1. Introduction
In this paper, we derive two new sufficiency theorems in optimal control problems as the parametric and nonparametric problems of Bolza with nonlinear dynamics, free initial and final states, and inequality and equality mixed time-state-control constraints. The fundamental components of the sufficiency theorems of this article are a similar version of the Pontryagin maximum principle, a hypothesis usually called the transversality condition, a crucial second order inequality arising from the original algorithm employed to prove one of the sufficiency theorems, a related hypothesis of the Legendre–Clebsh necessary condition, the positivity of a quadratic function on the cone of critical directions, and a fundamental integral Weierstrass inequality involving a function whose role is parallel to the Hamiltonian of the problem. Given an admissible process, its set of active indices of the inequality restrictions has to be piecewise constant on the underlying time interval, the Lagrange multipliers associated with the inequality mixed constraints must be nonnegative and in fact they have to be zero whenever the corresponding index is inactive. The optimal control of the proposed optimal process need not be continuous but only measurable, see, for example, [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17], where the authors study several optimal control problems having a degree of generality very similar to the one treated in this paper and where the continuity of the optimal controls is a crucial assumption in those sufficiency theories. In the first of the sufficiency theorems of this work, the deviation between optimal costs and feasible costs is estimated by quadratic functionals, two of them playing the role of the square of the norm of the classical Banach space
.
It is worth mentioning that second order sufficient conditions, as pointed out in [
15], are necessary in nonlinear problems when the extremal is not unique or when an existence theorem is not applicable. In addition, the sufficient treatments have shown to be of crucial relevance in some parametric optimal control problems studying the analysis of stability or sensitivity, see, for example, [
16,
17]. In the previous references, the initial or final states are free, but they are restricted to lie in some surfaces delimited by curves; in contrast, the initial and final states of the nonparametric optimal control problem studied in this article are completely free, in the sense that they are not necessarily restricted to a parametrization, but they only must belong to any sets belonging to the images of a surface determined by a
function. On the other hand, it is worth observing that all the crucial hypotheses of the sufficiency treatment studied in this article, are stated in the theorems, in contrast with other second order necessary and sufficiency theories that depend upon the verifiability of some crucial preliminary assumptions, see, for example, [
18,
19,
20], where the necessary second order conditions for optimality depend on some previous hypotheses involving some notions of normality or regularity of a solution; or [
11], where the corresponding sufficiency theory depends on the linear independence of some vectors whose role is the gradients of the active inequality and the equality restrictions. Finally, it is important to point out that, in [
21,
22], one can also find some sufficiency theories where the deviation between admissible and optimal costs around the optimal control has a quadratic growth.
The main novelties of this paper concern the facts that the sufficiency technique, used to prove Theorem 1, is independent of the standard hypothesis of continuity of the optimal controls, an assumption imposed in almost all the sufficiency theories having a similar degree of generality as the one studied in this article. In Corollary 1, the initial and final points of the states are not only variable, but they are completely free, in the sense that they may belong to any sets that must only be contained in a
manifold, the sufficiency method employed to prove one of the results of the paper does not invoke classical sufficiency tools such as bounded matrix-valued Riccati equations, Hamilton–Jacobi inequalities, generalized notions of conjugate points, the linear independence of the gradients involving the active inequality and the equality constraints, insertions of the original optimal control problem in an abstract optimization problem involving a Banach space, or certain techniques based on arguments of convexity, see [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17] for details. In the parametric sufficiency theorem of the article, if an admissible process satisfies all of its hypotheses, the former not only is a weak minimum, but the deviation between the optimal cost and the admissible costs is estimated by functionals playing similar roles of the squares of several norms.
The organization of the article is the following. In
Section 2, we state a parametric optimal control problem we shall be concerned with together with some elementary definitions, and we also pose one of the main results of the paper. In
Section 3, we establish a nonparametric optimal control problem we shall be interested in, some fundamental definitions, a corollary that forms one of the crucial results of the paper, and an example illustrating how one can apply the results of the article.
Section 4 is dedicated to state three auxiliary results in which the proof of one of the theorems is based and whose proof is referred to [
23].
Section 5 is dedicated to the proof of Theorem 1. Finally, in
Section 6, some conclusions and some future directions of open problems are briefly enunciated.
2. A Fundamental Theorem
Suppose an interval
in
is given, in which we have functions
,
,
,
, and
. Set
where
and
. If
, then
and we are not concerned with statements regarding
. Similarly, if
, then
, and we are not concerned with statements regarding
.
Let be a sequence of measurable functions and let be a measurable function. We shall denote uniform convergence of to by . Similarly, strong convergence in by and weak convergence in by .
We are going to assume throughout the article that L, f and have first and second continuous derivatives with respect to x and u on . Moreover, we shall suppose that the functions l and are of class on .
Let be the space of all absolutely continuous functions mapping T to and .
Define
, and keep in mind that the notation
means any member
. The parametric optimal control problem we shall be concerned with, denoted by (P), consists of minimizing a functional of the form
over all
satisfying the constraints
The elements (* means transpose) are called parameters, the members in are called processes, and a process is feasible or admissible if it satisfies the constraints. The notation means a member .
The following notation will allow us to introduce the main results of this section.
• A process
is a
solution of
if it is feasible and
for all feasible processes
. A feasible process
is a
weak minimum of (P) if it is a minimum of
I with respect to the norm
that is, if, for some
,
for all feasible processes
satisfying
. In other words, if
I affords a weak minimum at
, then, if
is admissible and it is sufficiently close to
, in the sense that the quantities
,
and
are sufficiently small, then
.
• For all
, define the Hamiltonian of the problem by
Given
and
set, for all
,
and let
• The
second variation of
J along
in the direction
, is given by
where for all
,
and the notation
refers to any element
in
. In addition,
is the second derivative of
l evaluated at
b.
• Given
and
, set
where
• For all
and all
, define
where
Finally, given
, denote by
the set of active indices of
with respect to the inequality restrictions. For all
, let
be the cone of all
verifying
The set is the cone of critical directions with respect to , and the symbol means the derivative of .
Theorem 1 below is a crucial tool in order to obtain Corollary 1, the latter being the main result of the article. Theorem 1 gives sufficiency for weak minima of problem (P). Hypothesis (i) of Theorem 1 is the transversality condition, hypothesis (ii) is an inequality relation that was found during the original proof of Theorem 1, hypothesis (iii) is a modified version of the Legendre–Clebsch condition, hypothesis (iv) is the positivity of a quadratic integral on the cone of critical directions, and hypothesis (v) involves a Weierstrass integral inequality hypothesis. A remarkable component of Theorem 1 concerns the fact that the optimal control is not necessarily continuous but only measurable. The notation, is the second derivative of along in the direction .
Theorem 1. Let be a feasible process. Suppose that is piecewise constant on T, that there exist , with , and , such that and the following is verified
- (i)
.
- (ii)
.
- (iii)
.
- (iv)
for all , .
- (v)
For all feasible with , .
Then, for some and all admissible processes satisfying , In particular, is a weak minimum of (P).
3. The Main Result
Suppose that we have an interval
in
, two sets
and functions
,
,
and
. Set
where
and
. If
, then
and we are not concerned with statements regarding
. Similarly, if
, then
and we are not concerned with statements regarding
.
We are going to assume throughout this section that , g and have first and second continuous derivatives with respect to x and u on . Additionally, we suppose that the function ℓ, is of class on .
Set , where as usual, denotes the space of absolutely continuous functions mapping T to and .
In this section, we shall be concerned with the nonparametric optimal control problem, denoted by
, of minimizing the functional
over all pairs
satisfying the restrictions
Members in A are called processes, and a process is feasible if it satisfies the restrictions.
A process solves if it is feasible and for all feasible processes . A feasible process is a weak minimum of if it is a minimum of relative to the essential supremum norm, that is, if for some , for all feasible processes satisfying .
Let
be any function of class
such that
. Relate the nonparametric optimal control problem
with the parametric optimal control problem given in
Section 2, denoted by
, that is,
is the parametric problem defined in
Section 2, with the following data;
,
,
,
,
and
the components of
, that is,
.
Lemma 1. The following is verified:
- (i)
is a feasible process of if and only if is a feasible process of and .
- (ii)
If is a feasible process of , then - (iii)
If solves , then solves .
Proof. This is precisely Lemma 1 of [
23]. □
Corollary 1 below is an immediate consequence of Theorem 1 and Lemma 1. It gives sufficiency conditions of problem . Once again, it is worthwhile observing that the optimal control is not necessarily continuous but only measurable.
Corollary 1. Let be any function of class such that and let be the parametric optimal control problem posed before enunciating Lemma 1. Let be a feasible process of . Suppose that is piecewise constant on T, there exist , with , and , , such thatand the following is verified: - (i)
.
- (ii)
.
- (iii)
.
- (iv)
for all , .
- (v)
For all feasible with , .
Then, is a weak minimum of .
Remark 1. It is worth observing that our sufficiency theory can also be applied to isoperimetric problems of Bolza with inequality and equality constraints.
In order to illustrate this fact for the nonparametric problem studied in this section, let
be functions of class
in
. In addition, let
be functions having first and second continuous derivatives with respect to
x and
u on
, and consider the isoperimetric nonparametric optimal control problem of minimizing
subject to
Additionally, set
,
,
, where
and
. Here, in
, there are
k intervals
and
singletons
. In
, there are
K singletons
. Remark 1 follows from the fact that the isoperimetric optimal control problem stated above is equivalent to the nonparametric optimal control problem of minimizing
subject to
Example 1 below shows how Corollary 1 can be applied. In the former, an inequality-equality constrained optimal control problem is solved by verifying that the first order sufficiency conditions
are satisfied by an element
. In addition,
satisfies hypotheses (i), (ii), (iii), (iv) and (v) of Corollary 1, and hence it is a weak minimum of
.
Example 1. Consider the nonparametric optimal control problem of minimizingover all satisfying the constraintswhere For this example, the data of the nonparametric problem are given by , , , , , , , , , , and .
It is straightforwardly verified that the functions , g, and their first and second derivatives with respect to x and u are continuous on . Moreover, the function ℓ is in .
Additionally, it is evident that the process is admissible of . Let be defined by . Clearly, is in and . The related parametric problem denoted by has the following data; , , , , , and , the components of , that is, with and .
Note that, if we set , then is feasible for . In addition, clearly, is constant on T. Let , and note that , and .
It is straightforwardly verified that, for all
,
and then
satisfies the first order sufficiency hypotheses of Corollary 1. As
,
,
, then
and hence hypothesis (i) of Corollary 1 is satisfied. In addition, one can easily verify that,
and so hypothesis (ii) of Corollary 1 is fulfilled.
Now, for all
,
and so, for all
,
which in turn implies that
satisfies hypothesis (iii) of Corollary 1.
In addition, observe that, for all
,
Thus,
is given by all
satisfying
Moreover, note that, for all
,
and, for all
,
Therefore, for all
,
Consequently,
for all
,
, and so hypothesis (iv) of Corollary 1 is satisfied.
Now, observe that, if
is feasible, for all
,
Therefore, if
is feasible,
In addition, if
is feasible,
Accordingly, for any
and for any
feasible with
,
Therefore, hypothesis (v) of Corollary 1 is fulfilled for any and . By Corollary 1, is a weak minimum of .
Remark 2. The reader can find a concrete example concerning the existence of a purely measurable optimal control in which one of its components satisfies a classical type of amplitude constraints on the controls u.
Indeed, see Example 1 of [
23], where one can find a concrete optimal control
with
and the feasible controls
satisfying the amplitude constraints
Remark 3. It would be of interest to see how the references quoted in this article or even the sufficiency theory presented in this paper can be generalized to the more complicated situation of the discrete-time case. See, for instance, [24], where time is measured in days in order to introduce a mathematical model to describe the outbreak of the Sars-Cov-2 in Ireland in March–May 2020. In the above reference, the optimal control treatment appeals to piecewise constant controls and state constraints for which a theoretical analysis is not amenable and hence a numerical approach is studied. It is worth mentioning that the optimal control model mentioned above saved lives and minimized the economical costs of the pharmaceutical interventions.