c 2012 Society for Industrial and Applied Mathematics
SIAM J. CONTROL OPTIM.
Vol. 50, No. 3, pp. 1139–1173
VARIATIONAL APPROACH TO SECOND-ORDER OPTIMALITY
CONDITIONS FOR CONTROL PROBLEMS WITH PURE STATE
CONSTRAINTS∗
DANIEL HOEHENER†
Abstract. For optimal control problems with set-valued control constraints and pure state
constraints we propose new second-order necessary optimality conditions. In addition to the usual
second-order derivative of the Hamiltonian, these conditions contain extra terms involving secondorder tangents to the set of feasible trajectory-control pairs at the extremal process under consideration. The second-order necessary optimality conditions of the present work are obtained by using
a variational approach. In particular, we present a new second-order variational equation. This
approach allows us to make direct proofs as opposed to the classical way of obtaining second-order
necessary conditions by using an abstract infinite dimensional optimization problem. No convexity
assumptions on the constraints are imposed and optimal controls are required to be merely measurable.
Key words. optimal control, second-order necessary optimality conditions, pure state constraints, variational equation, second-order variational equation, maximum principle, second-order
tangents
AMS subject classifications. 49K15, 49K21, 49K30, 34K27
DOI. 10.1137/110828320
1. Introduction. We consider the optimal control problem of the Bolza form
(P )
1
Minimize
l(t, x(t), u(t))dt
0
over measurable u and solutions x of the control system
ẋ(t) = f (t, x(t), u(t)) a.e. in [0, 1],
(1.1)
x(0) = x0
satisfying the constraints
x(t) ∈ K
∀t ∈ [0, 1]
and
u(t) ∈ U (t) a.e.,
where the maps f : [0, 1] × Rn × Rm → Rn and l : [0, 1] × Rn × Rm → R, the set-valued
map U : [0, 1] Rm , the subset K ⊂ Rn , and the initial state x0 ∈ Rn are given. The
special case when K = Rn will be denoted by (PN C ).
We refer to a measurable function u : [0, 1] → Rm as a control function or simply
control. The set of all control functions u such that u(t) ∈ U (t) a.e. will be denoted by
U. A process (x, u) comprises a control function u ∈ U and an absolutely continuous
function x ∈ W 1,1 ([0, 1]; Rn ) which satisfies the differential equation (1.1). A state
trajectory x is a solution of (1.1) for some control function u. A process (x, u) is
called feasible if x(t) ∈ K for all t ∈ [0, 1]. Under some standard assumptions, the
∗ Received
by the editors March 22, 2011; accepted for publication (in revised form) January
9, 2012; published electronically May 15, 2012. Financial support for this work by the European
Commission (FP7-PEOPLE-2010-ITN, grant 264735-SADCO) is gratefully acknowledged.
http://www.siam.org/journals/sicon/50-3/82832.html
† Combinatoire & Optimisation, Institut de Mathématiques de Jussieu (UMR 7586), Université
Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France ([email protected]).
1139
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DANIEL HOEHENER
state trajectory x is uniquely defined by a control function u and the control system
(1.1). Consequently, denoting by M([0, 1]; Rm ) the set of all measurable maps from
[0, 1] to Rm , we can define the cost functional J : M([0, 1]; Rm ) → R by
J(u) :=
0
1
l(t, xu (t), u(t))dt,
where xu ∈ W 1,1 ([0, 1]; Rn ) is the solution of (1.1). For ū ∈ M([0, 1]; Rm ) and
u ∈ L∞ ((0, 1); Rm ), let J (ū)u denote the directional derivative of J at ū in the
direction u. The Hamiltonian H : [0, 1] × Rn × Rn × Rm of problem (P ) is defined by
H(t, x, p, u) = f (t, x, u), p − l(t, x, u).
(1.2)
Let (x̄, ū) be an optimal solution of problem (P ) and assume that the constrained normal maximum principle holds true. That is, there exist p ∈ W 1,1 ([0, 1]; Rn ), a Borel
measurable ν : [0, 1] → Rn , and a positive Radon measure μ, satisfying the conclusion
of Theorem 3.4 (see section 3 below). It follows then that for ψ(t) := [0,t] ν(s)dμ(s)
whenever t ∈ ]0, 1] and for every pair (y, u) ∈ W 1,1 ([0, 1]; Rn ) × L∞ ((0, 1); Rm ) that
is “tangent” to the set of feasible processes of (P ), we have
(1.3) J (ū)u = −
0
1
Hu (t, x̄(t), p(t) + ψ(t), ū(t))u(t)dt −
[0,1]
ν(t), y(t)dμ(t) ≥ 0.
A more precise description of a suitable notion of tangency is given in section 4. A
proof of (1.3) is provided in section 5.
This paper is devoted to second-order necessary optimality conditions. Similar to
classical analysis, where second-order conditions are meaningful only when first-order
terms vanish, our second-order necessary conditions apply to “tangents” u such that
J (ū)u = 0. These second-order conditions are stated in Theorem 3.5 and constitute
the main contribution of this paper.
Second-order optimality conditions for optimal control problems have been studied for almost half a century, but it remains a challenge to formulate them in a very
general context. Earlier results concerning second-order necessary conditions for control problems without state constraints are due to Hestenes [28] and Warga [52, 53].
Then, Gilbert and Bernstein [27] generalized and simplified these results in order to
state second-order necessary conditions for problems with very general constraints
on controls. More precisely, they allowed U to be an arbitrary subset of the set of
measurable maps M([0, 1]; U ), where U is an open subset of Rm .
Let (x̄, ū) ∈ W 1,1 ([0, 1]; Rn ) × M([0, 1]; Rm ) be an extremal process for a control
problem and let C(x̄, ū) denote a set of critical directions associated with it. Ideally,
C(x̄, ū) should be the set of tangents to the set of feasible processes. However, since
in practice one needs to describe the set of critical directions analytically, C(x̄, ū) is
usually assumed to be a subset of the ideal set of critical directions. The choice of the
set C(x̄, ū) is therefore a part of the second-order statements and has an important
impact on their quality. One can distinguish two ways of formulating second-order
necessary conditions. The first common formulation is as follows:
For all (y, u) ∈ C(x̄, ū), there exist multipliers (adjoint state variable,
cost multipliers, etc.) such that some second-order conditions hold
true.
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1141
The second alternative, which in general is stronger, is as follows:
There exist multipliers such that some second-order conditions hold
true ∀(y, u) ∈ C(x̄, ū).
In the first case, multipliers depend on critical directions; in the second case they are
independent from them.
For instance, in [27, Thm. 3.1], multipliers depend on the critical direction which
makes statements weaker. Also, the authors of [27] impose that ∀(y, u) ∈ C(x̄, ū), we
have u ∈ U, where
(1.4)
U ⊂ L∞ ([0, 1]; Rm ),
U is convex,
U ⊂ U − ū,
0 ∈ U.
Consequently, in the case of a nonconvex U, there might be many feasible directions
which are not in C(x̄, ū).
Later on, Maruyama [36] imposed the additional hypothesis that the set U is
a closed, convex subset of L∞ ((0, 1); Rm ) with nonempty interior and obtained the
same results as in [27] but with multipliers independent from the critical directions. In
[54], Zeidan and Zezza derived results similar to those in [36], without the convexity
assumptions, for control sets described by equality and inequality constraints and
piecewise continuous controls (i.e., in P W C([0, 1]; Rm )). More precisely,
U = {u ∈ P W C([0, 1]; Rm ) | g(u(t)) = 0 and h(u(t)) ≤ 0 ∀t ∈ [0, 1]},
where g and h are some given smooth maps.
In [27, 36, 52, 53] the authors deduced the second-order optimality conditions
from an abstract infinite dimensional optimization problem. A different approach was
proposed by Hestenes [28] and followed more recently by Milyutin and Osmolovskii
[38, 39]. There, the authors worked directly in the state-control space. A particularity
of [38, 39] is that the main results are stated for the so-called Pontryagin minimum.
In general, the notion of the Pontryagin minimum is weaker than the notion of strong
local minima, investigated in the present paper. Indeed, every strong local minimum
is a Pontryagin minimum. Further, in [38, 39] the authors restrict their attention
to mixed state-control constraints. Let us underline that the hypotheses needed to
obtain their results, notably the full rank condition (see, for instance, [39, Ch. 3]), do
not hold for pure state constraints.
In Theorem 3.2 we state second-order necessary optimality conditions for the
Bolza problem (PN C ) involving a normal maximum principle (see, for instance, [51]).
Using second-order tangents, we define a set of admissible variations of a reference control which is in general larger than U from (1.4). Our result applies to any first-order
necessary optimality conditions in the form of the constrained maximum principle. In
addition, the use of second-order variations leads to a very geometric and, as we believe, quite direct proof of Theorem 3.2, in comparison to some very technical results
in second-order theory. In [54], the authors used a variational approach that is similar
to ours but they only considered first-order variations of controls and included the
control constraints into the definition of the Hamiltonian. Therefore their optimality
conditions are different from ours. For the sake of completeness, by using techniques
similar to those of [39], we deduce from Theorem 3.2 second-order necessary conditions
for a problem with mixed state-control equality constraints (Theorem 3.3) similar to
those in [38, 39] but with the additional advantage that the optimal control is assumed
to be essentially bounded instead of piecewise continuous as in [38, 39].
Note that there is a rich literature on problems with mixed equality and inequality
constraints; see, for instance, [16, 49] and the references therein. For problems that are
1142
DANIEL HOEHENER
linear in the control see Dmitruk [19, 20, 21, 22]. The goal of this paper is to provide
second-order necessary conditions that are applicable to optimal control problems
under pure state constraints. A comparison of our results with those on problems
with mixed constraints and/or particular dynamics is therefore beyond the scope of
this introduction.
Pure state constraints, which are important in applications, bring some additional
difficulties because the state is not linked to controls by a functional relation. Consider
the case where the state constraint is formulated as an inequality constraint, i.e.
(1.5)
g(x(t)) ≤ 0 ∀t ∈ [0, 1],
where x : [0, 1] → Rn denotes the state trajectory and g : Rn → Rk is a given (smooth)
map. A classical way of dealing with state constraints of this form is to differentiate
the components gi of the map g(x(·)) with respect to time, until an explicit dependence from the control appears, so that the problem can be treated as a problem
with mixed constraints. See [11, 12, 26, 31] for first-order necessary conditions using
this approach. Bonnans and Hermant [6, 7, 8] provided second-order necessary and
sufficient conditions for optimal control problems with state constraints of the form
(1.5). They used second-order necessary conditions due to Kawasaki for an abstract
optimization problem with infinitely many inequality constraints [32, 34]. The major
difficulty in this approach is that these second-order necessary conditions contain, in
the presence of pure state constraints, a so-called curvature term. By a careful analysis of junction points, using an extension of the junction conditions from [31], they
were able to characterize the curvature term in a way that led to no-gap second-order
optimality conditions, i.e., second-order necessary and sufficient conditions that are as
close as in classical analysis (see [8]). Thanks to the no-gap character of the optimality conditions, the second-order necessary conditions are very strong. On the other
hand, these results are proved under quite restrictive assumptions. One of these is
continuity of the optimal control (their assumptions do imply that the optimal control
is continuous). In addition, except in the case of mixed state-control constraints, they
do not allow constraints on the control, and finally, it is required that there exist only
finitely many boundary arcs, which for a constraint of order ≥3 in general is not true;
see, for instance, [37, 47].
Páles and Zeidan addressed the problem of pure state constraints in a series
of papers, see [40, 43, 44] and the references therein. The optimal control problem
considered in [40] is similar to the one of [8]. However, contrary to the strong regularity
conditions in [8], Páles and Zeidan provide necessary conditions for problems where the
map g from (1.5) is only locally Lipschitz continuous. The drawback of their approach
is that, as in [27], the multipliers depend upon critical directions. In [43] these results
were extended to problems with more general pure and mixed state constraints:
g(t, x(t), u(t)) ∈ Q(t) a.e.,
k(t, x(t)) ∈ S(t) ∀t,
where Q and S are set-valued maps with closed, convex values having nonempty interior. Finally, in [44] the second-order conditions were derived for time independent
state constraints S(t) = S ∀t ∈ [0, 1], where S is a closed convex subset of Rn with
nonempty interior, and control constraints are of the form u(t) ∈ U almost everywhere, where U is an arbitrary subset of Rm . Note that in both cases the pure state
constraints are more general than (1.5), but multipliers depend on critical directions.
In addition, to use the time transformation introduced by Dubovitskii and Milyutin
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1143
in [23], the authors impose in [44] differentiability of f with respect to the time variable. This approach has the advantage that their results can be extended to time
optimal problems; see [45]. However, for nonautonomous control systems arising in
applications, it often occurs that the dynamic of the system is less regular in the time
variable and may even be discontinuous.
In mathematical programming, second-order optimality conditions are classically
derived using second-order variations; see [24]. Since these early results, much work
has been done to extend this approach to a very general context, notably constrained
optimization problems in infinite dimensional spaces (see, for instance, [3, 5, 15, 30,
32, 41, 46] and the references therein). A detailed treatment of this topic can be
found in the book by Bonnans and Shapiro [9]. An important application of such
second-order necessary optimality conditions for abstract optimization problems are
second-order necessary optimality conditions for optimal control problems.
In the present work, unlike previous papers on second-order necessary optimality
conditions for control problems with pure state constraints, we do not reformulate
our optimal control problem in an abstract form. Indeed, some of our assumptions,
in particular the low regularity of the optimal control, are not covered by known
abstract general second-order theories. Therefore we provide a direct proof of secondorder necessary optimality conditions. For this aim we use second-order variations
of the state trajectory, which we obtain from second-order control variations and a
second-order variational equation; see section 4. Some of the results of section 4
(Propositions 4.2 and 4.4) are new. Further, in the difference with the approach
involving an abstract optimization problem, we separate proofs of first- and secondorder conditions, which allows us to benefit entirely from the most general first-order
conditions. See, for instance, [51] for an overview of first-order necessary optimality
conditions. In addition, our main result is a straightforward extension of the case when
state constraints are absent. In particular, the presence of state constraints does not
lead to any restrictions on control constraints and multipliers are independent from
critical directions. Finally, we do not assume any regularity of the optimal control
except the usual measurability and also dynamics of the control system are allowed
to be merely measurable in the time variable. The price to pay for dealing with
measurable optimal controls is that our results are stated for the subset A(1) (x̄, ū) of
the set of first-order tangents A(x̄, ū). (See section 3 for the definitions of these sets.)
Finally, normality of the maximum principle is important for second-order conditions,
and even though there exist some sufficient conditions for normality of the constrained
maximum principle with end point constraints (see, for instance, [25]), they are not
applicable in the case of a fixed initial state. For this reason, we do not consider end
point constraints here.
The paper is organized as follows. In section 2 we recall some definitions and
state the main assumptions. In section 3 we present the main results of this paper,
in particular, second-order necessary optimality conditions for problems (PN C ) and
(P ). First- and second-order variational equations for problems with and without
state constraints are provided in section 4. The proofs of the main results are given
in sections 5 and 6.
2. Notation and main assumptions.
2.1. Basic notation. We denote the norm in Rn by |·| and by ·, · the inner
product. B := {x ∈ Rn | |x| ≤ 1} denotes the closed unit ball, B̊ its interior,
i.e., the open unit ball, and S n−1 the unit sphere, i.e., the boundary of B. B(x, r)
denotes a closed ball with radius r > 0 and center x ∈ Rn , i.e., B(x, r) = x + rB.
1144
DANIEL HOEHENER
For an arbitrary set S ⊂ Rn , ∂S denotes its boundary, Int S its interior, and S c its
complement. The convex hull and the closed convex hull of the set S will be denoted
by co S and co S, respectively.
W 1,1 ([0, 1]; Rn ) stands for the space of absolutely continuous maps from [0, 1]
to Rn and M([0, 1]; Rm ) for the space of measurable maps from [0, 1] to Rm . For
1 ≤ p < ∞ (resp., p = ∞), Lp ((0, 1); Rm ) denotes the Lebesgue space; in particular,
for maps u ∈ Lp ((0, 1); Rm ) we have that |u|p is integrable, respectively, that |u|
is essentially bounded. N BV ([0, 1]; Rn ) denotes the space of normalized functions of
bounded variation on [0, 1] with values in Rn , i.e., functions of bounded total variation,
vanishing at 0 and right-continuous on ]0, 1[. For any ψ ∈ N BV ([0, 1]; Rn ), the right
(left) limit of ψ at t ∈ [0, 1[ (resp., t ∈ ]0, 1]) is denoted by ψ(t+) (resp., ψ(t−)). For
properties of the space N BV ([0, 1]; Rn ) see, for instance, [35].
For a continuous map f : [0, 1] → Rn we denote the supremum norm by f ∞ :=
maxt∈[0,1] |f (t)|. The norm of a map u ∈ Lp ((0, 1); Rm ) for 1 ≤ p < ∞ (resp., p = ∞)
is denoted by
1/p
up :=
[0,1]
|u(t)|p dt
,
respectively, u∞ := inf {C ≥ 0 | |u(t)| ≤ C for a.e. t ∈ [0, 1]}. Here and in the rest
of the paper a.e. means with respect to the Lebesgue measure.
Finally, let Y be a normed, finite dimensional vector space and L : Rn → Y a
linear map. Then the norm of L is the operator norm, i.e., L := inf {C ≥ 0 |
LxY ≤ C ∀x ∈ S n−1 }.
Let f : [0, 1]×Rn ×Rm → Rk be a map such that f (t, ·, ·) is differentiable for every
t ∈ [0, 1]. Then we denote by fx (t0 , x0 , u0 ) and fu (t0 , x0 , u0 ) the partial derivative of
f with respect to x, respectively, u, at (t0 , x0 , u0 ). Further, f (t, x0 , u0 ) denotes the
derivative of the map (x, u) → f (t, x, u) evaluated at (x0 , u0 ). Analogously, if f (t, ·, ·)
is twice differentiable for all t ∈ [0, 1], we denote the second-order partial derivatives with respect to x and/or u at (t0 , x0 , u0 ) by fxx (t0 , x0 , u0 ), fxu (t0 , x0 , u0 ), and
fuu (t0 , x0 , u0 ). Further, f (t, x0 , u0 ) denotes the Hessian of the map (x, u) → f (t, x, u)
evaluated at (x0 , u0 ). Second-order derivatives are bilinear maps and for u, v ∈ Rn
we write fxx (t0 , x0 , u0 )uv instead of fxx (t0 , x0 , u0 )(u, v). The same simplification will
be used for fxu and fuu .
When dealing with Lipschitz continuous functions that are not differentiable, we
will use the notion of the Clarke gradient.
Definition 2.1. Consider a function f : Rn → R and a point x ∈ Rn . Assume
that f is Lipschitz continuous on a neighborhood of x. The Clarke gradient of f at x
is the set
∂f (x) := co {ξ ∈ Rn | ∃xi → x, ∇f (xi ) exists and ∇f (xi ) → ξ}.
Let A be a measurable subset of [0, 1]. The characteristic function of A, denoted
by χA : [0, 1] → {0, 1}, is defined by
1 if t ∈ A,
χA (t) :=
0 otherwise.
For an interval I ⊂ R, B(I) denotes the Borel σ-algebra associated with the interval I.
Next, we recall some definitions concerning tangent sets to subsets of Rn . The
distance between a point x ∈ Rn and a subset K ⊂ Rn is defined by dist(x, K) :=
1145
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
inf k∈K |x−k|. We denote by TK (x) and CK (x), respectively, the Bouligand and Clarke
tangent cone of K at x. See [2] for their definitions. First- and second-order adjacent
subsets are defined next.
Definition 2.2. Let K be a closed subset of Rn and let x ∈ K. The adjacent
cone to K at x is the set defined by
dist(x + hu, K)
n =0 .
TK (x) := u ∈ R lim
h→0+
h
We recall that a set K is called sleek if the Bouligand and the Clarke tangent
cone to K coincide for all x ∈ K. In this case, also the adjacent tangent cone to K
coincides with the Clarke tangent cone to K for all x ∈ K and the set-valued map
(x) is lower semicontinuous at every point of K. For further properties of
x TK
sleek sets, see, for instance, [2].
Definition 2.3. Let K be a closed subset of Rn . Further, let x ∈ K and u ∈ Rn .
The second-order adjacent subset to K at (x, u) is the set defined by
dist(x + hu + h2 v, K)
(2)
n
=0 .
TK (x, u) := v ∈ R lim
h→0+
h2
(2)
(2)
(x). Furthermore, TK (x, 0) =
Note that TK (x, u) = ∅ implies that u ∈ TK
(2)
TK (x), and finally, if K is convex, then TK (x, u) is convex as well. The set K is
called twice derivable if for all (x, u) the second-order Bouligand and the second-order
adjacent tangent set to K at (x, u) coincide. See, for instance, [2] for more information
about derivable sets. We will also use polar and normal cones.
Definition 2.4. Let K be a closed subset of Rn . The (negative) polar cone to
K is the set defined by
K := {x ∈ Rn | x, k ≤ 0 ∀k ∈ K}.
The normal cone to K at x is the set NK (x) := CK (x) .
For basic properties of tangent sets and normal cones, see, for instance, [2]. Finally, we give a useful regularity property of a subset of Rn .
Definition 2.5. Let K be a closed subset of Rn and let r > 0. We say that K
satisfies the uniform interior sphere property of radius r if for every x ∈ ∂K there
exists yx ∈ Rn such that
B(yx , r) ⊂ K and |yx − x| = r.
The following proposition is an immediate consequence of [13, Prop. 2.2.2].
Proposition 2.6. Let r > 0, y ∈ Rn and x ∈ ∂B(y, r). Then for every u ∈
T∂B(y,r) (x),
dist(x + u, B(y, r)) ≤
|u|2
.
r
2.2. Main assumptions. In this subsection we give a list of the assumptions
that we will use in the remainder of this paper.
Definition 2.7. A process (x̄, ū) is a strong local minimizer if there exists δ > 0
such that (x̄, ū) minimizes J over all feasible processes (x, u) satisfying x̄−x∞ ≤ δ.
Let us consider an admissible process (x̄, ū) for either (P ) or (PN C ). Throughout
the paper we assume the following:
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DANIEL HOEHENER
(A1) The set-valued map U is measurable and has closed nonempty images.
(A2) (a) For a.e. t ∈ [0, 1], ∀u ∈ U (t), ∀x ∈ x̄(t) + δB, f (·, x, u) and l(·, x, u) are
measurable and f (t, ·, ·), l(t, ·, ·) are continuously differentiable.
(b) For all R > 0, there exists kR ∈ L1 ((0, 1); R+ ) such that for a.e. t ∈ [0, 1],
f (t, ·, u) and l(t, ·, u) are kR (t)-Lipschitz on RB for all u ∈ U (t).
(c) There exists γ ∈ L1 ((0, 1); R+ ) such that for a.e. t ∈ [0, 1] and all x ∈ Rn ,
|f (t, x, u)| ≤ γ(t)(1 + |x|)
∀u ∈ U (t).
(d) For all R > 0, the mapping t → sup |x|≤R |l(t, x, u)| is integrable on
u∈U(t)∩RB
[0, 1].
For the rest of the paper, we are going to abbreviate (t, x̄(t), ū(t)) by [t]; thus, for
instance, f [t] := f (t, x̄(t), ū(t)).
Some of the results of this paper will require additional assumptions. In order to
state them, let (x̄, ū) be an admissible process of either (P ) or (PN C ) and let δ > 0.
(H1) (a) fu [·] ∈ L1 ((0, 1); R+ ) and ∃a1 ∈ L1 ((0, 1); R+ ) such that for a.e. t ∈
[0, 1],
fu (t, x, u) − fu (t, y, v) ≤ a1 (t)(|x − y| + |u − v|)
∀x, y ∈ x̄(t) + δ B̊, ∀u, v ∈ ū(t) + δ B̊.
(b) For a.e. t ∈ [0, 1] ∀x ∈ x̄(t) + δB, f (t, x, U (t)) is a closed set.
(c) For all t ∈ [0, 1], f (t, ·, ·) is twice differentiable on (x̄(t)+δ B̊)×(ū(t)+δ B̊).
(d) f [·] ∈ L1 ((0, 1); R+ ) and ∃a2 ∈ L1 ((0, 1); R+ ) such that for a.e. t ∈
[0, 1],
f (t, x, u) − f (t, y, v) ≤ a2 (t)(|x − y| + |u − v|)
∀x, y ∈ x̄(t) + δ B̊, ∀u, v ∈ ū(t) + δ B̊
(H2) (a) For all t ∈ [0, 1], f (t, ·, ·) is twice differentiable on (x̄(t)+δ B̊)×(ū(t)+δ B̊).
(b) The mapping b1 : [0, 1] → R+ defined by
b1 (t) := f [t] + f [t]
is integrable on [0, 1].
(c) There exists b2 ∈ L1 ((0, 1); R+ ) such that for a.e. t ∈ [0, 1],
f (t, x, u)−f (t, y, v)+f (t, x, u)−f (t, y, v) ≤ b2 (t)(|x−y|+|u−v|)
∀x, y ∈ x̄(t) + δ B̊, ∀u, v ∈ ū(t) + δ B̊.
3. Main results. In this section we present the main results of the present work.
The proofs rely on the variational equations stated in section 4 and will be given in
section 6.
3.1. Second-order necessary optimality conditions for problem (PN C ).
In this subsection we present second-order necessary optimality conditions for the
problem (PN C ). We start with first-order necessary conditions, usually called the
maximum principle. The Hamiltonian H : [0, 1] × Rn × Rn × Rm → R is defined
by (1.2).
Theorem 3.1. Let (x̄, ū) be a strong local minimizer for problem (PN C ). Assume
(H1)(b) and that l satisfies the same assumptions as f in (H1)(b). Then there exists
p ∈ W 1,1 ([0, 1]; Rn ) such that
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1147
(i) −ṗ(t) = Hx (t, x̄(t), p(t), ū(t)) a.e.,
(ii) p(1) = 0,
(iii) H(t, x̄(t), p(t), ū(t)) = maxu∈U(t) H(t, x̄(t), p(t), u) a.e.
Proof. It is well known that by using an additional state variable, the Bolza
problem can be reduced to the Mayer one. Applying this to (PN C ) we get a problem
for which [51, Prop. 6.4.4] applies. By using this proposition, the statement of the
theorem can be easily deduced.
Let p be the adjoint state variable from the maximum principle. Below we will use
the abbreviation [t] also for (t, x̄(t), p(t), ū(t)). The precise meaning will be given by
the context. We introduce the following linear system, which will play an important
role throughout the paper:
ẏ(t) = fx [t]y(t) + fu [t]u(t) a.e.,
(3.1)
y(0) = 0.
Define A := {t ∈ [0, 1] | ū(t) ∈ ∂U (t), Hu [t] = 0} and
UA := u ∈ L∞ ((0, 1); Rm ) | ∃c > 0, h0 > 0 s.t.
∀h ∈ [0, h0 ], ∀t ∈ A, dist(ū(t) + hu(t), U (t)) ≤ ch2 .
The next theorem was announced in [29]. We will provide its detailed proof in section 6.
Theorem 3.2. Let (x̄, ū) be a strong local minimizer for problem (PN C ). Assume
(H1) and that l fulfills the same assumptions as f in (H1). Let p be as in the maximum
principle of Theorem 3.1. Then ∀u ∈ UA and v ∈ L∞ ((0, 1); Rm ) such that
(u(t)) a.e.
u(t) ∈ TU(t)
(3.2)
either
1
0
Hu [t]u(t)dt < 0 or
(3.3)
0
1
1
0
and
(2)
v(t) ∈ TU(t) (ū(t), u(t)) for a.e. t ∈ A,
Hu [t]u(t)dt = 0 and then the inequality
2Hu [t]v(t) + Hxx [t]y(t)y(t) + 2Hxu [t]y(t)u(t) + Huu [t]u(t)u(t) dt ≤ 0
holds true, where y ∈ W 1,1 ([0, 1]; Rn ) is the solution of the linear system (3.1).
Remark 3.1. Observe that if there exists r > 0 such that U (t) satisfies the
uniform interior sphere property of radius r for almost every t ∈ A, then for every
(ū(t)) and Hu [t]u(t) = 0, we
u ∈ L∞ ((0, 1); Rm ) satisfying for a.e. t ∈ A, u(t) ∈ TU(t)
have that u ∈ UA . Indeed, let t ∈ A. It follows from the maximum principle that
(3.4)
Hu [t]ũ ≤ 0
∀ũ ∈ TU(t)
(ū(t)).
By the uniform interior sphere property there exists yt ∈ Rm such that Bt :=
B(yt , r) ⊂ U (t) and |yt − ū(t)| = r. Since Hu [t] = 0, we deduce from (3.4) that
Hu [t] ∈ TU(t)
(ū(t)) . Then obviously Hu [t] ∈ TB t (ū(t)) . This implies that
TB t (ū(t)) = {λHu [t] | λ ≥ 0}.
Next, since Hu [t]u(t) = 0, we have that u(t) ∈ T∂B
(ū(t)). Furthermore,
t
dist(ū(t) + hu(t), U (t)) ≤ dist(ū(t) + hu(t), Bt ),
and therefore we deduce from Proposition 2.6 that dist(ū(t)+hu(t), U (t)) ≤ 1r u2∞ h2 .
1148
DANIEL HOEHENER
3.2. Second-order necessary optimality conditions for problems with
mixed constraints. Theorem 3.2 can be applied to deduce second-order necessary
conditions for problems with mixed equality constraints. Consider problem (PN C ),
where U (t) = Rm ∀t ∈ [0, 1], and suppose that we have the additional mixed statecontrol equality constraint
(M C)
γ(t, x(t), u(t)) = 0
∀t ∈ [0, 1],
where γ : [0, 1] × Rn × Rm → Rk is given. We denote this problem by (PMC ). Let Q
be an open subset of [0, 1] × Rn × Rm satisfying
Γ := {(t, x, u) ∈ [0, 1] × Rn × Rm | γ(t, x, u) = 0} ⊂ Q.
Before stating the second-order necessary conditions, we introduce the following Hamiltonian:
H(t, x, p, u, b) := p, f (t, x, u) − l(t, x, u) − b, γ(t, x, u).
Analogously to the previous subsection, [t] abbreviates (t, x̄(t), p(t), ū(t), b(t)) as well
as (t, x̄(t), ū(t)). The precise meaning will be given by the context.
Theorem 3.3 (see [29, Thm. 15]). Let (x̄, ū) be a strong local minimizer for
problem (PMC ). Assume the following hypotheses:
(a) f , l, and γ are twice continuously differentiable on Q.
(b) γu satisfies the full rank condition, rank γu (t, x, u) = k ∀(t, x, u) ∈ Γ.
(c) ū ∈ L∞ ((0, 1); Rm ).
Then there exist p ∈ W 1,1 ([0, 1]; Rn ) and an essentially bounded b ∈ L∞ ((0, 1); Rk )
satisfying
(i) −ṗ(t) = Hx (t, x̄(t), p(t), ū(t), b(t)) a.e.,
(ii) p(1) = 0,
(iii) Hu (t, x̄(t), p(t), ū(t), b(t)) = 0 a.e.
and such that the inequality
0
1
Hxx [t]y(t)y(t) + 2Hxu [t]y(t)u(t) + Huu [t]u(t)u(t)dt ≤ 0
holds for every u ∈ L∞ ((0, 1); Rm ) satisfying
γx [t]y(t) + γu [t]u(t) = 0
∀t ∈ [0, 1],
where y ∈ W 1,1 ([0, 1]; Rn ) is the solution of the linear system (3.1).
Remark 3.2. Results of the same nature for problems in the Mayer form can be
found in [38] and [39], where the authors require ū to be piecewise continuous.
Remark 3.3. Assuming that an appropriate full rank condition holds, problems
with mixed inequality constraints can be reduced to problems with mixed equality
constraints; see [50]. Thus, the above theorem can also be used to find second-order
necessary conditions for problems with mixed inequality constraints.
Proof of Theorem 3.3. Similar to [39], the main idea of the proof is to reduce
problem (PMC ) to a problem without the mixed constraints, i.e., a problem to which
Theorem 3.2 can be applied. See [29] for details.
3.3. Second-order necessary optimality conditions for problem (P ). Below we state necessary optimality conditions for problem (P ). We start by presenting
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1149
a normal maximum principle for problem (P ). This result is based on the maximum
principle [51, Thm. 9.5.1] and a normality result due to Bettiol and Frankowska [4].
In this subsection we will make the following assumptions on the set of state
constraints K ⊂ Rn :
r
(SC) (a) There exist g1 , . . . , gr ∈ C 2 (Rn ; R) such that K = i=1 Ki , where for
i = 1, . . . , r, Ki := {x ∈ Rn | gi (x) ≤ 0}.
(b) For all x ∈ ∂K, 0 ∈
/ co {∇gi (x) | i ∈ I(x)}, where I(x) := {i ∈
{1, . . . , r} | gi (x) = 0}.
Remark 3.4. Note that assumption (SC) implies that the sets Ki are of class
1,1
Cloc
. See, for instance, [17, 18] for properties of smooth sets. In particular, it follows
that ∂Ki = gi−1 (0) := {x ∈ Rn | gi (x) = 0} ∀i.
In addition, (SC) implies that the set K is sleek and twice derivable (see [2,
Thm. 4.7.4, Prop. 4.7.5]). Finally, one has the following characterizations of the
tangent and the normal cone:
(x) = {w ∈ Rn | ∇gi (x), w ≤ 0 ∀i ∈ I(x)},
TK
NK (x) = R+ co {∇gi (x) | i ∈ I(x)},
where x ∈ ∂K.
In addition, we require that the set K ⊂ Rn satisfy the following inward pointing
condition:
(IP) For all R > 0, ∃ηR > 0, MR > 0, ρR > 0 such that ∀x ∈ ∂K ∩ RB, ∀z ∈
K ∩ B(x, ηR ), ∀s ∈ [0, 1], ∃uz ∈ U (s) satisfying
|f (s, z, uz )| ≤ MR ,
sup
n∈NK (x)∩S n−1
n, f (s, z, uz ) ≤ −ρR .
The Hamiltonian H is defined by (1.2). We provide the proof of the following theorem
in section 5.
Theorem 3.4. Let x0 ∈ Int K and (x̄, ū) be a strong local minimizer for the
problem (P ). Assume (SC) and (IP). If x̄ is Lipschitz continuous, then there exist
p ∈ W 1,1 ([0, 1]; Rn ), a positive Radon measure μ, and a Borel measurable ν : [0, 1] →
Rn satisfying
ν(t) ∈ NK (x̄(t)) ∩ B,
μ-a.e.,
such that for ψ(t) := [0,t] ν(s)dμ(s) for t ∈ ]0, 1] and q := p + ψ we have
(i) −ṗ(t) = Hx (t, x̄(t), q(t), ū(t)) a.e.,
(ii) q(1) = 0,
(iii) H(t, x̄(t), q(t), ū(t)) = maxu∈U(t) H(t, x̄(t), q(t), u) a.e.,
(iv) supp μ ⊂ {t ∈ [0, 1] | I(x̄(t)) = ∅}.
Furthermore, the following jump conditions hold true:
ψ(t) − ψ(t−) ∈ NK (x̄(t)) ∀t ∈ ]0, 1].
(3.5)
Remark 3.5. It is not difficult to show that there exist positive Radon measures
μi , i = {1, . . . , r}, such that
ψ(t) =
ν(s)dμ(s) =
[0,t]
r i=1
[0,t]
∇gi (x̄(s))dμi (s) ∀t ∈ ]0, 1].
1150
DANIEL HOEHENER
Next we state second-order necessary conditions for problem (P ). First, we introduce some notation. Let (x̄, ū) be an admissible process of problem (P ). We set
H[t] := H(t, x̄(t), q(t), ū(t)), where q = p+ψ is as in Theorem 3.4. Hu [t], Hx [t], Hxx [t],
etc., are defined similarly. The set of tangent directions A(x̄, ū) for problem (P ) at
(x̄, ū) is defined as follows:
A(x̄, ū) := {(y, u) ∈ W 1,1 ([0, 1]; Rn ) × L∞ ((0, 1); Rm ) |
u(t) ∈ TU(t)
(ū(t)) a.e., y is solution of (3.1) s.t. y(t) ∈ TK
(x̄(t)) ∀t ∈ [0, 1]}.
Under all the assumptions of Theorem 3.4, ∀(y, u) ∈ A(x̄, ū) we have
1
0
Hu [t]u(t)dt ≤ 0
and
[0,1]
ν(t), y(t)dμ(t) ≤ 0.
For second-order conditions, we have to use a smaller set of tangents in order to ensure
that the tangent variations to the control remain uniformly bounded:
A(1) (x̄, ū) := (y, u) ∈ A(x̄, ū) ∃c > 0, h0 > 0 s.t. for a.e. t ∈ [0, 1],
dist(ū(t) + hu(t), U (t)) ≤ ch2 ∀h ∈ ]0, h0 ] .
Remark 3.6. Consider the set U ∞ := {u ∈ L∞ ((0, 1); Rm ) | u(t) ∈ U (t) a.e.},
(2)
ū ∈ U ∞ , and u ∈ L∞ ((0, 1); Rm ). If the second-order adjacent set TU ∞ (ū, u) defined
in the L∞ topology is nonempty, then there exist c > 0, h0 > 0 such that for almost
every t ∈ [0, 1],
dist(ū(t) + hu(t), U (t)) ≤ ch2
∀h ∈ [0, h0 ].
Note that the converse statement is not true in general. That is, the above inequality
(2)
does not imply nonemptyness of TU ∞ (ū, u).
(1)
For (y, u) ∈ A (x̄, ū) and v ∈ L∞ ((0, 1); Rm ), consider the following secondorder approximation of the nonlinear control system (1.1):
(3.6)
⎧
ẇ(t) = fx [t]w(t) + fu [t]v(t)
⎪
⎪
⎨
1
1
+ fxx [t]y(t)y(t) + fxu [t]y(t)u(t) + fuu [t]u(t)u(t) a.e.,
⎪
2
2
⎪
⎩
w(0) = 0.
For i ∈ {1, . . . , r}, we set
Ti := {t ∈ [0, 1] | ∃sj → t such that ∀j, gi (x̄(sj )) < 0, ∇gi (x̄(sj )), y(sj ) > 0}.
Then, for t ∈ [0, 1] we define
⎧
⎪
⎪
⎨
ci (t) :=
lim sup
s→t
gi (x̄(s))<0
⎪
(x̄(s)),y(s)>0
∇g
⎪ i
⎩
∇gi (x̄(s)), y(s)2
4|gi (x̄(s))|
if t ∈ Ti ,
0
otherwise.
We say that w ∈ W 1,1 ([0, 1]; Rn ) satisfies (TV) if
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1151
(TV) For all i ∈ {1, . . . , r} and ∀t ∈ [0, 1] such that gi (x̄(t)) = 0 and ∇gi (x̄(t)),
y(t) = 0 we have
1
∇gi (x̄(t)), w(t) + gi (x̄(t))y(t)y(t) + ci (t) ≤ 0.
2
(3.7)
We say that w satisfies (TV) strictly if (3.7) holds with strict inequality.
Remark 3.7. Note that by [34, Thm. 3.2], the condition (TV) is equivalent to the
statement that the map t → g (x̄(t))w(t) + 12 g (x̄(t))y(t)y(t) is in the second-order
adjacent set to the set of nonpositive continuous functions at (g(x̄(·)), g (x̄(·))y(·)).
The set of second-order tangent directions A(2) (x̄, ū, u, y) of problem (P ) at the
process (x̄, ū) and the first-order tangent (y, u) ∈ A(1) (x̄, ū) is defined by
A(2) (x̄, ū, y, u) := (w, v) ∈ W 1,1 ([0, 1]; Rn ) × L∞ ((0, 1); Rm ) (2)
v(t) ∈ TU(t) (ū(t), u(t)) a.e., w is solution of (3.6) s.t. (TV) is satisfied .
We can state now the second-order conditions. Their proof will be given in section 6.
Theorem 3.5. Assume (SC) and (IP), x0 ∈ Int K, and let (x̄, ū) be a strong
local minimizer such that (H2) holds true and x̄ is Lipschitz. Further suppose that l
fulfills the same assumptions as f in (H2). Let q, μ, and ν be as in the maximum
principle of Theorem 3.4 and let (y, u) ∈ A(1) (x̄, ū) be such that
1
Hu [t]u(t)dt = 0 and
ν(t), y(t)dμ(t) = 0.
0
[0,1]
(2)
Let (w̃, ṽ) ∈ A(2) (x̄, ū, y, u) satisfy (TV) strictly and D(t) ⊂ TU(t) (ū(t), u(t)) be convex
sets such that ṽ(t) ∈ D(t) for a.e. t ∈ [0, 1]. Then for every (w, v) ∈ A(2) (x̄, ū, y, u)
with v(t) ∈ D(t) a.e., we have
(3.8)
[0,1]
w(t), ν(t)dμ(t) +
1
+
0
0
1
Hu [t]v(t)dt
1
1
Hxx [t]y(t)y(t) + Hxu [t]y(t)u(t) + Huu [t]u(t)u(t) dt ≤ 0.
2
2
Note that if the set-valued map U (·) has convex images, then one can define
(2)
D(t) = TU(t) (ū(t), u(t)) ∀t.
Second, we would like to underline that the above result is derived for any q, μ,
ν satisfying the constrained maximum principle.
Remark 3.8. Consider (y, u) ∈ A(1) (x̄, ū) and for t ∈ [0, 1] set
B(t) :=
1
1
fxx [t]y(t)y(t) + fxu [t]y(t)u(t) + fuu [t]u(t)u(t).
2
2
The set of second-order active constraints is defined as follows:
I (2) (x̄(t), y(t)) := {i ∈ {1, . . . , r} | gi (x̄(t)) = 0 and ∇gi (x̄(t)), y(t) = 0} .
Observe that every solution w of (3.6) can be represented by w = w1 + w2 , where
w1 ∈ W 1,1 ([0, 1]; Rn ) is the solution of
ẇ1 (t) = fx [t]w1 (t) + B(t) a.e. and w1 (0) = 0,
1152
DANIEL HOEHENER
and w2 ∈ W 1,1 ([0, 1]; Rn ) is the solution of
(3.9)
ẇ2 (t) = fx [t]w2 (t) + fu [t]v(t)
a.e. and w2 (0) = 0.
For t ∈ [0, 1], we define the reachable set at t by
(2)
R(t) := w(t) w solves (3.9), v ∈ L∞ ((0, 1); Rm ) s.t. v(s) ∈ TU(s) (ū(s), u(s)) a.e. .
Consequently, if there exists a unique t ∈ [0, 1] such that I (2) (x̄(t), y(t)) = ∅, then
there exists (w, v) ∈ A(2) (x̄, ū, y, u) satisfying (TV) strictly if and only if
inf
r∈R(t)
max
1
∇gi (x̄(t)), r + w1 (t) + gi (x̄(t))y(t)y(t) + ci (t) < 0.
2
i∈I (2) (x̄(t),y(t))
It would be interesting to study more general conditions for the existence of pairs
(w, v) ∈ A(2) (x̄, ū, y, u) satisfying (TV) strictly. This problem, however, is out of the
scope of this paper.
3.4. Examples. We conclude this section with two examples. In the first example we consider the case without pure state constraints and we compare Theorem 3.2
with other known results. The second example illustrates the application of Theorem 3.5.
Example 1. We consider problem (PN C ), where n = m = 2, x = (x1 , x2 )T ,
u = (u1 , u2 )T , x0 = 0, and f : R4 → R2 , l : R4 → R are defined by f (x1 , x2 , u1 , u2 ) =
(u1 , u1 + u2 )T , l(x1 , x2 , u1 , u2 ) = −x1 x2 + u21 − u2 , respectively. The set-valued map
U : [0, 1] R2 is time independent,
U (t) := {(u1 , u2 ) ∈ R2 | |u1 | ≤ 1, 0 ≤ u2 ≤ 1 and g(u1 , u2 ) ≤ 0} ∀t ∈ [0, 1],
where g : R2 → R is given by g(u1 , u2 ) = −u21 + u2 .
Consider the feasible process (x̄, ū) defined by (x̄(t), ū(t)) = (0, 0) ∀t ∈ [0, 1].
Clearly this process is not optimal. In what follows, we show that one can use the
second-order necessary conditions of Theorem 3.2 in order to prove that (x̄, ū) is not
a strong local minimum. We will also show that the necessary conditions stated in
[27] and [43] do not allow a similar conclusion.
First, note that for the above data all our assumptions are satisfied and the statements of Theorem 3.2 and Theorem 3.5 are equivalent. The Hamiltonian H : R6 → R
is defined by the following:
H(x1 , x2 , p1 , p2 , u1 , u2 ) = p1 u1 + p2 (u1 + u2 ) + x1 x2 − u21 + u2 .
Thus, the adjoint equation of the maximum principle corresponding to (x̄, ū) = 0 is
ṗ(t) = 0 and p(1) = 0.
Consequently, p ≡ 0 and it is not difficult to verify that
sup H(x̄(t), p(t), u) = H(x̄(t), p(t), ū(t)) = 0
u∈U(t)
∀t ∈ [0, 1];
hence the maximum principle holds true.
Consider u(t) = (1, 0)T ∀t ∈ [0, 1]. Then A = [0, 1], u ∈ UA , and v(t) := (0, 1)T ∈
(2)
TU(t) (ū(t), u(t)) ∀t ∈ [0, 1]. The linear system (3.1) for u(t) is
ẏ(t) = (1
1)T
and y(0) = 0.
1153
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
Thus y(t) = (t, t)T . The partial derivatives of the Hamiltonian are as follows:
−2 0
0 1
, Hxx [t] =
, and Hxu [t] = 0.
Hu [t] = (0 1), Huu [t] =
0 0
1 0
Then,
0
1
1
0
Hu [t]u(t)dt = 0 and
2Hu [t]v(t) + Hxx [t]y(t)y(t) + 2Hxu y(t)u(t) + Huu [t]u(t)u(t) dt =
1
2t2 dt > 0,
0
which contradicts the second-order conditions (3.3).
On the other hand, observe that U (t) = {(u1 , u2 ) ∈ R2 | g̃(u1 , u2 ) ∈ Q}, where
g̃(u1 , u2 ) = (−u21 + u2
u1 − 1
− u2 )T
− u1 − 1 u2 − 1
and Q = R5− .
However, [43, Thm. 4.1] cannot be applied since the required normality condition is
not satisfied by the data.
We show next that necessary optimality conditions of [27, Thm. 3.1] are satisfied
by (x̄, ū). Applying these conditions to our problem, we find again that p(·) ≡ 0 and
the Hamiltonian H : R6 → R is defined by
H(x1 , x2 , p1 , p2 , u1 , u2 ) = p1 u1 + p2 (u1 + u2 ) − x1 x2 + u21 − u2 .
It is obvious that all u ∈ Ũ, where Ũ is as in (1.4), must satisfy u(t) ∈ R × {0}.
1
Consequently, ∀u ∈ Ũ , 0 Hu [t]u(t)dt = 0, i.e., the first-order conditions are satisfied.
Further, the linearized system considered in [27] is
1 0
u1 (t)
y1 (0)
0
ẏ1 (t)
=
and y(0) =
=
,
ẏ(t) =
ẏ2 (t)
u2 (t)
y2 (0)
1 1
0
where u(·) = (u1 (·), u2 (·))T ∈ U. The second-order conditions derived in [27] are as
follows:
(3.10)
1
1
Hxx [t]y(t)y(t)+ 2Hxu [t]y(t)u(t)+ Huu [t]u(t)u(t)dt =
−2y1 (t)2 + 2u1 (t)2 dt ≥ 0,
0
0
where u1 ∈ L∞ ((0, 1); R) and y1 (t) =
1
that y1 (t)2 ≤ t 0 u1 (s)2 ds. Hence,
0
1
2y1 (t)2 dt ≤
0
1
t
0
u1 (s)ds. By the Hölder inequality we have
tdt ·
1
0
2u1 (t)2 dt ≤
0
1
2u1 (t)2 dt.
Thus (3.10) holds true and consequently the necessary optimality conditions from
[27, Thm. 3.1] are satisfied. Therefore we cannot deduce that the process (x̄, ū) is not
optimal.
Example 2. We consider problem (P ), where again n = m = 2, and we use the
same notation as in the previous example. Let C, M > 0 be given constants and
assume that the dynamics of the control system are as follows:
2Cu1
u2
if t ≤ 12 ,
if t ≤ 12 ,
and f2 (t, x, u) =
f1 (t, x, u) =
(u1 + u2 )x1 + u2 otherwise,
−x2 otherwise,
1154
DANIEL HOEHENER
where f (t, x, u) = (f1 (t, x, u), f2 (t, x, u))T . The cost is given by
−M x1 − 2Cu1
if t ≤ 12 ,
l(t, x, u) =
2
2
2
2
u1 (x1 + x2 ) + u2 − x2 otherwise.
The control constraints are given by the set-valued map U : [0, 1] R2 defined by
U (t) = U = [−1, 1] × [−1, 1] ∀t ∈ [0, 1].
In order to formulate the state constraints we define the map g : R2 → R4 by
T
g(x) = (g1 (x), g2 (x), g3 (x), g4 (x))T = (x1 , −x1 − 2C, −C − x2 , x2 − C) ,
and we set K = {x ∈ R2 | g(x) ≤ 0}. Finally we assume that x0 = (−C, 0)T . It is
easy to see that this data satisfies all the assumptions of Theorem 3.5.
Consider the reference control ū = (ū1 , ū2 )T given by
1 if t ≤ 12 ,
and ū2 (t) ≡ 0.
ū1 (t) =
0 otherwise,
This leads to the following reference trajectory x̄ = (x̄1 , x̄2 )T :
−C + 2Ct if t ≤ 12 ,
and x̄2 (t) ≡ 0.
x̄1 (t) =
0
otherwise,
(x̄, ū) is obviously a feasible process. Next, we are going to show that it is an extremal.
First-order optimality conditions. First, notice that NK (x̄(t)) ∩ B = [0, 1] ×
{0} ∀t ≥ 1/2 and I(x̄(t)) = ∅ ∀t ∈ [0, 1/2[. Thus if we set
1
(0, 0)T if t < 12 ,
ν(t) ≡
, μ = δ{ 1 } , and ψ(t) =
2
0
(1, 0)T otherwise,
then ν(t) ∈ NK (x̄(t)) ∩ B μ-a.e. The corresponding solution p of the adjoint equation
(see Theorem 3.4(i)) for q := p + ψ is given by
1
1
−M t + M
−M t + M
2 − 1 if t ≤ 2 ,
2 − 1 if t < 2 ,
q1 (t) =
p1 (t) =
−1
otherwise,
0
otherwise,
and p2 (t) = q2 (t) ≡ 0. It remains to check the maximum principle. If t < 1/2, then
M
H(t, x̄(t), q(t), u) = 2q1 (t)Cu1 +q2 (t)u2 +M x̄1 (t)+2Cu1 = 2C
− M t u1 +M x̄1 (t).
2
Hence, maxu∈U(t) H(t, x̄(t), q(t), u) = H[t]. If t > 1/2, then
H(t, x̄(t), q(t), u) = −q1 (t)(u1 x̄1 (t) + u2 x̄1 (t) + u2 ) − q2 (t)x̄2 (t)
− u1 (x̄1 (t)2 + x̄2 (t)2 ) − u22 + x̄2 (t)2
= −u22 ≤ 0.
Thus the maximum principle holds true for this choice of p, ν, μ and consequently
(x̄, ū) is an extremal. We show next, using the second-order necessary conditions,
that it is not a strong local minimizer.
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1155
Second-order necessary conditions. We consider the following first-order control
variations:
√
2 4 + M if t ≤ 12 ,
u1 (t) ≡ 0 and u2 (t) =
0
otherwise.
It is obvious that u(t) ∈ TU (ū(t)) a.e. Furthermore, this control variation leads to the
following solution of the linearized system (3.1):
√
2t 4 + M
if t ≤ 12 ,
y1 (t) ≡ 0 and y2 (t) = −t+ 1 √
2
e
4 + M otherwise.
Clearly, y(t) ∈ TK
(x̄(t)) ∀t ∈ [0, 1]. Furthermore, it is easy to see that
[0,1]
ν(t), y(t)dμ(t) = 0
and
0
1
Hu [t]u(t)dt = 0,
and, consequently, the direction (y, u) is critical. Let us now consider the secondorder control variation v(·) and the corresponding solution w(·) of the second-order
approximation (3.6),
T
1
v(t) ≡ − , 0
C
(−2t, 0)T
and w(t) =
(−1, 0)T
if t ≤ 12 ,
otherwise.
(2)
Then v(t) ∈ TU (ū(t), u(t)). The only active constraint is g1 (x) ≤ 0. However, since
y1 (t) ≡ 0, it follows immediately that ci (t) ≡ 0. In addition, w satisfies (TV) strictly
since
1
∇g1 (x̄(t)), w(t) + g1 (x̄(t))y(t)y(t) = w1 (t) = −1 ∀t ∈ [1/2, 1].
2
On the other hand, the second-order inequality (3.8) does not hold because
[0,1]
w(t), ν(t)dμ(t) +
0
1
Hu [t]v(t)dt
1
1
Hxx [t]y(t)y(t) + Hxu [t]y(t)u(t) + Huu [t]u(t)u(t) dt
2
2
0
M
M
1
1
M
M
+ 2+
= 0.
= −1 −
1−
>
2+
−1−
4
2
e
2
2
4
1
+
We conclude that the extremal (x̄, ū) is not a strong local minimizer.
4. Variational equations. In real analysis, second-order necessary conditions
for a local minimum of a twice continuously differentiable function ϕ : Rn → R are
valid for critical points, i.e., points x ∈ Rn , where the first-order necessary condition
ϕ (x) = 0 holds, and are based on the fact that if the critical point x is a local
minimum then for all perturbations δx ∈ Rn with |δx| small enough, the inequality
ϕ(x + δx) − ϕ(x) ≥ 0
holds. The same approach can be used to derive second-order necessary conditions
for optimal control problems. However, there are three questions to be answered:
1156
DANIEL HOEHENER
1. What are the admissible perturbations of controls under control constraints?
2. What are the associated perturbations of trajectories?
3. How do we deal with problems involving state constraints?
The purpose of this section is to answer these questions.
It is natural to investigate perturbations of the control, since they imply corresponding perturbations of the state trajectories. Forgetting the state constraints for a
moment, an intuitive choice for the control perturbations is essentially bounded maps
that are “tangent” to the set of admissible controls. This intuition is confirmed by
the following proposition, which is a slight modification of [2, Thm. 8.5.1].
Proposition 4.1. Consider ū ∈ U and let u ∈ L∞ ((0, 1); Rm ) be such that
(ū(t)) a.e. Then ∀h > 0, ∃uh ∈ L∞ ((0, 1); Rm ) satisfying
u(t) ∈ TU(t)
(i) uh ∞ ≤ 2u∞,
(ii) ū(t) + huh (t) ∈ U (t) a.e.,
a.e.
(iii) uh −→ u, when h → 0+.
We would like to underline that in general the statement of Proposition 4.1 is not
true when we replace the adjacent tangent cone by the (larger) Bouligand tangent
cone.
Proof. The proof is a simple modification of the proof of [2, Thm. 8.5.1] and very
similar to the proof of Proposition 4.2 below and therefore it is omitted.
In order to obtain second-order conditions, we need second-order perturbations.
For second-order tangents we have the following result, similar to Proposition 4.1.
Proposition 4.2. Consider ū ∈ U and let u, v ∈ L∞ ((0, 1); Rm ) be such that
(a) ∃c, h0 > 0, such that ∀h ∈ ]0, h0 ], dist(ū(t) + hu(t), U (t)) ≤ h2 c a.e.,
(2)
(b) v(t) ∈ TU(t) (ū(t), u(t)) a.e.
Then, ∀h ∈ ]0, h0 ], ∃vh ∈ L∞ ((0, 1); Rm ) satisfying
(i) vh ∞ ≤ 2v∞ + c,
(ii) ū(t) + hu(t) + h2 vh (t) ∈ U (t) a.e.,
a.e.
(iii) vh −→ v, when h → 0+.
(2)
(ū(t)), then TU(t) (ū(t), u) = Rm .
Remark 4.1. If U (t) is sleek and u ∈ Int TU(t)
Indeed, for sleek U (t), u ∈ Int TU(t)
(ū(t)) implies by [48, Thm. 2] that ∃ > 0 such
that
ū(t) + [0, ]B(u, ) ⊂ U (t),
from which we deduce readily the statement.
Remark 4.2. It is possible to prove that if for a.e. t ∈ [0, 1] the set U (t) is
twice derivable, then assumption (a) is a sufficient condition for the nonemptyness of
(2)
TU(t) (ū(t), u(t)).
Remark 4.3. It is easy to verify that any pair (u, v) ∈ L∞ ((0, 1); Rm )2 with
(ū(t)) a.e. satisfies the assumptions of Proposition 4.2 and that
u ≡ 0 and v(t) ∈ TU(t)
in this case, the statement of Proposition 4.2 is a consequence of Proposition 4.1.
However, as the following example shows, in general there are other possible choices
than u ≡ 0.
Example. Let U = {(x, y) ∈ R2 | x2 ≤ y}, ū = (ū1 , ū2 ) = (1, 1). Then, u =
(u1 , u2 ) = (1, 2) and v = (v1 , v2 ) = (0, 1) are such that
(2)
v ∈ TU (ū, u) and dist(ū + hu, U ) ≤ h2 .
Proof. Let x → ϕ(x) := x2 . Then ϕ (ū1 ) = 2 and ϕ (ū1 ) = 2. First, we show
that ∀hn → 0+ there exists a sequence vn → v such that ū + hn u + h2n vn ∈ U . By
1157
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
the Taylor formula,
1
ϕ(ū1 + hn u1 ) = ϕ(ū1 ) + hn ϕ (ū1 )u1 + h2n ϕ (ū1 )u21 + Rn
2
= ū2 + hn u2 + h2n v2 + Rn ,
where Rn /h2n → 0, when n → ∞. Hence, by defining vn = (0, v2 +
the desired result. On the other hand,
Rn
h2n ),
we obtain
dist(ū + hu, U ) ≤ |ϕ(ū1 + hu1 ) − ū2 − hu2 | ≤ h2 .
This completes the proof.
Proof of Proposition 4.2. Let h ∈ ]0, h0 ]. We define
ah (t) := dist(ū(t) + hu(t) + h2 v(t), U (t)).
Then ah is measurable. Further, there exists a measurable yh such that yh (t) ∈ U (t) ∀t
and such that
ah (t) = |ū(t) + hu(t) + h2 v(t) − yh (t)|.
Similarly there exists a measurable zh such that zh (t) ∈ U (t) ∀t and such that
bh (t) := |ū(t) + hu(t) − zh (t)| = dist(ū(t) + hu(t), U (t)).
It follows from assumption (a) that for almost every t in [0, 1], bh (t) ≤ h2 c. Thus, by
definition of ah , we have obviously
ah (t) ≤ |ū(t) + hu(t) + h2 v(t) − zh (t)|≤ h2 c + h2 |v(t)|
a.e. in [0, 1].
Therefore,
ah (t)
≤ c + v∞
h2
a.e. in [0, 1].
Next, we define vh : [0, 1] → Rm by
vh (t) :=
yh (t) − hu(t) − ū(t)
.
h2
With this definition we find
|v(t) − vh (t)| =
ah (t)
,
h2
∀t ∈ [0, 1].
Therefore, for almost every t ∈ [0, 1], |vh (t)| ≤ 2v∞ + c and consequently vh ∈
L∞ ((0, 1); Rm ). Further, it follows directly from the definition of vh that ū(t) +
(2)
hu(t) + h2 vh (t) ∈ U (t) a.e. Finally, since v(t) ∈ TU(t) (ū(t), u(t)) a.e., ah (t)/h2 → 0
a.e., when h → 0+.
The two previous propositions provide an answer to the first of the three questions
stated at the beginning of this section for the case of unconstrained state trajectories.
In the following, we remain in this particular case and respond to the second question.
First, let us recall the following well-known fact, which can be found, for example,
in [1, pp. 255–256].
1158
DANIEL HOEHENER
Proposition 4.3 (variational equation). Let (x̄, ū) be an admissible process for
(PN C ) and suppose that assumption (H1)(a) is satisfied. Let u ∈ L∞ ((0, 1); Rm ) be
(ū(t)) and let y ∈ W 1,1 ([0, 1]; Rn ) be
such that for almost every t ∈ [0, 1], u(t) ∈ TU(t)
the solution of the linearized control system (3.1) corresponding to u. For all h > 0,
let uh be as in Proposition 4.1 and let xh ∈ W 1,1 ([0, 1]; Rn ) be the solution of
ẋh (t) = f (t, xh (t), ū(t) + huh (t)) a.e.,
xh (0) = x0 .
Then,
xh − x̄ unif
−→ y
h
when h → 0 + .
Remark 4.4. It follows from assumption (A2) and the Picard–Lindelöf theorem
that ∀h > 0, xh exists and is unique.
Remark 4.5. The hypothesis (H1)(a) could be replaced by the following weaker
hypothesis:
∃ku ∈ L1 ((0, 1); R+ ) such that for a.e. t ∈ [0, 1] and ∀x ∈ x̄(t) + δB,
f (t, x, ·) is ku (t)-Lipschitz on ū(t) + δB.
Next, we present a second-order analogue of Proposition 4.3. To the best of our
knowledge, this is a new result dealing with a second-order variational equation.
Proposition 4.4 (second-order variational equation). Let (x̄, ū) be an admissible process for (PN C ) and assume that (H2) is satisfied. Let u, v ∈ L∞ ((0, 1); Rm )
be such that assumptions (a) and (b) of Proposition 4.2 are verified, and let y ∈
W 1,1 ([0, 1]; Rn ) be the solution of the linear system (3.1). For all h ∈ ]0, h0 ], let vh
be as in Proposition 4.2 and let xh ∈ W 1,1 ([0, 1]; Rn ) be the solution of
ẋh (t) = f (t, xh (t), ū(t) + hu(t) + h2 vh (t)) a.e.,
(4.1)
xh (0) = x0 .
Then,
xh − x̄ − hy unif
−→ w
h2
when h → 0+,
where w ∈ W 1,1 ([0, 1]; Rn ) is the solution of the second-order approximation (3.6).
Proof. We define zh ∈ W 1,1 ([0, 1]; Rn ) by
zh (t) = x̄(t) + hy(t) + h2 w(t)
and Rh : [0, 1] → Rn by
(4.2)
Rh (t) = żh (t) − f (t, x̄(t) + hy(t) + h2 w(t), ū(t) + hu(t) + h2 vh (t)).
Step 1. We prove that ∀h > 0 small enough, the map Rh (·)/h2 is integrably
bounded by a mapping independent from h.
In order to simplify the notation, we define
f [t, h, τ ] := f (t, x̄(t) + τ y(t) + τ 2 w(t), ū(t) + τ u(t) + τ 2 vh (t)).
fx [t, h, τ ], fu [t, h, τ ], fxx [t, h, τ ], fxu [t, h, τ ], and fuu [t, h, τ ] are defined in a similar
way.
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1159
Before we start the actual estimation of Rh (·), observe that
d
f [t, h, τ ] = fx [t, h, τ ](y(t) + 2τ w(t)) + fu [t, h, τ ](u(t) + 2τ vh (t)).
dτ
Therefore,
h
fx [t, h, s] (y(t) + 2sw(t)) + fu [t, h, s] (u(t) + 2svh (t)) ds = f [t, h, h] − f [t].
(4.3)
0
Also,
d
fx [t, h, τ ]y(t) = fxx [t, h, τ ]y(t) (y(t) + 2τ w(t)) + fxu [t, h, τ ]y(t) (u(t) + 2τ vh (t)) ,
dτ
d
fu [t, h, τ ]u(t) = fuu [t, h, τ ]u(t) (u(t) + 2τ vh (t)) + fux [t, h, τ ]u(t) (y(t) + 2τ w(t)) .
dτ
Thus,
s
fxx [t, h, τ ]y(t) (y(t) + 2τ w(t)) + fxu [t, h, τ ]y(t) (u(t) + 2τ vh (t)) dτ
0
(4.4)
= fx [t, h, s]y(t) − fx [t]y(t),
s
0
fuu [t, h, τ ]u(t) (u(t) + 2τ vh (t)) + fux [t, h, τ ]u(t) (y(t) + 2τ w(t)) dτ
= fu [t, h, s]u(t) − fu [t]u(t).
From (4.3) and (4.4) we deduce that
Rh (t) = h2 ẇ(t) −
h
−
0
0
0
0
h
−
s
0
h
fx [t, h, s]2sw(t) + fu [t, h, s]2svh (t) ds
fxx [t, h, τ ]y(t) (y(t) + 2τ w(t)) + fxu [t, h, τ ]y(t) (u(t) + 2τ vh (t)) dτ ds
s
fuu [t, h, τ ]u(t) (u(t) + 2τ vh (t)) + fux [t, h, τ ]u(t) (y(t) + 2τ w(t)) dτ ds.
Using this, assumption (H2), the fact that max{w∞ , u∞ , 2v∞ + c} < ∞, and
the fact that vh ∞ ≤ 2v∞ + c ∀h > 0 small enough, we deduce that for some
C > 0 independent from t and ∀h > 0 small enough,
|Rh (t)| ≤ h2 |ẇ(t)| + h2 C(b1 (t) + b2 (t)),
and consequently we have shown that ∀h > 0 small enough, |Rh (·)|/h2 is integrably
bounded by a function that is independent from h.
Step 2. We prove that
Rh (t) = h2 fu [t](v(t) − vh (t)) + o(t, h2 ) a.e.,
where |o(t, h2 )|/h2 → 0 when h → 0+.
Indeed, it suffices to consider the second-order Taylor development of
f (t, x̄(t) + hy(t) + h2 w(t), ū(t) + hu(t) + h2 vh (t)).
1160
DANIEL HOEHENER
Then the statement follows readily from the definitions of y and w. By Proposition 4.2
a.e.
we know that vh −→ v when h → 0+ and therefore,
|Rh (t)|
−−−−→ 0
h→0+
h2
(4.5)
a.e.
Step 3. We show that |xh (t) − zh (t)|/h2 converges uniformly to 0.
Using assumption (A2), it is not difficult to show that there exist r > 0 and
kr ∈ L1 ((0, 1); R+ ) such that
|f (t, xh (t), ū(t) + hu(t) + h2 vh (t)) − f (t, zh (t), ū(t) + hu(t) + h2 vh (t))|
≤ kr (t)|xh (t) − zh (t)|
∀h > 0 small enough for a.e. t ∈ [0, 1]. Then, since by Step 1 Rh (·) is integrable, we
deduce from (4.2) that
t
|f (t, xh (s), ū(s) + hu(s) + h2 vh (s)) − żh (t)|ds
|xh (t) − zh (t)| ≤
0
≤
t
0
kr (s)|xh (s) − zh (s)| + |Rh (s)|ds.
Hence, by the Gronwall lemma, ∃M > 0, independent from h, such that
t
|xh (t) − zh (t)| ≤ M
|Rh (s)|ds ∀t ∈ [0, 1].
0
Using this and (4.5), it follows from the first step and the Lebesgue dominated convergence theorem that
1
|Rh (t)|
|xh (t) − zh (t)|
dt ≥ lim sup
0 = lim M
2
h→0+
h→0+
h
h2
t∈[0,1]
0
xh (t) − x̄(t) − hy(t)
= lim sup − w(t) .
2
h→0+
h
t∈[0,1]
Consequently (xh − x̄ − hy)/h2 converges to w uniformly when h → 0+.
The above result answers the second question. In the state constrained case, we
are going to show that the trajectories xh constructed as in Proposition 4.4 remain in
the set K if K satisfies (SC) and if we impose that the solution w of the second-order
approximation (3.6) satisfies (TV) strictly.
Proposition 4.5. Let (x̄, ū) be a feasible process for (P ). Assume that (H2) and
(SC) are satisfied. Let u, v, y, w, and xh be as in Proposition 4.4. In addition assume
that
(4.6)
(x̄(t))
y(t) ∈ TK
∀t ∈ [0, 1]
and that ∀i ∈ {1, . . . , r} and ∀t ∈ [0, 1] such that
gi (x̄(t)) = 0
and
∇gi (x̄(t)), y(t) = 0,
we have
(4.7)
1
∇gi (x̄(t)), w(t) + gi (x̄(t))y(t)y(t) + ci (t) < 0,
2
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1161
where ci (·) is defined as in section 3.3. Then there exists η > 0 such that ∀h ∈ ]0, η],
xh (t) ∈ K ∀t ∈ [0, 1].
Proposition 4.5 can be proved using relation (4.13) below and [42, Lemma 3.3]
(see also [33] and [41]). For the sake of completeness we provide its direct proof.
Proof. It suffices to prove that ∀i ∈ {1, . . . , r}, there exists η > 0 such that
∀h ∈ ]0, η],
gi (xh (t)) ≤ 0 ∀t ∈ [0, 1].
(4.8)
Therefore we fix an arbitrary i ∈ {1, . . . , r} and prove that (4.8) is satisfied.
We start by stating some implications of the assumptions and introducing some
unif
notation. First, note that since xh −→ x̄, when h → 0+, for some constant C > 0,
C := (x̄∞ + C) B xh (t) ∀t ∈ [0, 1]
for all small h > 0. Since C is compact and gi ∈ C 2 (Rn ; R), gi is Lipschitz on C with
Lipschitz constant Lgi .
Define BT i := {t ∈ [0, 1] | gi (x̄(t)) = 0}. By Remark 3.4 and (4.6), ∀t ∈ BT i ,
∇gi (x̄(t)), y(t) ≤ 0.
(4.9)
As in the proof of Proposition 4.4, we define
zh (t) = x̄(t) + hy(t) + h2 w(t)
(4.10)
∀t ∈ [0, 1].
unif
Obviously, zh −→ x̄ when h → 0+. Hence, we have that zh (t) ∈ C ∀t ∈ [0, 1] and for
all small h > 0. Therefore, the following inequality holds:
(4.11)
gi (xh (t)) ≤ gi (zh (t)) + Lgi zh − xh ∞ .
Also, by Proposition 4.4,
(4.12)
zh − xh ∞ /h2 → 0 when h → 0 + .
Thus, using the Taylor formula and (4.10)–(4.12), we obtain that ∀h > 0 small enough
and ∀t ∈ [0, 1],
(4.13)
gi (xh (t)) ≤ gi (x̄(t)) + h∇gi (x̄(t)), y(t)
1 2
+h
g (x̄(t))y(t)y(t) + ∇gi (x̄(t)), w(t) + o(h2 ),
2 i
where o(h2 )/h2 → 0, when h → 0+ and o(h2 ) is independent from t.
Our first goal is to show that (4.8) holds on (BT i + 1 B̊) ∩ [0, 1] for some 1 > 0.
We consider t ∈ N := {s ∈ BT i | ∇gi (x̄(s)), y(s) = 0}. We claim that there exist
t > 0 and ht > 0 such that ∀h ∈ ]0, ht ],
(4.14)
gi (xh (s)) ≤ 0 ∀s ∈ ]t − t , t + t [ ∩ [0, 1].
Indeed, assume to the contrary that the claim is false. Then there exist hn > 0
converging to 0 and a sequence S := {sn }n∈N converging to t when n → ∞ such that
(4.15)
gi (xhn (sn )) > 0 ∀n.
1162
DANIEL HOEHENER
We distinguish two cases. First assume that the sequence S has a subsequence
S̃ := {s̃n } with corresponding subsequence {h̃n } such that ∀s ∈ S̃,
gi (x̄(s)) < 0 and ∇gi (x̄(s)), y(s) > 0.
(4.16)
We show that this leads to a contradiction with (4.15). For all s ∈ S̃ and all h > 0
we have
1 gi (x̄(s))y(s)y(s) + ∇gi (x̄(s)), w(s) + o(h2 )
gi (x̄(s)) + h∇gi (x̄(s)), y(s) + h2
2
1 o(h2 )
∇gi (x̄(s)), y(s)2
gi (x̄(s))y(s)y(s) + ∇gi (x̄(s)), w(s) −
+
= h2
2
4gi (x̄(s))
h2
2
h∇gi (x̄(s)), y(s)
.
+ gi (x̄(s)) 1 +
2gi (x̄(s))
≤0
Thus, it follows from (4.13) that ∀s ∈ S˜ and ∀h > 0 small enough,
gi (xh (s))
1 o(h2 )
∇gi (x̄(s)), y(s)2
gi (x̄(s))y(s)y(s) + ∇gi (x̄(s)), w(s) +
+
≤ h2
.
2
4|gi (x̄(s))|
h2
Then, by (4.7), there exists δ̃ > 0 such that for some ñ0 ∈ N,
∀n ≥ ñ0
gi (xh (s̃n )) ≤ h2 −δ̃ + o(h2 )/h2
∀h > 0 small enough. Let ñ1 ∈ N be such that o(h̃2n )/h̃2n ≤ δ̃/2 ∀n ≥ ñ1 . Then,
gi (xh̃n (s̃n )) ≤ −h̃2n
δ̃
≤0
2
∀n ≥ max{ñ0 , ñ1 },
which contradicts (4.15).
Consider now the case when (4.16) is not satisfied for s = sn and all large n.
Then there exists a subsequence Ŝ := {ŝn } such that ∀s ∈ Ŝ either gi (x̄(s)) = 0 and
thus by (4.9), ∇gi (x̄(s)), y(s) ≤ 0 or gi (x̄(s)) < 0 and ∇gi (x̄(s)), y(s) ≤ 0. In both
cases it is obvious that ∀s ∈ Ŝ,
gi (x̄(s)) + h∇gi (x̄(s)), y(s) ≤ 0
∀h > 0.
Hence by (4.13) ∀h > 0 small enough,
gi (xh (s)) ≤ h
2
1 o(h2 )
gi (x̄(s))y(s)y(s) + ∇gi (x̄(s)), w(s) +
2
h2
∀s ∈ Ŝ.
Finally, using (4.7), continuity, and the fact that ci (t) ≥ 0, we find that there exists
δ̂ > 0 such that for some n̂0 ∈ N,
gi (xh (ŝn )) ≤ h2 −δ̂ + o(h2 )/h2
∀n ≥ n̂0
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1163
∀h > 0 small enough. Analogously to the previous case, we denote the subsequence
corresponding to Ŝ by {ĥn }. Thus, choosing n̂1 > 0 such that o(ĥ2n )/ĥ2n ≤ δ̂/2 ∀n ≥
n̂1 , we obtain that
gi (xĥn (ŝn )) ≤ −ĥ2n
δ̂
≤0
2
∀n ≥ max{n̂0 , n̂1 },
which also contradicts (4.15), and consequently we have established that (4.14) holds.
Let us now consider t ∈ BT i \ N . By the definition of N and (4.9), there exists
δt > 0 such that
∇gi (x̄(t)), y(t) ≤ −δt ,
and therefore there exists t > 0 such that ∀s ∈ ]t − t , t + t [ ∩ [0, 1],
∇gi (x̄(s)), y(s) ≤ −
δt
.
2
Then, by (4.13),
gi (xh (s)) ≤ −h
δt
+ o(h)
2
∀s ∈ ]t − t , t + t [ ∩ [0, 1],
where o(h)/h → 0 when h → 0+ and o(h) is independent from s. Consequently, we
can find ht > 0 such that ∀h ∈ ]0, ht ],
(4.17)
gi (xh (s)) ≤ −
hδt
≤0
4
∀s ∈ ]t − t , t + t [ ∩ [0, 1].
Next we show that there exist 1 > 0, h1 > 0 such that (4.8) holds on (BT i +
1 B̊) ∩ [0, 1] ∀h ∈ ]0, h1 ]. Thanks to the compactness of BT i , (4.14), and (4.17), we
know that there exists a finite family T := {t1 , . . . , tq } such that
BT i ⊂
q t
t
]tk − k , tk + k [ ∩ [0, 1]
2
2
k=1
and such that ∀k ∈ {1, . . . , q},
(4.18)
gi (xh (s)) ≤ 0
tk
2
> 0 and h1 := mink htk > 0. We claim that ∀h ∈ ]0, h1 ],
gi (xh (t)) ≤ 0 ∀t ∈ BT i + 1 B̊ ∩ [0, 1].
Then we set 1 := mink
(4.19)
∀s ∈ ]tk − tk , tk + tk [, h ∈ ]0, htk ].
Indeed, let t ∈ (BT i + 1 B̊) ∩ [0, 1]. Then there exists tk ∈ T such that |t − tk | ≤
1 + tk /2 ≤ tk . Then the claim follows from (4.18), since htk ≥ h1 .
It remains to prove that (4.8) holds on M := [0, 1] \ (BT i + 1 B̊). Since M is
compact, there exists δ > 0 such that
gi (x̄(t)) ≤ −δ ∀t ∈ M.
Then, using the uniform convergence of xh to x̄, when h → 0+, we deduce the
existence of h2 > 0 such that ∀h ∈ ]0, h2 ],
xh − x̄∞ ≤
δ
.
Lgi
1164
DANIEL HOEHENER
Consequently, ∀h ∈ ]0, h2 ],
(4.20)
gi (xh (t)) ≤ gi (x̄(t)) + Lgi xh − x̄∞ ≤ 0
∀t ∈ M.
Using (4.19) and (4.20), it follows for η := min{h1 , h2 } that ∀h ∈ ]0, η], (4.8)
holds.
5. First-order necessary conditions. In this section we provide a proof of
Theorem 3.4. Further, we will use this theorem to deduce the relation (1.3). Finally,
we state a condition similar to (1.3) for problem (PN C ).
We start with the proof of Theorem 3.4. All notation and definitions are as in
section 3.3.
Proof of Theorem 3.4. We start by proving a (not necessarily normal) maximum
principle and then deduce normality from a theorem due to Bettiol and Frankowska
[4].
Let us define g : Rn → R by
g(x) =
max
i∈{1,...,r}
gi (x)
∀x ∈ Rn .
With this definition, g is locally Lipschitz. For λ ≥ 0, let us define the unnormalized
Hamiltonian Hλ : [0, 1] × Rn × Rn × Rm → R as follows:
Hλ (t, x, p, u) = p, f (t, x, u) − λl(t, x, u).
Using a standard argument, we deduce from [51, Thm. 9.5.1] that there exist p ∈
W 1,1 ([0, 1]; Rn ), λ ≥ 0, a positive Radon measure μ̃, and a Borel measurable function
ν̃ : [0, 1] → Rn satisfying
(5.1)
ν̃(t) ∈ ∂g(x̄(t)),
μ̃-a.e.,
such that
(i) (p, μ̃, λ) = 0,
(ii) supp μ̃ ⊂ {t | g(x̄(t)) = 0},
(iii) −ṗ(t) = Hxλ (t, x̄(t), q̃(t), ū(t)) a.e.,
(iv) q̃(1) = 0,
(v) Hλ (t, x̄(t), q̃(t), ū(t)) = maxu∈U(t) Hλ (t, x̄(t), q̃(t), u) a.e.,
where q̃(t) = p(t) + ψ̃(t) with
ν̃(s)dμ̃(s) for t ∈ [0, 1[,
ψ̃(t) := [0,t[
ν̃(s)dμ̃(s) for t = 1.
[0,1]
Next, we show that (iii)–(v) remain true if we replace q̃(t) by q(t) = p(t) + ψ(t), where
ψ ∈ N BV ([0, 1]; Rn ) and
ν̃(s)dμ̃(s) ∀t ∈ ]0, 1].
(5.2)
ψ(t) =
[0,t]
Indeed, since ν̃ is bounded and μ̃ is a finite measure, ψ̃ is a function of bounded
variation. This implies that ψ̃ has a countable number of discontinuities. Thus we
can define
⎧
ψ̃(t)
if ψ̃ is continuous at t ∈ ]0, 1[,
⎪
⎪
⎨
ψ̃(t+) if ψ̃ is discontinuous at t ∈ ]0, 1[,
ψ(t) :=
0
if t = 0,
⎪
⎪
⎩
ψ̃(1)
if t = 1.
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1165
Note that with this definition ψ satisfies (5.2) and is of bounded variation. In addition,
it is right continuous and vanishes at zero. Hence ψ ∈ N BV ([0, 1]; Rn ) and ψ̃(t) =
ψ(t) a.e., because they differ only at a countable number of points, i.e., on a set of
zero Lebesgue measure. Therefore, if we define q(t) = p(t) + ψ(t) as above, then also
q̃(t) = q(t) a.e. and consequently (iii)–(v) are also satisfied for q.
Let us now show that there exist a Borel measurable ν : [0, 1] → Rn and a positive
Radon measure μ such that
(5.3) ν(t) ∈ NK (x̄(t)) ∩ B, μ-a.e.
and
ψ(t) =
ν(s)dμ(s) ∀t ∈ ]0, 1].
[0,t]
First, we show that (5.1) implies that
ν̃(t) ∈ NK (x̄(t)),
(5.4)
μ̃-a.e.
Indeed, it follows directly from the maximum rule [14, Prop. 2.3.12] that
∂g(x̄(t)) = co {∇gi (x̄(t)) | gi (x̄(t)) = g(x̄(t))}
∀t ∈ [0, 1].
For t ∈ [0, 1] such that g(x̄(t)) = 0, this condition can be rewritten
∂g(x̄(t)) = co {∇gi (x̄(t)) | i ∈ I(x̄(t))} ⊂ NK (x̄(t)).
The above set inclusion is a consequence of Remark 3.4. Thus, since supp μ̃ ⊂ {t ∈
[0, 1] | g(x̄(t)) = 0}, (5.4) holds true. Finally, because there exists M > 0 such that
ν̃(t) ≤ M μ̃-a.e., we define
∀t ∈ [0, 1], ν(t) =
Then ∀t ∈ ]0, 1],
ν̃(t)
,
M
and ∀S ∈ B([0, 1]), μ(S) = M μ̃(S).
ν(s)dμ(s) =
[0,t]
ν̃(s)dμ̃(s) = ψ(t),
[0,t]
and we have
ν(t) ∈ NK (x̄(t)) ∩ B,
μ-a.e.
Next, let us prove that ψ satisfies the jump conditions (3.5). By [25, Prop. 2.5], we
know that if the set-valued map t NK (x̄(t)) has a closed graph, then ψ satisfies
the jump conditions. Let us therefore prove that t NK (x̄(t)) has a closed graph.
Note that since K is sleek, the set-valued map CK (·) is lower semicontinuous and
(·) = CK (·). Now consider a sequence (tj , yj ) → (t, y) such that yj ∈ NK (x̄(tj )) ∀j.
TK
We have to show that y ∈ NK (x̄(t)).
(x̄(t)). Since TK
(·) is lower semicontinuous,
For this, consider an arbitrary v ∈ TK
for every neighborhood V of v there exists η > 0 such that ∀x ∈ K ∩ B(x̄(t), η),
TK
(x ) ∩ V = ∅.
In particular, it follows that yj , w ≤ 0 for j such that x̄(tj ) ∈ B(x̄(t), η),
yj ∈ NK (x̄(tj )), and w ∈ TK
(x̄(tj )) ∩ V . Since the neighborhood V can be chosen arbitrarily small, by continuity of the scalar product we find y, v ≤ 0.
(x̄(t)), we have established
Finally, because the above is true for arbitrary v ∈ TK
that y ∈ NK (x̄(t)) and hence NK (x̄(·)) has a closed graph.
It remains to show that λ can be chosen equal to 1. This follows directly from [4,
Cor. 3.6] if we can show that the following regularity condition is verified:
1166
DANIEL HOEHENER
For any x ∈ ∂K and > 0 there exists ηx > 0 such that ∀y, z ∈
K ∩ B(x, ηx ),
sup
n∈NK (y)∩S n−1
n, z − y ≤ |z − y|.
To prove this, let x ∈ ∂K and > 0 be fixed. Then ∀i = 1, . . . , r, by continuity of
∇gi (·), there exists an ηxi > 0 such that ∀x ∈ B(x, ηxi ) we have
|∇gi (x ) − ∇gi (x)| ≤
(5.5)
.
2
Define ηx := mini ηxi . Then, consider an arbitrary pair y, z ∈ K ∩ B(x, ηx ). We can
assume that y ∈ ∂K; otherwise NK (y) ∩ S n−1 = ∅ and the supremum equals, by
convention, −∞. Further, y = z; otherwise the statement is trivial. Fix i ∈ I(y). As
gi (y) = 0 we find
gi (z) − gi (y) ≤ 0.
(5.6)
Then, by the mean-value theorem, there exists ξ ∈ B(x, ηx ) such that
gi (z) − gi (y) = ∇gi (ξ), z − y.
(5.7)
It follows from the definition of ηx and (5.5)–(5.7) that
∇gi (y), z − y ≤ ∇gi (y) − ∇gi (ξ), z − y ≤ |z − y|.
This allows us to conclude, since NK (y) = R+ co {∇gi (y) | i ∈ I(y)} and 0 ∈
/
co {∇gi (y) | i ∈ I(y)}. The proof is complete.
The remainder of this section is dedicated to statement (1.3) from the introduction. Let ū ∈ M([0, 1]; Rm ) and u ∈ L∞ ((0, 1); Rm ). The directional derivative of
J(·) at ū in the direction u is defined by
J(ū + hu) − J(ū)
.
h→0+
h
J (ū)u := lim
Further, the set of tangent directions A(x̄, ū) for problem (P ) at (x̄, ū) is defined as
in section 3.3. We have the following corollary of Theorem 3.4.
Corollary 5.1. Let x0 ∈ Int K and let (x̄, ū) be a strong local minimizer
for problem (P ). Suppose that (SC) and (IP) are satisfied, f satisfies (H1)(a), and
l satisfies the same assumptions as f in (H1)(a). If x̄ is Lipschitz continuous and
p, μ, ν, ψ are as in Theorem 3.4, then ∀(y, u) ∈ A(x̄, ū),
1
Hu (t, x̄(t), p(t) + ψ(t), ū(t))u(t)dt −
ν(t), y(t)dμ(t) ≥ 0.
J (ū)u = −
0
[0,1]
Proof. It suffices to show that
1
Hu (t, x̄(t), p(t) + ψ(t), ū(t))u(t)dt −
J (ū)u = −
0
[0,1]
ν(t), y(t)dμ(t).
a.e.
For h > 0, let uh be such that {uh } is bounded in L∞ ((0, 1); Rm ) and uh −→ u. By
Proposition 4.1 such sequence uh does exist. Then, by Proposition 4.3, the solution
xh ∈ W 1,1 ([0, 1]; Rn ) of
ẋh (t) = f (t, xh (t), ū(t) + huh (t)) a.e.,
xh (0) = x0
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
1167
satisfies
yh :=
xh − x̄ unif
−→ y
h
when h → 0 + .
Hence, using the above definitions and the Lipschitz continuity of f and l, we have
that ∀(y, u) ∈ A(x̄, ū),
J (ū)u = lim
(5.8)
h→0+
0
1
l(t, x̄(t) + hyh (t), ū(t) + huh (t)) − l[t] + R(h, t)
dt,
h
where for almost every t ∈ [0, 1], R(h, t)/h → 0 when h → 0+. It is not difficult to
show that the integrand on the right-hand side of (5.8) is bounded by an integrable
function that is independent of h. Thus, the Lebesgue dominated convergence theorem
applies and it remains to show that
1
l(t, x̄(t) + hyh (t), ū(t) + huh (t)) − l[t]
dt
h→0+
h
1
=−
Hu (t, x̄(t), p(t) + ψ(t), ū(t))u(t)dt −
lim
0
0
[0,1]
ν(t), y(t)dμ(t).
Indeed, this can be proved using the definition of the Hamiltonian, the Taylor formula,
and integration by parts. See the beginning of the proof of Theorem 3.5 in section 6
for details.
It is obvious that the same line of argument leads to the following statement for
the case of problem (PN C ).
Corollary 5.2. Let (x̄, ū) be a strong local minimizer for problem (PN C ). Assume that (H1)(a)–(b) are satisfied and that l satisfies the same assumptions as f in
(A1), (A2), and (H1)(a)–(b). Let p ∈ W 1,1 ([0, 1]; Rn ) be as in the maximum principle.
Then ∀u ∈ L∞ ((0, 1); Rm ) satisfying u(t) ∈ TU(t)
(ū(t)) a.e.,
J (ū)u = −
0
1
Hu (t, x̄(t), p(t), ū(t))u(t)dt ≥ 0.
6. Second-order necessary conditions. In this section, we provide proofs of
Theorems 3.2 and 3.5. We start with the proof of Theorem 3.2. We use the same
notation as in section 3.1. In addition, we introduce the critical set CN C (x̄, ū) for
problem (PN C ) at the process (x̄, ū) which is defined as follows:
CN C (x̄, ū) :=
(y, u, v) ∈ W 1,1 ([0, 1]; Rn ) × UA × L∞ ((0, 1); Rm ) (3.1) and (3.2) are satisfied and
0
1
Hu [t]u(t)dt = 0 .
The critical set has the property that if (y, u, v) ∈ CN C (x̄, ū) and λ > 0, then
(λy, λu, λ2 v) ∈ CN C (x̄, ū).
Proof of Theorem 3.2. Since u(t) ∈ TU(t)
(ū(t)) a.e., by Corollary 5.2 we know
1
that 0 Hu [t]u(t)dt ≤ 0. Thus, all we have to prove is that the inequality (3.3) holds
∀(y, u, v) ∈ CN C (x̄, ū). Fix (y, u, v) ∈ CN C (x̄, ū).
1168
DANIEL HOEHENER
First step. Let χA denote the characteristic function of A. We prove that ∀h > 0,
a.e.
a.e.
there exist uh , vh ∈ L∞ ((0, 1); Rm ) such that uh −→ u, vh −→ χA v when h → 0+
and such that ∀h > 0 small enough,
(6.1)
(6.2)
ū(t) + huh (t) + h2 vh (t) ∈ U (t)
uh ∞ ≤ 2u∞ ,
(6.3)
vh ∞ ≤ 2v∞ + c.
a.e.,
(ū(t)) a.e., there exists a sequence ûh ∈
By Proposition 4.1, since u(t) ∈ TU(t)
L∞ ((0, 1); Rm ), converging a.e. to u, when h → 0+ and such that ∀h > 0,
ū(t) + hûh (t) ∈ U (t) for a.e. t ∈ [0, 1],
ûh ∞ ≤ 2u∞.
With this, one defines uh as follows:
u(t)
uh (t) :=
ûh (t)
if t ∈ A,
otherwise.
(2)
Further, since 0 ∈ TU(t) (ū(t), 0) a.e. and u ∈ UA , the pair (χA u, χA v) satisfies
all the assumptions of Proposition 4.2. Hence, there exists a sequence of essentially
bounded maps vh such that ∀h > 0 small enough
a.e.
vh −→ χA v,
vh ∞ ≤ 2v∞ + c,
ū(t) + hu(t) + h2 vh (t) ∈ U (t)
for a.e. t ∈ A.
Hence, vh , uh satisfy (6.1)–(6.3).
Second step. We prove that ∀h > 0, the solution xh ∈ W 1,1 ([0, 1]; Rn ) of
ẋh (t) = f (t, xh (t), ū(t) + huh (t) + h2 vh (t)) a.e.,
xh (0) = x0 ,
satisfies
xh − x̄ unif
−→ y
h
(6.4)
when h → 0 + .
To show this, we set ũh := uh + hvh . Then, ∀h > 0 small enough, the sequence
{ũh }h>0 satisfies the hypotheses of Proposition 4.3. Hence, we can deduce (6.4).
Third step. We show that ∀h > 0,
(6.5) J(ū + huh + h2 vh ) − J(ū)
1
Hx [t]hyh (t) + H[t] − H(t, xh (t), p(t), ū(t) + huh (t) + h2 vh (t)) dt,
=
0
where yh := (xh − x̄)/h. This is a direct consequence of integration by parts of
1
p(t), f (t, xh (t), ū(t) + huh (t) + h2 vh (t)) − f [t]dt,
0
and the maximum principle (Theorem 3.1).
1169
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
unif
Last step. Since xh −→ x̄, when h → 0+ and x̄ is a strong local minimizer, we
deduce from (6.5) that
(6.6)
0≤
1
0
Hx [t]hyh (t) + H[t] − H(t, xh (t), p(t), ū(t) + huh (t) + h2 vh (t)) dt
∀h > 0 small enough. Then we use the Taylor formula and the results of the second
and the third steps in order to deduce (3.3) from (6.6). This is very similar to what
is done in the proof of Theorem 3.5 below and therefore we omit the details of this
step.
Next we are going to prove Theorem 3.5. The proof is based on the first-order conditions from Theorem 3.4 and the second-order variational equation, Proposition 4.5.
We use the same notation as in section 3.3. In addition, we define the set of critical
directions C(x̄, ū) of problem (P ) at the process (x̄, ū) by
C(x̄, ū) :=
(y, u) ∈ A(1) (x̄, ū) 0
1
Hu [t]u(t)dt +
[0,1]
ν(t), y(t)dμ(t) = 0 ,
where μ and ν are as in the maximum principle (Theorem 3.4). Finally, let us recall
the following integration by parts formula, which follows from [10, Thm. 6]: If f ∈
W 1,1 ([0, 1]; Rn ) and ϕ ∈ N BV ([0, 1]; Rn ), then
[0,1]
f (t)dϕ(t) = f (1), ϕ(1) −
0
1
ϕ(s), f (s)ds.
Proof of Theorem 3.5. Fix (y, u) ∈ C(x̄, ū). Note that it follows from the maximum
principle that
Hu [t]u(t) = 0
(6.7)
a.e.
and
ν(t), y(t) = 0
μ-a.e.
Let (w̃, ṽ) ∈ A(2) (x̄, ū, y, u) be such that w̃ satisfies (TV) strictly. First we show
that (3.8) holds for (w̃, ṽ). By Propositions 4.4 and 4.5, for h > 0, there exist w̃h ∈
unif
a.e.
W 1,1 ([0, 1]; Rn ) and ṽh ∈ L∞ ((0, 1); Rm ) such that w̃h −→ w̃ and ṽh −→ ṽ when
h → 0+ and ∀h > 0 small enough,
(x̃h , ũh ) := (x̄ + hy + h2 w̃h , ū + hu + h2 ṽh )
is an admissible process of (P ). Since (x̄, ū) is a strong local minimizer, ∀h > 0 small
enough, the inequality
J(ũh ) − J(ū) ≥ 0
holds true. Using the very definition of the Hamiltonian, this can be reformulated as
follows:
1
(6.8)
0
H[t] − H(t, x̃h (t), q(t), ũh (t)) + q(t), f (t, x̃h (t), ũh (t)) − f [t] dt ≥ 0.
1170
DANIEL HOEHENER
Integrating by parts, we find
1
p(t), f (t, x̃h (t), ũh (t)) − f [t]dt
0
= p(1), hy(1) + h2 w̃h (1) +
(6.9)
0
1
1
0
ψ(t), f (t, x̃h (t), ũh (t)) − f [t]dt
= ψ(1), hy(1) + h2 w̃h (1) −
Hx [t] hy(t) + h2 w̃h (t) dt,
[0,1]
ν(t), hy(t) + h2 w̃h (t)dμ(t).
Then, since p(1) + ψ(1) = q(1) = 0, we deduce from (6.7)–(6.9) that
(6.10)
0
1
H[t] + Hx [t] hy(t) + h2 w̃h (t) − H(t, x̃h (t), q(t), ũh (t)) dt
−
ν(t), h2 w̃h (t)dμ(t) ≥ 0
[0,1]
∀h > 0 small enough.
Next, we are going to show that the map
1
(H(t, x̃h (t), q(t), ũh (t)) − H[t] − Hx [t]hy(t))
h2
is integrably bounded by a map which does not depend from h. In order to do this,
we use the same principle as in the proof of Proposition 4.4. Set
H[t, τ, h] := H(t, x̄(t) + τ y(t) + τ 2 w̃h (t), q(t), ū(t) + τ u(t) + τ 2 ṽh (t)).
Hx [t, τ, h], Hu [t, τ, h], Hxx [t, τ, h], etc., are defined in a similar way. This leads to the
following:
1
(H(t, x̃h (t), q(t), ũh (t)) − H[t] − Hx [t]hy(t))
h2
h
1
= 2
(Hx [t, s, h]2sw̃h (t) + Hu [t, s, h]2sṽh (t)) ds
h 0
h s
1
+ 2
Hxx [t, τ, h]y(t) (y(t) + 2τ w̃h (t)) + Hxu [t, τ, h]y(t) (u(t) + 2τ ṽh (t)) dτ ds
h 0 0
h s
1
+ 2
Huu [t, τ, h]u(t) (u(t) + 2τ ṽh (t)) + Hux [t, τ, h]u(t) (y(t) + 2τ w̃h (t)) dτ ds.
h 0 0
Then, using the assumptions, it is not difficult to see that the map on the right-hand
side is integrably bounded by a function independent from h. Furthermore, by the
Taylor formula we have
H[t, h, h] − H[t] − Hx [t] hy(t) + h2 w̃h (t)
1
= Hu [t]h2 ṽh (t) + Hxx [t]h2 y(t)y(t) + Hxu [t]h2 y(t)u(t)
(6.11)
2
1
+ Huu [t]h2 u(t)u(t) + Rh (t),
2
1171
SECOND-ORDER NECESSARY OPTIMALITY CONDITIONS
where Rh (t)/h2 → 0 when h → 0+. Therefore, since w̃h converges uniformly to w̃
when h → 0+, we can apply Lebesgue’s dominated convergence Theorem to (6.10):
1
1 lim 2 H[t] + Hx [t] hy(t) + h2 w̃h (t) − H[t, h, h] dt
0 h→0+ h
−
lim ν(t), w̃h (t)dμ(t)
(6.12)
1
h→0+ h2
= lim
0
[0,1] h→0+
1
H[t] + Hx [t] hy(t) + h2 w̃h (t) − H[t, h, h] dt
−
2
[0,1]
ν(t), h w̃h (t)dμ(t)
≥ 0.
Consequently, (3.8) can be deduced from (6.11) and (6.12).
(2)
Now let D(t) ⊂ TU(t) (ū(t), u(t)) be convex subsets such that ṽ(t) ∈ D(t) for
almost every t ∈ [0, 1] and consider (w, v) ∈ A(2) (x̄, ū, x, u) with v(t) ∈ D(t) a.e. For
λ ∈ ]0, 1[, we define
wλ := λw̃ + (1 − λ)w
and v λ := λṽ + (1 − λ)v.
It is obvious that (wλ , v λ ) ∈ A(2) (x̄, ū, y, u) and wλ satisfies (TV) strictly. Hence,
∀λ ∈ ]0, 1[,
(6.13)
[0,1]
wλ (t), ν(t)dμ(t) +
+
0
1
0
1
Hu [t]v λ (t)dt
1
1
Hxx [t]y(t)y(t) + Hxu [t]y(t)u(t) + Huu [t]u(t)u(t) dt ≤ 0.
2
2
unif
a.e.
Finally, since obviously wλ −→ w and v λ −→ v when λ → 0 and {v λ }λ is bounded in
L∞ ((0, 1); Rm ), we deduce (3.8) from (6.13) by taking the limit when λ →
0+.
Acknowledgments. The author is grateful to H. Frankowska for providing very
helpful advice and to anonymous referees for their constructive comments.
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