Admissible directions in optimal control

under uniqueness assumptions

JAVIER FROSENBLUETH

IIMAS, Universidad Nacional Aut´

onma de M´

exico

Apartado Postal 20-126, CDMX 01000

MEXICO

Abstract: It is well-known that, for a mathematical programming problem involving equality and inequality con-

straints, the uniqueness of a Lagrange multiplier associated with a local solution implies, under certain smoothness

assumptions, second order necessary optimality conditions. Those conditions hold on a set of critical directions de-

ﬁned by those points satisfying the constraints and for which the minimizing function and the standard Lagrangian

coincide. No similar links between uniqueness of multipliers and second order conditions seem to have been

established for optimal control problems. In this paper, we provide some results in this direction. In particular,

we study and completely solve a natural conjecture which provides, under uniqueness assumptions, nonnegative

second variations on a classical cone of admissible directions.

Key–Words: Optimal control, uniqueness of Lagrange multipliers, second order conditions, normality

Received: January 10, 2023. Revised: September 13, 2023. Accepted: October 16, 2023. Published: November 15, 2023.

1 Introduction

It is well-known that, for a mathematical program-

ming problem involving equality and inequality con-

straints, a second order necessary condition for a local

solution x0holds, under certain smoothness assump-

tions, on the tangent cone TS1(x0)at x0of the set S1

of points satisfying the constraints and for which the

minimizing function and the standard Lagrangian co-

incide. Thus, if x0is a regular point of S1(in the sense

that the tangent cone and the set of tangential con-

straints coincide), the second order condition holds on

the set RS1(x0)of tangential constraints (or critical

directions) at x0with respect to S1.

Regularity can be achieved in different ways. In

particular it holds if there is only one Lagrange multi-

plier associated with the local minimizer. A simple

line of reasoning, to support this statement, can be

given as follows. First of all, the uniqueness of the

multiplier is equivalent to the Mangasarian-Fromovitz

constraint qualiﬁcation with respect to S1at x0. Sec-

ond, this constraint qualiﬁcation is equivalent to the

condition of normality of x0relative to the set S1.

Finally (and this is a crucial result in the theory of

mathematical programming), normality implies regu-

larity. Thus, if the set of Lagrange multipliers associ-

ated with x0is a singleton, then x0is a regular point

of S1and a second order condition holds on RS1(x0).

For clarity of exposition, and for comparison rea-

sons, we shall ﬁnd convenient to “unfold” in the next

few lines some of these concepts, main characters,

statements and implications.

Consider the nonlinear programming problem,

which we label (N), of minimizing fon the set S,

where f, gi:Rn→R(i∈A∪B)are given func-

tions, A={1, . . . , p},B={p+ 1, . . . , m}, and

S:= {x∈Rn|gα(x)≤0 (α∈A),

gβ(x) = 0 (β∈B)}.

We assume, as in [1], [2], [3], that the functions delim-

iting the problem are continuously differentiable and,

when second derivatives occur, they are twice contin-

uously differentiable (for weaker assumptions see, for

example, [4], [5], [6], [7]).

Let us begin with the Karush-Kuhn-Tucker

(KKT) conditions or ﬁrst order Lagrange multiplier

rule. Denote by Λ(f, x0)the set of all λ=

(λ1, . . . , λm)∈Rmsatisfying

i. λα≥0and λαgα(x0) = 0 (α∈A).

ii. If F(x) := f(x) + hλ, g(x)ithen F0(x0) = 0.

Here, the function Fis the standard Lagrangian,

gis the function mapping Rnto Rmwhose compo-

nents are g1, . . . , gm,hλ, g(x)i=Pm

1λigi(x)is the

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standard inner product, and λ1, . . . , λmare the Kuhn-

Tucker or Lagrange multipliers.

The KKT conditions become ﬁrst order neces-

sary conditions for a local solution if a certain con-

straint qualiﬁcation is imposed. In other words, if x0

is a local solution to the problem, the nonemptiness of

Λ(f, x0)can be assured if some extra assumption on

x0and the constraints is added to the hypothesis.

Denote by

I(x) := {α∈A|gα(x)=0}(x∈S)

the set of active (or effective or binding)indices at x.

From the theory of convex cones (see, for example,

[6], [7]) or using the Farkas-Minkowski theorem of

the alternative (see [5]), it can be shown that, if

RS(x0) := {h∈Rn|g0

α(x0;h)≤0 (α∈I(x0)),

β(x0;h) = 0 (β∈B)}

denotes the set of vectors satisfying the tangential

constraints at x0(see [6], [7]), also called the lin-

earized tangent cone or the cone of locally con-

strained directions (see [5]), then

Λ(f, x0)6=∅ ⇔ f0(x0;h)≥0for all h∈RS(x0)

or, equivalently, Λ(f, x0)6=∅⇔−f0(x0)∈R∗

S(x0)

where B∗:= {z∈Rn| hy, zi ≤ 0for all y∈B}is

the (closed convex) dual or polar cone of B⊂Rn.

One can ﬁnd in the literature different answers to

the question of how, if x0is a local minimum, the re-

lation −f0(x0)∈R∗

S(x0)can be assured. In other

words, there are different assumptions on the con-

straints which ensure that the condition −f0(x0)∈

R∗

S(x0)is a necessary optimality condition for our

problem. Not all constraint qualiﬁcations coincide,

indeed, and a rather intricate web of implications and

equivalences has been established (see, for example,

[5]).

We shall ﬁrst give to that question a simple an-

swer (that is, a constraint qualiﬁcation) by making use

of the tangent cone. Moreover, as we shall see, this

approach will lead us also to the derivation of second

order conditions.

The deﬁnition we choose of tangent cone is the

one given by Hestenes in [6]. As shown in [5], it is

equivalent to the one introduced by Bouligand (also

known as contingent cone). A sequence {xq} ⊂ Rn

is said to converge to x0in the direction hif his a unit

vector, xq6=x0, and

lim

q→∞ |xq−x0|= 0,lim

q→∞

xq−x0

|xq−x0|=h.

The tangent cone of Sat x0, denoted by TS(x0), is

the (closed) cone determined by the unit vectors hfor

which there exists a sequence {xq}in Sconverging to

x0in the direction h. Equivalently (see [7]), TS(x0)is

the set of all h∈Rnfor which there exist sequences

{xq}in Sand {tq}of positive numbers, such that

lim

q→∞ tq= 0,lim

q→∞

xq−x0

=h.

Note that, if {xq}converges to x0in the direction

hand fhas a differential at x0, then

lim

q→∞

f(xq)−f(x0)

|xq−x0|=f0(x0;h).

Also, if fhas a second differential at x0, then

lim

q→∞

f(xq)−f(x0)−f0(x0;xq−x0)

|xq−x0|2=1

2f00(x0;h).

Suppose that x0solves (N) locally, h∈TS(x0)is a

unit vector, and {xq} ⊂ Sis a sequence converging to

x0in the direction h. Hence, f(xq)≥f(x0)for large

values of qand, therefore,

0≤lim

q→∞

f(xq)−f(x0)

|xq−x0|=f0(x0;h).

If also f0(x0)=0, then

0≤lim

q→∞

f(xq)−f(x0)

|xq−x0|2=1

2f00(x0;h).

This proves the following basic result on ﬁrst and

second order necessary conditions (see, for example,

[5, p 303], [6, p 27-28], [7, p 214, 220]).

Theorem 1. Suppose x0solves (N) locally. Then

the following holds.

a. f0(x0;h)≥0for all h∈TS(x0).

b. If f0(x0) = 0, then f00(x0;h)≥0for all h∈

TS(x0).

The dual cone T∗

S(x0)of the tangent cone of S

at x0is called the normal cone of Sat x0. By the

ﬁrst part of Theorem 1, if x0is a local minimum point

of a C1function fon a set S(actually, merely dif-

ferentiability at x0is required) then the negative gra-

dient −f0(x0)is an outer normal of Sat x0, that is,

−f0(x0)∈T∗

S(x0).

Note that TS(x0)⊂RS(x0)since, if h∈TS(x0)

is a unit vector, and {xq} ⊂ Sa sequence converging

to x0in the direction h, then

lim

q→∞

gγ(xq)−gγ(x0)

|xq−x0|=g0

γ(x0;h) (γ∈A∪B)

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and, therefore, g0

α(x0;h)≤0 (α∈I(x0)) and

β(x0;h) = 0 (β∈B). Of course, the converse may

not hold. For example, if S={x:x3≤0}, then

TS(0) = {h:h≤0}but RS(0) = R.

Hence, we have R∗

S(x0)⊂T∗

S(x0). If these two

dual cones coincide and x0solves (N) locally, then

−f0(x0)∈T∗

S(x0) = R∗

S(x0)

and so Λ(f, x0)6=∅. The condition

R∗

S(x0) = T∗

S(x0)

is referred to in [7] as quasi-regularity and in [5] as the

Guignard-Gould-Tolle constraint qualiﬁcation. Ac-

cording to [5, p 264], it “enjoys the best situation

with respect to the problem, being the most general

constraint qualiﬁcation for the feasible set of problem

(N).”

The notion of “regularity” as deﬁned in [6, p 35],

[7, p 221], known also as Abadie’s constraint qualiﬁ-

cation (see, for example, [5]), is a crucial one. A point

x0∈Swill be said to be regular with respect to Sif

TS(x0)and RS(x0)coincide. Clearly, regularity im-

plies quasi-regularity, but one may ﬁnd quasi-regular

points of Swhich are not regular. These cases, how-

ever, are exceptional. To give a simple example (see

[7]), this occurs with the origin with respect to the set

S={(x, y) : x≤0, x +y≤0, x2=y2}.

The same arguments given above yield the fol-

lowing result, the ﬁrst order Lagrange multiplier rule

for a regular solution. It is a direct consequence of

Theorem 1(a) and the deﬁnition of regularity.

Theorem 2. If x0solves (N) locally and is a reg-

ular point of S, then Λ(f, x0)6=∅.

The second order Lagrange multiplier rule is a

straightforward consequence of Theorem 1(b). In

what follows, F(x) = f(x) + hλ, g(x)idenotes (as

before) the Lagrangian with respect to λ∈Λ(f, x0).

Theorem 3. Suppose that x0∈Sand λ∈

Λ(f, x0). If x0solves (N) locally, then F00(x0;h)≥0

for all h∈TS1(x0)where S1:= {x∈S|F(x) =

f(x)}. In particular, if x0is a regular point of S1, then

F00(x0;h)≥0for all h∈RS1(x0).

Note that, if λ∈Λ(f, x0)and Γ := {α∈A|

λα>0}, then

S1[ = S1(λ) ]

={x∈Rn|gα(x)≤0 (α∈A, λα= 0),

gβ(x) = 0 (β∈Γ∪B)}

={x∈S|gα(x) = 0 (α∈Γ)}.

Therefore, by deﬁnition of tangential constraints, we

have

RS1(x0)

={h∈Rn|g0

α(x0;h)≤0 (α∈I(x0), λα= 0),

β(x0;h) = 0 (β∈Γ∪B)}

={h∈RS(x0)|g0

α(x0;h) = 0 (α∈Γ)}

={h∈RS(x0)|f0(x0;h)=0}.

In general, it may be difﬁcult to test for quasi-

regularity or even regularity, and some criteria imply-

ing these conditions is required. As pointed out in [6],

“it is customary in the calculus of variations to call a

condition on the gradients g0

1(x0), . . . , g0

m(x0)anor-

mality condition if it implies regularity at x0.”

In [7], a point x0∈Sis said to be normal relative

to Sif λ= 0 is the only solution of

i. λα≥0and λαgα(x0) = 0 (α∈A).

ii. Pm

1λig0

i(x0)=0.

Normality implies regularity. A proof of this fact

can be found in [7] where another condition, called

properness, is used to establish this implication. A

point x0∈Sis said to be proper relative to Sif the

set {g0

β(x0)|β∈B}is linearly independent and, if

p > 0, there exists hsuch that

α(x0;h)<0 (α∈I(x0)), g0

β(x0;h) = 0 (β∈B).

As shown in [7, p 241], properness and normality are

equivalent. This is also proved in [5, p 256] by using

theorems of alternative (see Motzkin in [5, Theorem

2.4.19]), and in [4, p 43] for inequality constraints, in

terms of positive linear independence, by an applica-

tion of the Hahn-Banach separation theorem.

Since normality is equivalent to properness and

they imply regularity, Theorem 2 tells us that both

are constraint qualiﬁcations. They are also referred

to in the literature as the Cottle-Dragomirescu and

the Mangasarian-Fromovitz constraint qualiﬁcations

respectively. In view of Theorem 2, we have the fol-

lowing classical result.

Theorem 4. If x0solves (N) locally and is a nor-

mal point of S, then Λ(f, x0)6=∅.

It is important to point out that the conclusion that

normality is a constraint qualiﬁcation can be reached

without making use of Theorem 2 and the notion of

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regularity. The well-known Fritz John necessary op-

timality condition (see, for example, [1], [5], [7], [8]

and, for a nonsmooth multiplier rule, see [4]) allows

the cost multiplier to vanish. It states that, if x0solves

(N) locally, then there exist λ0≥0and λ∈Rm, not

both zero, such that

i. λα≥0and λαgα(x0) = 0 (α∈A).

ii. F0(x) := λ0f(x) + hλ, g(x)i ⇒ F0

0(x0)=0.

In view of this result, if x0solves (N) locally and

is a normal point of S, then λ0>0(since, by normal-

ity relative to S, if λ0= 0 then λ= 0, contradicting

the nontriviality condition) and the multipliers can be

chosen so that λ0= 1, thus implying nonemptiness of

Λ(f, x0).

We are now in a position to state a fundamental

result on second order necessary conditions under nor-

mality assumptions. It is an immediate consequence

of Theorem 3 and the well-known link between nor-

mality and regularity. For the application of this result

note that, given λ∈Λ(f, x0),x0is a normal point of

S1(λ)if µ= 0 is the only solution of

i. µα≥0and µαgα(x0) = 0 (α∈A, λα= 0).

ii. Pm

1µig0

i(x0) = 0.

Theorem 5. Suppose that x0∈Sand λ∈

Λ(f, x0). If x0solves (N) locally and is a normal point

of S1(λ), then F00(x0;h)≥0for all h∈RS1(x0).

Let us now introduce one of the main characters

which plays a leading role in the theory: the linear

independence constraint qualiﬁcation (LI). For a given

point x0∈S, it asks the set {g0

γ(x0)|γ∈I(x0)∪B}

to be linearly independent.

This is, by the way, the deﬁnition of normal-

ity given in [6], in contrast with the deﬁnition in [7]

which we introduced before. As one readily veriﬁes,

it corresponds to normality of x0with respect to (nei-

ther Snor S1but) the set of equality constraints for

active indices:

S0[ = S0(x0) ]

:= {x∈Rn|gγ(x) = 0 (γ∈I(x0)∪B)}.

This follows since, by deﬁnition, x∈S0(x0)is nor-

mal relative to S0(x0)if λ= 0 is the only solution

i. λαgα(x0) = 0 (α∈A).

ii. Pm

1λig0

i(x) = 0.

This is equivalent to LI, the condition that the lin-

ear equations g0

γ(x0;h) = 0 (γ∈I(x0)∪B)in hbe

linearly independent.

Note that, given x0∈Sand λ∈Rmwith λα≥0

(α∈A), we have RS0(x0)⊂RS1(x0)⊂RS(x0)

where, by deﬁnition,

RS0(x0) = {h∈Rn|g0

γ(x0;h)=0

(γ∈I(x0)∪B)}.

Also, if x0is a normal point of S0, then it is a normal

point of S1, and hence a normal point of S.

In most textbooks (see a thorough explanation

and references in [1]) the main result on second or-

der necessary conditions differs from Theorem 5 in

two fundamental aspects: it is derived under the as-

sumption of the linear independence constraint quali-

ﬁcation (that is, normality relative to S0instead of S1)

and the second order condition holds on the set of crit-

ical directions RS0(x0)(instead of RS1(x0)). Thus, in

this result, the assumptions are stronger and the con-

clusions weaker than those of Theorem 5 since, usu-

ally, normality relative to S0is stronger than normality

relative to S1and RS1(x0)contains properly the set

RS0(x0). As pointed out in [1], “The source of this

weaker result can be attributed to the traditional way

of treating the active inequality constraints as equality

constraints.” Explicitly, this rather “well-worn result”

(see [1]) is the following.

Theorem 6. Suppose x0∈Sand λ∈Λ(f, x0).

If x0solves (N) locally and is a normal point of

S0(x0), then F00(x0;h)≥0for all h∈RS0(x0).

The set of critical directions in Theorem 5 is then,

in general, bigger than that of Theorem 6. How-

ever, as the following examples show, even under the

strong assumption of Theorem 6 (normality relative

to S0(x0)or the LI constraint qualiﬁcation), the set of

critical directions cannot be “too big”: we may very

well have negative second variations on RS(x0).

Example 1. Consider the problem of minimizing

f(x1, x2) = x1subject to

g1(x1, x2) = −x2

1−x2≤0,

g2(x1, x2) = x1+x2= 0.

Clearly x0= (1,−1) is a local solution to

the problem and LI is satisﬁed since the gradients

1(x0) = (−2,−1) and g0

2(x0) = (1,1) are linearly

independent. We have

F(x1, x2) = x1+λ1(−x2

1−x2) + λ2(x1+x2)

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and so F0(x1, x2) = (1 −2λ1x1+λ2,−λ1+λ2).

Thus F0(x0)=0implies λ1=λ2= 1. Also,

F00(x1, x2) = −2λ10

0 0 =−2 0

0 0 

and so F00(x0;h, k) = −2h2. Now, we have

1(x1, x2)=(−2x1,−1), g0

1(x1, x2) = (1,1)

and so

RS(x0) = {(h, k)| −2h−k≤0, h +k= 0}.

Therefore (1,−1) ∈RS(x0)but F00(x0; 1,−1) =

−2<0.

Example 2. Consider the problem of minimizing

f(x) = −x3subject to g(x) = x−1≤0.

Clearly x0= 1 is a (global) solution and LI holds

since g0(x0)=16= 0. We have

F(x) = f(x) + λg(x) = −x3+λx −λ

and so F0(x) = −3x2+λand F0(x0)=0implies

that λ= 3. Also F00(x0;h) = −6h2. Since

RS(x0) = {h|g0(x0;h) = h≤0}

we have −1∈RS(x0)and F00(x0;−1) = −6<0.

We can now explain, in a clear and succinct way,

the role played by the uniqueness of Lagrange multi-

pliers in the intricate web of constraint qualiﬁcations.

Let us begin with a simple argument which im-

plies uniqueness. Suppose x0∈Sand λ∈Λ(f, x0).

If also ¯

λ∈Λ(f, x0), set µ:= ¯

λ−λand observe that

µαgα(x0) = ¯

λαgα(x0)−λαgα(x0) = 0 (α∈A),

µig0

i(x0) =

λig0

i(x0)−

λig0

i(x0) = 0.

Thus, if x0is normal relative to S0(x0), then µ= 0

and Λ(f, x0)would be a singleton. In other words, the

following result holds.

Theorem 7. Suppose x0∈Sand λ∈Λ(f, x0).

If x0is normal relative to S0(x0), then Λ(f, x0) =

{λ}.

In view of Theorem 4, if x0affords a local mini-

mum to fon Sand x0is normal relative to S0(and so

normal relative to S) then there exists λ∈Λ(f, x0).

We have just proved that, in this event, λ∈Λ(f, x0)

is unique. As one readily veriﬁes, if normality is as-

sumed relative to S, the existence of λ∈Λ(f, x0)can

be assured, but it may not be unique. Here we arrive

at a crucial result in the theory.

In [2] it is shown that uniqueness of the La-

grange multiplier associated with the point x0can be

achieved by an assumption weaker than that of nor-

mality relative to S0(x0), namely, normality relative

to S1(λ). Moreover, this assumption is not only sufﬁ-

cient for the uniqueness of the multiplier but also nec-

essary. This is the content of the following result.

Theorem 8. Suppose that x0∈Sand λ∈

Λ(f, x0). Then x0is normal relative to S1(λ)if and

only if Λ(f, x0) = {λ}.

In [2], the statement of this result is expressed in

terms of a condition called the strict Mangasarian-

Fromovitz constraint qualiﬁcation which is no other

than properness relative to S1(λ). The proof given

in that paper, however, relies precisely on the equiva-

lence between normality and properness.

Combining this result with Theorem 5, we ﬁnally

reach the main result on second order necessary con-

ditions under uniqueness assumptions.

Theorem 9. Suppose that x0∈Sand λ∈

Λ(f, x0). If x0solves (N) locally and Λ(f, x0) =

{λ}, then F00(x0;h)≥0for all h∈RS1(x0).

It is worth mentioning that an entirely different

approach, found in [3], is based on the idea that, since

constraint qualiﬁcations are independent of the objec-

tive function f, if a constraint qualiﬁcation implies

a certain property for the Lagrange multipliers, this

property will hold for all objective functions for which

x0affords a local minimum. If we deﬁne F(x0)as the

set of all f∈C1(Rn,R)such that x0affords a local

minimum to fon S, the result on uniqueness of La-

grange multipliers given in [3] states that, if x0∈S,

then Λ(f, x0)is a singleton for all f∈ F(x0)⇔x0

is normal relative to S0(x0).

Uniqueness of multipliers in different areas of

constrained optimization has been studied from differ-

ent points of view in order to deal with, to mention a

few, problems subject to cone constraints [9], sensitiv-

ity analysis of optimization problems [10], composite

optimization [11], sets of functions with a common

local minimum [3], or derivation of second order con-

ditions [2], [5], [6], [7], [12], [13], [14].

In this paper we shall be concerned with unique-

ness of multipliers for certain classes of optimal con-

trol problems. Our main objective will be to state the

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main aspects of the theory in that context and give

some answers to questions related to possible links

between uniqueness of multipliers, admissible direc-

tions, and second order optimality conditions.

The stage for such problems, involving equality

and inequality constraints in the control functions, is

rather different than that for the ﬁnite dimensional

case. In particular, it has been proved that the cor-

responding Mangasarian-Fromovitz constraint quali-

ﬁcation (also called properness) and the condition of

normality are equivalent [15], as in the mathemat-

ical programming problem, but these conditions do

not necessary imply regularity [16]. Moreover, the

Mangasarian-Fromovitz constraint qualiﬁcation with

respect to S1implies uniqueness of the Lagrange mul-

tiplier, but the converse may not hold [17], [18].

It is of interest to know if, for optimal control

problems, the uniqueness of Lagrange multipliers im-

plies or not second order conditions on a speciﬁc set of

critical directions. Let us move forward to that ques-

tion.

2 Statement of the problem

In the remaining of this paper we shall deal with an

optimal control problem with ﬁxed endpoints, posed

over piecewise C1trajectories and piecewise continu-

ous controls, and involving inequalities and equalities

in the control functions.

We believe this problem, though (relatively) sim-

ple compared with other formulations, captures the

essence of the question connecting uniqueness of mul-

tipliers with second order conditions. The problem is

not simple, by any means. Actually, the difﬁculties

encountered for this kind of optimal control problems

are of a much subtler nature than those for (N). As ex-

plained in [4, p 335], the type of constraints we shall

now deal with (even for the classical Lagrange prob-

lem in the calculus of variations with equality con-

straints) “make the constrained optimal control prob-

lem much more complex than (N), or even the isoperi-

metric problem. In part, this is because we now have

inﬁnitely many constraints, one for each t.”

To state the problem, suppose we are given an in-

terval T:= [t0, t1]in R, two points ξ0,ξ1in Rn, and

functions Land fmapping T×Rn×Rmto Rand

Rnrespectively, and ϕ= (ϕ1, . . . , ϕq)mapping Rm

to Rq. Denote by Xthe space of piecewise C1func-

tions mapping Tto Rn, and by Ukthe space of piece-

wise continuous functions mapping Tto Rk(k∈N).

Let Z:= X× Um, and set

D:= {(x, u)∈Z|˙x(t) = f(t, x(t), u(t)) (t∈T),

x(t0) = ξ0, x(t1) = ξ1},

S:= {(x, u)∈D|ϕα(u(t)) ≤0,

ϕβ(u(t)) = 0 (α∈R, β ∈Q, t ∈T)}

where R={1, . . . , r}and Q={r+ 1, . . . , q}. For

all (x, u)in Z, let

I(x, u) := Zt1

L(t, x(t), u(t))dt.

The problem we shall deal with, which we label (P),

is that of minimizing Iover S.

Acontrol function uis an element of Umand a

state trajectory xcorresponding to a control function

uis an element of Xwhich solves the differential

equation ˙x(t) = f(t, x(t), u(t)) (t∈T). A pair

(x, u)comprising a state trajectory xand an associ-

ated control function uis referred to as a process. If

usatisﬁes the control constraints u(t)∈U(t∈T),

where

U:= {u∈Rm|ϕα(u)≤0 (α∈R),

ϕβ(u) = 0 (β∈Q)},

and xsatisﬁes the endpoint constraints x(t0) = ξ0and

x(t1) = ξ1, then the process (x, u)is said to be ad-

missible, and it solves (P) if it achieves the minimum

value of Iover all admissible processes.

With respect to the functions delimiting the prob-

lem, we assume that, if F:= (L, f), then F(t, ·,·)is

C2for all t∈Tand ϕis C2;F(·, x, u)and its deriva-

tives in (x, u)are piecewise continuous and there ex-

ists an integrable function α:T→Rsuch that, at any

point (t, x, u)∈T×Rn×Rm,

|F(t, x, u)|+|∇(x,u)F(t, x, u)|+

|∇2

(x,u)F(t, x, u)| ≤ α(t).

Moreover, the q×(m+r)-dimensional matrix

∂ϕi

∂ukδiαϕα

(i= 1, . . . , q;α= 1, . . . , r;k= 1, . . . , m), has

rank qon U(here δαα = 1,δαβ = 0 (α6=β)). This

condition is equivalent to the condition that, at each

point uin U, the matrix

∂ϕi

∂uk(i=i1, . . . , ip;k= 1, . . . , m)

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has rank p, where i1, . . . , ipare the indices i∈

{1, . . . , q}such that ϕi(u) = 0 (see [12] for details).

Finally, given (x0, u0)∈S, we shall denote by [t]

the point (t, x0(t), u0(t)) and set A(t) := fx[t]and

B(t) := fu[t]. In what follows, the notation ‘∗’ will

be used to denote transpose.

3 Normality and ﬁrst order condi-

tions

First order necessary conditions are well established

and one version (consequence of the Maximum Prin-

ciple) can be stated as follows (see, for example, [6]).

This result can be seen as the analogous of the Fritz

John necessary optimality condition in mathematical

programming mentioned in the introduction.

For all (t, x, u, p, µ, λ)in T×Rn×Rm×Rn×

Rq×Rdeﬁne the Hamiltonian function as

H(t, x, u, p, µ, λ) :=

hp, f(t, x, u)i − λL(t, x, u)− hµ, ϕ(u)i.

Theorem 10. Suppose (x0, u0)solves (P). Then

there exist λ0≥0,p∈X, and µ∈ Uq, not vanishing

simultaneously on T, such that

a. µα(t)≥0and µα(t)ϕα(u0(t)) = 0 (α∈

R, t ∈T);

b. ˙p(t) = −H∗

x(t, x0(t), u0(t), p(t), µ(t), λ0)on

every interval of continuity of u0;

c. Hu(t, x0(t), u0(t), p(t), µ(t), λ0) = 0 (t∈T).

In this theorem, the case λ0= 1, as in the theory

of mathematical programming, is particuarly relevant.

Denote by Ethe set of all (x0, u0, p, µ)∈Z×X×Uq

satisfying

a. µα(t)≥0and µα(t)ϕα(u0(t)) = 0 (α∈

R, t ∈T);

b. ˙p(t) = −A∗(t)p(t) + L∗

x[t] (t∈T);

c. B∗(t)p(t) = L∗

u[t] + ϕ0∗(u0(t))µ(t) (t∈T).

Given (x0, u0)∈S, let Λ(x0, u0)be the set of all

(p, µ)∈X× Uqsuch that (x0, u0, p, µ)∈ E. The el-

ements of Ewill be called extremals and of Λ(x0, u0)

Lagrange multipliers (no confusion should arise with

the notation and terminology used in Section 1).

As for the ﬁnite dimensional case, given a so-

lution (x0, u0)to the problem, nonemptiness of

Λ(x0, u0)requires the assumption of a certain con-

straint qualiﬁcation applied to the constraints and the

admissible process. By analogy with the mathemati-

cal programming problem, we deﬁne normality rela-

tive to Sby imposing the null solution as the unique

solution to the system given in Theorem 10 when the

cost multiplier vanishes. This is a natural extension of

the deﬁnition of normality given for problem (N).

We shall say that (x0, u0)∈Sis normal relative

to Sif, given (p, µ)∈X× Uqsatisfying

i. µα(t)≥0and µα(t)ϕα(u0(t)) = 0 (α∈

R, t ∈T);

ii. ˙p(t) = −A∗(t)p(t)

[ = −H∗

x(t, x0(t), u0(t), p(t), µ(t),0) ] (t∈T);

iii. 0 = B∗(t)p(t)−ϕ0∗(u0(t))µ(t)

[ = H∗

u(t, x0(t), u0(t), p(t), µ(t),0) ] (t∈T),

then p≡0. Note that, in this event, also µ≡0.

From Theorem 10 we conclude that, if (x0, u0)

solves (P) and is a normal process relative to S, then

Λ(x0, u0)is nonempty, that is, there exists (p, µ)∈

X× Uqsuch that (x0, u0, p, µ)is an extremal.

The sets S0and S1, which played a fundamen-

tal role before, have also their counterpart in optimal

control. Denote the set of active indices at uby

Ia(u) := {α∈R|ϕα(u)=0}(u∈Rm).

Given u0∈ Um, let

S0[ = S0(u0) ] := {(x, u)∈D|ϕi(u(t)) = 0

(i∈Ia(u0(t)) ∪Q, t ∈T)}.

For µ∈ Uqwith µα(t)≥0 (α∈R, t ∈T), deﬁne

S1[ = S1(µ) ] := {(x, u)∈D|

ϕα(u(t)) ≤0 (α∈R, µα(t) = 0, t ∈T),

ϕβ(u(t)) = 0 (β∈Rwith µβ(t)>0,

or β∈Q, t ∈T)}.

Note that

S1(µ) = {(x, u)∈S|

ϕα(u(t)) = 0 (α∈R, µα(t)>0, t ∈T)}.

Since normality is deﬁned relative to any set of pro-

cesses given by inequality and equality constraints, it

can be applied to the sets S0(u0)and S1(µ). By deﬁ-

nition, we obtain the following conditions.

An admissible process (x0, u0)is normal relative

to S0(u0)if, given (p, µ)∈X× Uqsatisfying

i. µα(t)ϕα(u0(t)) = 0 (α∈R, t ∈T);

ii. ˙p(t) = −A∗(t)p(t) (t∈T);

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iii. B∗(t)p(t) = ϕ0∗(u0(t))µ(t) (t∈T),

then p≡0. As before, this implies that µ≡0.

Similarly, an admissible process (x0, u0)is nor-

mal relative to S1(µ)if, given (q, ν)∈X× Uqsatis-

fying

i. να(t)≥0and να(t)ϕα(u0(t)) = 0 (α∈

R, µα(t)=0, t ∈T);

ii. ˙q(t) = −A∗(t)q(t) (t∈T);

iii. B∗(t)q(t) = ϕ0∗(u0(t))ν(t) (t∈T),

then q≡0. In this event, again, we also have ν≡0.

As one readily veriﬁes, if (x0, u0)∈Sis normal

relative to S0(u0), then Λ(x0, u0)is a singleton. This

assumption, however, can be weakened, and the same

conclusion will follow if, given (p, µ)∈Λ(x0, u0),

the admissible process is normal relative to S1(µ).

The converse, however, may not hold. There are prob-

lems for which the pair (p, µ)is unique in Λ(x0, u0)

but (x0, u0)fails to be normal relative to S1(µ). A

full account of these results can be found in [18]. For

a characterization of the uniqueness of multipliers in

terms of normality of yet another set of constraints we

refer to [17].

4 Uniqueness and second order con-

ditions

We shall ﬁnd convenient to express the normality con-

ditions of the previous section in terms of certain con-

vex cones. Given µ∈Rqdeﬁne the following subsets

of indices of R:

Γ0(µ) := {α∈R|µα= 0},

Γp(µ) := {α∈R|µα>0}.

For all u∈Rmand µ∈Rq, consider the following

sets

τ0(u) := {h∈Rm|ϕ0

i(u)h= 0 (i∈Ia(u)∪Q)},

τ1(u, µ) := {h∈Rm|ϕ0

i(u)h≤0

(i∈Ia(u)∩Γ0(µ)),

ϕ0

j(u)h= 0 (j∈Γp(µ)∪Q)},

τ(u) := {h∈Rm|ϕ0

i(u)h≤0 (i∈Ia(u)),

ϕ0

j(u)h= 0 (j∈Q)}.

As shown in [13], [14], (x0, u0)is normal with

respect to S0(u0)if z≡0is the only solution of the

system

˙z(t) = −A∗(t)z(t), z∗(t)B(t)h= 0

for all h∈τ0(u0(t)) (t∈T).

Similarly, (x0, u0, µ)is normal with respect to S1(µ)

if z≡0is the only solution of the system

˙z(t) = −A∗(t)z(t), z∗(t)B(t)h≤0

for all h∈τ1(u0(t), µ(t)) (t∈T).

Finally, (x0, u0)is normal with respect to Sif z≡0

is the only solution of the system

˙z(t) = −A∗(t)z(t), z∗(t)B(t)h≤0

for all h∈τ(u0(t)) (t∈T).

For second order necessary conditions, we

consider the quadratic integral deﬁned, for all

(x, u, p, µ)∈Z×X× Uq, by

J((x, u, p, µ); (y, v)) :=

Zt1

2Ω(t, y(t), v(t))dt ((y, v)∈Z)

where, for all (t, y, v)∈T×Rn×Rm,

2Ω(t, y, v) :=

−[hy, Hxx(t)yi+ 2hy, Hxu(t)vi+hv, Huu(t)vi]

and H(t)denotes H(t, x(t), u(t), p(t), µ(t),1).

The next result is well-known in the literature

(see, for example, [12]). It guarantees uniqueness

of the Lagrange multiplier and provides second order

necessary conditions.

Theorem 11. Suppose (x0, u0)solves (P)

and (p, µ)∈Λ(x0, u0). If (x0, u0)is nor-

mal relative to S0(u0)then (p, µ)is unique and

J((x0, u0, p, µ); (y, v)) ≥0for all (y, v)∈Zsat-

isfying

i. ˙y(t) = A(t)y(t) + B(t)v(t) (t∈T);

ii. y(t0) = y(t1)=0;

iii. v(t)∈τ0(u0(t)) (t∈T).

This result is, clearly, the counterpart of Theorem

6 which deals with normality with respect to S0and

guarantees nonnegativity on the set RS0.

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It is worth mentioning that some examples found

in the literature (see [13], [14]) show that we may have

a solution (x0, u0)to the problem which is normal rel-

ative to S, a pair (p, µ)such that (x0, u0, p, µ)∈ E,

and (y, v)∈Zsatisfying (i), (ii) and (iii) of this theo-

rem, but J((x0, u0, p, µ); (y, v)) <0. In other words,

the conclusion of Theorem 11 may not hold if we as-

sume that the solution to the problem is normal with

respect to S.

Those references provide also examples where a

solution which is normal relative to S1(µ)may yield

a negative second variation on τ(u0(t)). In other

words, there exists an admissible process (x0, u0)

which solves the problem and is normal with re-

spect to S1(µ)with (p, µ)∈Λ(x0, u0)but, for some

(y, v)∈Zsatisfying (i), (ii) of Theorem 11 with

v(t)∈τ(u0(t)), we have J((x0, u0, p, µ); (y, v)) <

0. It is natural to ask if the same conclusion can be

reached assuming only uniqueness of the multiplier.

More generally, it is of interest to know if the sec-

ond order necessary condition given in Theorem 11

holds in a larger set and/or under weaker assumptions.

In particular, we would like to know if the theorem re-

mains valid assuming uniqueness of the multiplier in-

stead of the strong normality assumption on (x0, u0).

Explicitly, the question is if the following result holds.

Conjecture 1. Suppose (x0, u0)solves (P) and

(p, µ)∈Λ(x0, u0). If (p, µ)is unique, that is, if

Λ(x0, u0) = {(p, µ)}, then J((x0, u0, p, µ); (y, v)) ≥

0for all (y, v)∈Zsatisfying

i. ˙y(t) = A(t)y(t) + B(t)v(t) (t∈T);

ii. y(t0) = y(t1)=0;

iii. v(t)∈τ(u0(t)) (t∈T).

5 The example

In this ﬁnal section we provide a new and illustra-

tive result in the direction signaled by the conjec-

ture. It corresponds to an example where a solu-

tion (x0, u0)to the problem has a unique multiplier

(p, µ)∈Λ(x0, u0)but J((x0, u0, p, µ); (y, v)) <0

for some (y, v)∈Zsatisfying

˙y(t) = A(t)y(t) + B(t)v(t) (t∈T),

y(t0) = y(t1)=0,

v(t)∈τ(u0(t)) (t∈T).

In other words, the conclusion of the conjecture may

not be true.

Example 3. Consider the problem of minimizing

I(x, u) = R1

−1b(t)u3(t)dt subject to

˙x(t) = u(t) (t∈[−1,1]), x(−1) = x(1) = 0,

u2(t)≤1 (t∈[−1,1])

where

b(t) = 0if t∈[−1,0]

t2if t∈[0,1].

In this case T= [−1,1],n=m= 1,ξ0=ξ1=

0and, for all (t, x, u)∈T×R×R,

L(t, x, u) = b(t)u3, f(t, x, u) = u, ϕ(u) = u2−1.

Let

x0(t) := t+ 1 if t∈[−1,0]

1−tif t∈[0,1]

u0(t) := 1if t∈[−1,0]

−1if t∈(0,1].

Clearly (x0, u0)is a solution to the problem.

Now, if (x0, u0, p, µ)∈ E, then µ(t)≥0,˙p(t) =

p(t) = 3b(t)u2

0(t)+2u0(t)µ(t) (t∈[−1,1]).

Thus pis a constant satisfying

p=2µ(t)if t∈[−1,0]

3t2−2µ(t)if t∈(0,1].

Since µ(t)≥0for all t∈T, from the ﬁrst relation

we have p≥0and, from the second, p≤3t2for all

t∈(0,1] and so p≤0. Thus p≡0and therefore

µ(t) = 0if t∈[−1,0]

(3/2)t2if t∈[0,1].

This implies that (p, µ)is the only pair such that

(x0, u0, p, µ)∈ E.

We have H(t, x, u, p, µ) = pu −b(t)u3−(u2−

1)µand so

Hu(t, x, u, p, µ) = p−3b(t)u2−2µu,

Huu(t, x, u, p, µ) = −6b(t)u−2µ.

Therefore

Huu(t, x0(t), u0(t), p(t), µ(t))

=0if t∈[−1,0]

3t2if t∈[0,1].

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This implies that, for all (x, u, p, µ)∈Z×X× U1

and (y, v)∈Z,

J((x, u, p, µ); (y, v)) = −Z1

3t2v2(t)dt.

Since ϕ0(u) = 2u, we have

τ0(u0(t)) = {h|2u0(t)h= 0}={0},

τ1(u0(t), µ(t)) is given by

{h|2u0(t)h≤0}={h|h≤0}if t∈[−1,0],

{h|2u0(t)h= 0}={0}if t∈(0,1],

and τ(u0(t)) is given by

{h|2u0(t)h≤0}={h|h≤0}if t∈[−1,0],

{h|2u0(t)h≤0}={h|h≥0}if t∈(0,1].

Let

y(t) := −t−1if t∈[−1,0]

t−1if t∈[0,1]

v(t) := −1if t∈[−1,0]

1if t∈(0,1].

Then (y, v)solves ˙y(t) = v(t) (t∈T),y(−1) =

y(1) = 0,v(t)∈τ(u0(t)), and

J((x0, u0, p, µ); (y, v)) = −Z1

3t2v2(t)dt < 0.

Let us end this paper with an open question, a new

conjecture. As mentioned before, it is of interest to

know if Theorem 11 holds in a larger set and/or under

weaker assumptions. In particular, we would like to

know if Conjecture 1 is valid if we replace condition

(iii) by

v(t)∈τ1(u0(t), µ(t)) (t∈T).

Explicitly, the question is if the following result

holds.

Conjecture 2. Suppose (x0, u0)solves (P) and

(p, µ)∈Λ(x0, u0). If (p, µ)is unique, that is, if

Λ(x0, u0) = {(p, µ)}, then J((x0, u0, p, µ); (y, v)) ≥

0for all (y, v)∈Zsatisfying

i. ˙y(t) = A(t)y(t) + B(t)v(t) (t∈T);

ii. y(t0) = y(t1) = 0;

iii. v(t)∈τ1(u0(t), µ(t)) (t∈T).

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