1. Introduction
It is common practice in the fields of the calculus of variations and optimal control to extend the space of solutions for problems that cannot be solved in, say, an ordinary space, or if the solution is difficult to find, even with numerical approximation. This process, known as
extension, involves compactifying and regularizing the problem, resulting in a more manageable structure and the possibility of obtaining necessary and sufficient conditions for optimality. However, in order for an extension to be well posed, it is fundamental that the infimum value achievable in the original problem coincides with that of the extended problem. Otherwise, the extended problem will not provide any useful information about the original problem, which is the only one whose strategies we actually want or can implement. So, for instance, determining an extended minimizer and the solution to the Hamilton–Jacobi equation associated with the extended problem (analytically or using numerical methods) is useful only if from them, we can derive a quasi-optimal control and the value function for the original problem, respectively. Clearly, this is only possible if there is no gap between infima. However, in the presence of endpoint and state constraints, a gap often occurs, even in situations where the set of strict-sense original solutions is
-dense in the set of extended paths. In particular, this problem arises when all strict-sense solutions close to an extended trajectory that satisfies the constraints, for instance, a local minimizer, fail to meet them in turn. Criteria for avoiding an infimum gap have, therefore, been extensively investigated in the literature. In the calculus of variations, for example, it is well known that, in the absence of suitable coercivity assumptions, the minimum of an integral cost over Lipschitz-continuous functions with assigned initial and final points may not exist or may be greater than the minimum assumed in the largest set of absolutely continuous functions. In this context, the gap issue is called the Lavrentiev phenomenon, and it is still widely studied (see, e.g., [
1] and the comprehensive bibliography therein). As far as optimal control is concerned, a classical extension involves relaxation, obtained by either convexifying the set of admissible velocities or introducing relaxed controls that take values in a set of probability measures. Another extension is the impulsive one, in which a non-coercive problem with unbounded controls, i.e., where minimizing sequences of solutions may have increasing velocities and tend in the limit to discontinuous paths, is extended by admitting functions of bounded variation as solutions. A detailed description of these well-known extensions and a wide bibliography can be found, e.g., in [
2,
3]. The gap phenomenon has also been studied extensively in optimal control, often in correspondence with necessary optimality conditions, known in the literature as the Pontryagin Maximum Principle (see [
4,
5] for its original formulation and applications). In particular, starting from the seminal work by Warga [
6] in the early 1970s, criteria for excluding an infimum gap for different problems and extensions have been expressed in terms of normality conditions for some versions of the Pontryagin Maximum Principle, where normality means that all sets of multipliers have the cost multiplier different from zero (see, e.g., [
7,
8,
9,
10,
11,
12,
13]).
This paper focuses on the connection between the presence of an infimum gap, at least in a local sense, and a nonsmooth version of the maximum principle satisfied in abnormal form, i.e., not normal, for the following problem and its extension .
Given
and
, we introduce the constrained control system
Here,
is a closed subset of
, which we call the target;
is the state constraint function; and
is the dynamics function, where the compact subset
and the bounded subset
are the sets of control values. Indeed, with
denoting the closure of
V, let us define the sets
,
, and
of admissible control functions as follows
We call an
extended process any triple
that satisfies the dynamic constraint (
1). If
in particular, then we refer to
as a
strict-sense process. Any process (either extended or strict-sense) that additionally fulfills the endpoint and the state constraint in (2) is said to be
feasible. The sets of feasible strict-sense and feasible extended processes are denoted by
and
, respectively.
Given a cost function
, we introduce the
strict-sense optimal control problem
and the
extended optimal control problemNote how the controls
and
play different roles, given that only the control set
, to which
belongs, is extended. The opportunity to consider both arises from applications. For example, in impulsive problems, it is common that only certain control components can take values in an unbounded set. In this case, which we clarify in
Section 4,
represents these components while
represents the remaining ones (see, e.g., the model example in [
14]). Incidentally, this distinction is reflected in the hypotheses on the dynamics
, which require continuity in
and, instead, a form of uniform continuity for both
and its Clarke-generalized Jacobian
in the variable
, as specified in
Section 2.
Since
, it immediately follows that
. In fact, this inequality might be strict, in which case we say that
there is an infimum gap. In order to introduce the notion of a
local infimum gap, for any pair of extended processes
,
, we define the following control distance:
where
ℓ is the Lebesgue measure. Hence, a feasible strict-sense [resp., extended] process
is a
local minimizer for
[resp.,
] if there exists some
such that
for any
in
[resp.,
], satisfying
.
We distinguish the following two types of local infimum gaps according to whether we focus on the strict-sense problem or the extended problem:
Type-E local infimum gap, when the cost of a local minimizer of is strictly smaller than the infimum of in a -neighborhood of ,
Type-S local infimum gap, if a local minimizer of is not a local minimizer of .
Assuming the hypotheses provided in
Section 2, and with reference to the maximum principle of Definition 3 below, our main results are the following:
- (i)
If at , there is a type-E local infimum gap, then satisfies the maximum principle in abnormal form, i.e., for a set of multipliers with cost multiplier equal to zero;
- (ii)
If is a local minimizer of , then it satisfies the same maximum principle as the extended problem. If, in addition, at , there is a type-S local infimum gap, then it is an abnormal extremal.
We emphasize that the choice of the distance above, which plays a fundamental role in the proof of these results, represents a novelty compared to the works we quoted above. In fact, in these papers, they always consider local minimizers, where the distance between the trajectories is used instead of .
Furthermore, in
Section 4, we illustrate a relevant application of the above results to impulsive optimal control. There are significant examples in aerodynamics [
14,
15], mechanics [
16,
17], and biology [
18,
19] where the evolution of the involved variables can be modeled as a control system, in which controls can reach very high intensity in a very short time interval, resulting in an abrupt change in the state of the system. The impulsive extension is, therefore, a limit problem, in which the previous controls and trajectories are replaced with their (suitably defined) limits. It is worth emphasizing that in the above-mentioned applications to real-world problems, impulsive controls are only idealizations of the original controls so results in relation to the impulsive problem are of interest only if they provide information on the original problem, namely only if no gap of any type occurs.
As already mentioned, Warga was the first to study the correlation between the presence of an infimum gap and the validity of the maximum principle in abnormal form for a classical extension through relaxation in the measure of the controls. Specifically, he announced the result for a type-S
-local infimum gap in his early paper [
6], which focused on state constraint-free optimal control problems with smooth data. Then, in his monograph [
13], Warga proved the relationship between the gap and abnormality for a type-E
-local infimum gap in optimal control problems with state constraints (see also [
11]). His subsequent work [
12] extended this result to include nonsmooth data, utilizing the results in [
20]. Vinter and Palladino [
10] proved the above-mentioned correlation in the case of both type-E and type-S
-local infimum gaps for the classical extension through convex relaxation of a class of nonsmooth state-constrained optimal control problems, which subsumed those considered by Warga and under less restrictive hypotheses on data. Their techniques differed significantly from those of Warga, reflecting different approaches to the maximum principle. In more detail, the method adopted in [
11,
12,
20] involved constructing approximating cones to reachable sets and using set separation arguments, whereas the technique adopted in [
10] utilized perturbation and penalization procedures as well as Ekeland’s variational principle. When applied to nonsmooth optimal control problems, it is difficult to compare these methods as they require different assumptions on the dynamics and target. More importantly, they give rise to distinct abnormality conditions. Indeed, following Warga’s method, these conditions involve the use of ‘derivative containers’ as generalized gradients from [
12], whereas the second method relies on Clarke’s version of the maximum principle, in which subdifferentials are considered (see [
21,
22]). More recently, following the latter approach, results similar to those in [
10] were established in [
7] for the impulsive extension of optimal control problems with unbounded dynamics and state constraints (see also the references therein). Additionally, in [
8,
9], an abstract extension including both relaxation and impulsive extension as special cases was addressed. In particular, in [
7,
8], for the first time, we also provided sufficient conditions for the nondegeneracy of the abnormality condition related to a type-E
-local infimum gap.
However, besides considering
-local minimizers, all these works focused primarily on the type-E local infimum gap. Specifically, apart from Warga’s initial work, the type-S local infimum gap was only studied in [
10] for the extension through convexification of the dynamics and in [
9] for a more general extension. In both papers, the results were not entirely satisfactory; however, it was shown that a strict-sense
-local minimizer that is not also an extended local minimizer satisfies, in abnormal form, an ‘averaged version’ of the maximum principle, which is much less informative than the actual maximum principle.
In this paper, for the extension under consideration, on the one hand, we fill the gap in the previous literature regarding the results obtained for the type-E and type-S local infimum gaps by showing that in both cases, the local minimizer is abnormal for the maximum principle associated with the extended problem. On the other hand, we extend the previous results for the type-E
-local infimum gap to the case of the local minimizer based on the distance
described above. Note that from the continuity property of the input-output map associated with the control system, it follows that the present results imply the previous ones. With regard to the techniques used, we are inspired by the approach proposed in [
10], as generalized to the case of an abstract extension in [
9]. In particular, this allows us to consider rather weak assumptions, including nonsmooth dynamics and state constraint functions, and a target that is simply a closed set (see
Section 2).
This paper is organized as follows. In
Section 2, we present the notations used, some useful definitions, and precise assumptions. In
Section 3, we rigorously introduce the concepts of type-E and type-S local infimum gaps and state our main results, which are proved in
Section 5.
Section 4 is devoted to applying these results to the impulsive extension of a control-affine system with unbounded controls. We also give an example.
Section 6 contains some concluding remarks.
4. An Application: The Impulsive Extension
In this section, we describe how the previous results can be used to investigate the gap phenomenon in a case relevant to applications: the impulsive extension of an optimal control problem with endpoint and state constraints. We also provide an example of an impulsive problem in which both a type-E and a type-S local infimum gap occur, and we explicitly show the abnormality condition in this case.
4.1. An Impulsive Optimization Problem
Let us consider the following free end-time optimization problem with
unbounded, control-affine dynamics:
in which
,
,
,
for any
,
, and
.
We make the following assumptions on the data:
H4. (i.e., K might be ); the (unbounded) set of control values U is a closed cone; the target is a closed set; and the dynamics functions f, , the constraint function h, and the cost function Ψ are locally Lipschitz continuous.
Note that
, sometimes called
fuel or
energy, is simply the
-norm of the control function
u on
. Assuming, as usual, that the function
is merely monotone nondecreasing, this problem is non-coercive, i.e., there are no conditions that prevent a minimizing sequence of trajectories from having increasing velocities and converging to a discontinuous path. It is well known that it is possible to embed the original problem
into the
space-time or
extended problem
below, where the time becomes a new state variable and the trajectories are reparameterizations of the limits of the graphs of the trajectories of
in the
-norm [
25,
26,
27,
28] (we recall that
can be analyzed using a distributional approach, meaning that
u is substituted by a Radon measure, only if the coefficients
are autonomous and commute, i.e., the Lie brackets
are equal to 0 for any
(see, e.g., [
25,
29])):
where
, with
W the set of control values given by
Let
be an
original process, i.e., it satisfies the dynamics constraint together with the initial condition of problem
, and let
be defined as follows
We observe that
results in an
extended process, i.e., it satisfies the dynamics constraint together with the initial condition of problem
, and
a.e. Actually, the map that associates with each original process an extended process with
a.e. turns out to be a bijection, so that problem
is in correspondence with the
strict-sense problem, namely the optimal control problem that arises when in
, we limit ourselves to consider
strict-sense processes only, i.e., extended processes with
a.e. Therefore, the extension involves allowing the control variable
to vanish on some non-trivial intervals contained in
. There,
remains constant, whereas
y evolves instantaneously according to
. This is the reason why
, despite being an ordinary optimal control problem with controls taking values in compact sets, is usually labeled as the
impulsive extension of
. Indeed, problem
is also equivalent to another generalization of
where the controls are vector-valued measures and the trajectories are bounded variation paths [
14,
30,
31,
32,
33,
34].
Adopting the terminology of the present paper, we say that an extended or strict-sense process
is
feasible [resp. an original process
is
feasible] if it additionally fulfills all the endpoint and the state constraint of
[resp.
]. The sets of feasible original, feasible extended, and feasible strict-sense processes are denoted by
,
, and
, respectively. Given
and
, we define the distance:
Note that
is equivalent to the distance obtained by replacing
with
in the
-norm (possibly extending the controls to
constantly equal to 0), as
for some constant
. At this point, the definitions of the local minimizer and type-E and type-S local
-infimum gaps (see Definitions 1 and 2) can be easily adapted to the impulsive extension by replacing the distance
defined in (
5) with the distance
given in (
13). The unmaximized Hamiltonian associated with problem
above is given by
for all
.
Definition 4. We say that is a Ψ
-extremal if there exists a path , , , , and Borel-measurable and μ-integrable functions satisfying the following conditions: where is given by Moreover, if and , then . Furthermore, if , then (14) can be strengthened with We say that is normal if all sets of multipliers , as described above, have . Conversely, we say that is abnormal when it is not normal.
From Theorem 1, we deduce the following result.
Theorem 3. Let , and assume that hypothesis (H4) holds. Then, consider the following statements:
- (i)
If is a local Ψ-minimizer for , then is a Ψ-extremal. If at , there is a type-E local Ψ-infimum gap, then is an abnormal extremal;
- (ii)
If is a local Ψ-minimizer for , then is a Ψ-extremal. If at , there is a type-S local Ψ-infimum gap, then is an abnormal extremal.
Proof. The impulsive extended problem
has a free end time, so the results of the previous sections concerning fixed end-time problems do not apply straightforwardly. However, through a standard time-rescaling procedure that applies to free end-time problems with Lipschitz-continuous time dependence, we can embed problem
into a fixed end-time optimization problem, satisfying all the assumptions of Theorem 1 and for which, for example,
is still a local minimizer if it was so for
. Precisely, let
,
, and consider the
rescaled problem:
where, for any
, we have the set
Any element
satisfying all constraints in
is referred to as a feasible rescaled extended process. If
a.e., then
is called a feasible rescaled strict-sense process. For any pair of feasible rescaled extended processes
,
, we define the distance as
Let us associate the (feasible) rescaled process
with the given reference process
. From a straightforward application of the chain rule and standard calculations, we deduce that for any
, there exists some
such that with each feasible rescaled extended process
satisfying
, using the time change
we can associate the following feasible extended process
satisfying
. Moreover,
.
As a consequence, if
is a local
-minimizer for
, then
is a local
-minimizer for
, at which there is a type-E local infimum gap as soon as at
, there is a type-E local infimum gap. At this point, the proof of Theorem 3 can be derived by applying Theorem 1 to the rescaled problem. We omit the details, which follow the same line as the proofs in [
22] (Theorem 8.7.1), and [
8] (Theorem 4.1). □
Remark 7. Using similar arguments to those in [8], what we have done in this section can be easily generalized to control-polynomial impulsive problems, by which we mean that the dynamics of the original problem can be replaced withwhere d is an integer . This generalization may be relevant for some applications to Lagrangian mechanics, where dynamics are usually control-polynomial with a degree of (see [17]). 4.2. An Example
The following example tells us that both a type-S local infimum gap and a type-E local infimum gap may occur. Moreover, we exhibit sets of abnormal multipliers, which exist in accordance with Theorem 3.
Consider the optimization problem with scalar, unbounded controls:
Let
. Then, the space-time extension of the above problem is given by
Type-S local infimum gap. Let
be the following strict-sense process, where
, the control
is given by the constant pair
and
It is easy to see that
, which corresponds to the process of
associated with the control
, is trivially a strict-sense minimizer, as
is the unique feasible strict-sense trajectory. However,
is not a local minimizer for the extended problem
. Indeed, let us fix
sufficiently small, and let us consider the extended process
, where
and
is given by
so that one has
For any
, this is the description in the state space of a discontinuous state trajectory
for problem
, which first reaches the point
using the control
and then jumps to the position
with an impulse. Note that
is a feasible extended process that satisfies
whose cost is strictly less than the cost corresponding to
because it holds that
Thus, by the arbitrariness of , at , there is a type-S local infimum gap. Indeed, a set of abnormal multipliers corresponding to is given by , where , , , , , and for any .
Type-E local infimum gap. Now consider the following extended process
, where
and
is given by
so that one has
It is easy to see that is a minimizer for as it is feasible, and its corresponding cost is equal to zero. Moreover, at , there is a type-E local infimum gap since defined in the previous step is the unique feasible strict-sense process. Indeed, a set of abnormal multipliers corresponding to is given by , where , , , , , , , and for any .