ,xˆ ) = 0.
j j 0 T
j=1
X
Set λ := (p,α,β) with p 0 and α = 0. Then both λ and λ are in Λ.
≡ −
Observe that
dη
Ω[λ](x¯,u¯,v¯)= 1 β D2η (xˆ ,xˆ )(x¯ ,x¯ )2.
2 j j 0 T 0 T
j=1
X
Thus,eitherΩ[λ](x¯,u¯,v¯)orΩ[ λ](x¯,u¯,v¯)isnonnegative. Thedesiredresult
follows. − (cid:3)
Let us now deal with the qualified case. Take a critical direction w¯ =
(x¯,u¯,v¯)andconsidertheprobleminthevariablesζ Randr = (r ,r ,r )
Rn given by ∈ x0 u v ∈
×U ×V
min ζ
(QPw¯) s.t. Dη¯(xˆ0,uˆ,vˆ)r+D2η¯(xˆ0,uˆ,vˆ)(x¯0,u¯,v¯)2 =0,
Dϕ¯ (xˆ ,uˆ,vˆ)r+D2ϕ¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)2 ζ, i = 0,...,d .
i 0 i 0 0 ϕ
≤
Proposition 3.6. Assume that wˆ is a weak solution of (P) such that the
endpoint equality constraints are qualified at wˆ. Let w¯ be a critical
∈ C
direction. Then the problem (QP ) is feasible and has nonnegative value.
w¯
SINGULAR SOLUTIONS IN OPTIMAL CONTROL 9
Proof. Step I. Let us first show feasibility. Since Dη¯(xˆ ,uˆ,vˆ) is onto, there
0
exists r Rn such that the equality constraint in (QP ) is satisfied.
w¯
∈ ×U×V
Set
ζ := max Dϕ¯ (xˆ ,uˆ,vˆ)r+D2ϕ¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)2 .
i 0 i 0 0
0≤i≤dϕ{ }
Then (ζ,r) is feasible for (QP ).
w¯
Step II. Let us now prove that (QP ) has nonnegative value. Supposeon
w¯
the contrary that there is (ζ,r) R Rn feasible for (QP ) with
w¯
∈ × ×U ×V
ζ < 0. We look now for a family of feasible solutions of (P) that we will
denote by r(σ) . It shall be defined for small positive σ and satisfy
σ
{ }
(31) r(σ) (xˆ ,uˆ,vˆ), ϕ¯ (r(σ)) < ϕ¯ (xˆ ,uˆ,vˆ).
0 0 0 0
→
The existence of r(σ) will contradict the local optimality of (xˆ ,uˆ,vˆ).
σ 0
{ }
Consider hence
r˜(σ) := (xˆ ,uˆ,vˆ)+σ(x¯ ,u¯,v¯)+ 1σ2r.
0 0 2
Let 0 i d and observe that
ϕ
≤ ≤
ϕ¯ (r˜(σ)) =ϕ¯ (xˆ ,uˆ,vˆ)+σDϕ¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)
i i 0 i 0 0
(32) + 1σ2 Dϕ¯ (xˆ ,uˆ,vˆ)r+D2ϕ¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)2 +o(σ2)
2 i 0 i 0 0
ϕ¯ (xˆ ,uˆ,vˆ)+ 1σ2ζ +o(σ2).
≤ i 0 2
Analogously,
(33) î η¯(r˜(σ)) = o(σ2). ó
SinceDη¯(xˆ ,uˆ,vˆ)isonto,thereexistsr(σ)suchthat r(σ) r˜(σ) = o(σ2)
0 ∞
k − k
and η¯(r(σ)) = 0. This follows by applying the Implicit Function Theorem
to the mapping
(r,σ) η¯ (xˆ ,uˆ,vˆ)+σ(x¯ ,u¯,v¯)+ 1σ2r = η¯(r˜(σ)).
7→ 0 0 2
On the other hand, by taking σ sufficiently small in estimate (32), we can
obtain
(34) ϕ¯ (r(σ)) < ϕ¯ (xˆ ,uˆ,vˆ),
Ä i i 0 ä
sinceζ < 0.Hencer(σ)isfeasiblefor(AP)andverifies(31). Thiscontradicts
the optimality of (xˆ ,uˆ,vˆ). We conclude then that all the feasible solutions
0
of (QP ) have ζ 0, and therefore its value is nonnegative. (cid:3)
w¯
≥
We shall now go back to the proof of Theorem 3.3.
Proof. [of Theorem 3.3] The not qualified case is covered by Lemma 3.5
above. Hence, for this proof, assume that (30) holds.
Given w¯ , Proposition 3.6 implies that there cannot exist (ζ,r)
R Rn ∈ C such that ∈
× ×U ×V
Dη¯(xˆ ,uˆ,vˆ)r+D2η¯(xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)2 = 0,
0 0 0
Dϕ¯ (xˆ ,uˆ,vˆ)r+D2ϕ¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)2 ζ, i = 0,...,d ,
i 0 i 0 0 ϕ
ζ < 0. ≤

10 M.S.ARONNA
Therefore, the Dubovitskii-Milyutin Theorem (see [23]) guarantees the ex-
istence of (α,β) R1+dϕ+dη satisfying,
∈
dϕ dη
α Dϕ¯ (xˆ ,uˆ,vˆ)+ β Dη¯ (xˆ ,uˆ,vˆ) = 0,
i i 0 j j 0
i=0 j=1
(35) X X
dϕ dη
α D2ϕ¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)2+ β D2η¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)2 0
i i 0 0 j j 0 0
≥
i=0 j=1
X X
Let p be the solution of (9)-(10) associated with (α,β). Let us show that
λ := (α,β,p) is a Lagrange multiplier for wˆ. In fact, observe that
dϕ dη
0= α Dϕ¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)+ β Dη¯ (xˆ ,uˆ,vˆ)(x¯ ,u¯,v¯)
i i 0 0 j j 0 0
i=0 j=1
X X
= Dℓ(xˆ ,xˆ )(x¯ ,x¯ )= p x¯ +p x¯
0 T 0 T 0 0 T T
−
T T
= (p˙x¯+px¯˙)dt = (D Hu¯+D Hv¯)dt,
u v
Z0 Z0
where we used (9)-(10) and (12). Hence, necessarily one has
(36) D H = 0, D H = 0, a.e. on [0,T].
u v
This implies that λ Λ. On the other hand, simple computations yield that
∈
the second line of (35) is equivalent to
(37) Ω[λ](x¯,u¯,v¯) 0,
≥
and, therefore, the result follows. (cid:3)
Remark 3.7. Observe that condition (27) can be extended to the cone by
2
C
the continuity of Ω[λ] and the compactness of Λ.
In the sequel we aim to strengthen previous necessary condition by prov-
ing that the maximum in (27) remains nonnegative when taken in a smaller
set of multipliers. We shall first give a description of the subset of Lagrange
multipliers we work with. Set
(38) := (x¯,u¯,v¯) :(12) holds ,
2 2
H { ∈W }
and consider the subset of Λ given by
Λ# := λ Λ :Ω[λ] is weakly-l.s.c. on .
2
{ ∈ H }
Lemma 3.8 below provides a characterization of Λ# and Theorem 3.9 after
gives a new necessary optimality condition. Recall firstthe definitions of R
0
and K given in (21).
Lemma 3.8.
(39) Λ# = λ Λ:R [λ] 0 and K[λ] 0 .
0
{ ∈ (cid:23) ≡ }