Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems

By Carlo Mariconda

Abstract

This article deals with the Lipschitz regularity of the “approximate” minimizers for the Bolza type control functional of the form

among the pairs satisfying a prescribed initial condition , where the state is absolutely continuous, the control is summable and the dynamic is controlled-linear of the form . For the above becomes a problem of the calculus of variations. The Lagrangian is assumed to be either convex in the variable on every half-line from the origin (radial convexity in ), or partial differentiable in the control variable and satisfies a local Lipschitz regularity on the time variable, named Condition (S). It is allowed to be extended valued, discontinuous in or in , and non convex in . We assume a very mild growth condition, actually a violation of the Du Bois-Reymond–Erdmann equation for high values of the control, that is fulfilled if the Lagrangian is coercive as well as in some almost linear cases. The main result states that, given any admissible pair , there exists a more convenient admissible pair for where is bounded, is Lipschitz, with bounds and ranks that are uniform with respect to in the compact subsets of . The result is new even in the superlinear case. As a consequence, there are minimizing sequences that are formed by pairs of equi-Lipschitz trajectories and equi-bounded controls. A new existence and regularity result follows without assuming any kind of Lipschitzianity in the state variable. We deduce, without any need of growth conditions, the nonoccurrence of the Lavrentiev phenomenon for a wide class of Lagrangians containing those that satisfy Condition (S), are bounded on bounded sets “well” inside the effective domain and are radially convex in the control variable. The methods are based on a reparametrization technique and do not involve the Maximum Principle.

1. Introduction

The main object of the article concerns the existence of “nice” pairs of approximate solutions to an optimal control problem. For the sake of clarity, we motivate the core of the paper by means of the basic problem of the calculus of variations.

The classical problem of the calculus of variations consists of minimizing an integral functional

where is a positive, Lebesgue–Borel Lagrangian. The main ingredients in order to obtain the existence of a solution are summarized in Tonelli’s theorem:

Lower semicontinuity of with respect to ;

Convexity of with respect to ;

Superlinearity of with respect to :

where and .

When a minimizer of exists, a first step towards regularity is looking at its Lipschitzianity. When is autonomous, superlinearity alone suffices to ensure the Lipschitz continuity of the minimizers (see Reference 2Reference 24Reference 25). Weaker growth conditions were considered in the last decades, requiring a specific behavior of the Hamiltonian associated with , as belongs to the convex subdifferential of and . The essential idea of using such indirect growth conditions for the purposes of existence and regularity is due to F. Clarke, who introduced Condition (H) in his seminal paper Reference 21 of 1993 for Lagrangians possibly nonautonomous, extended valued, with state and velocity constraints. Few years later, with different methods, A. Cellina and his school began working around a growth condition (G), formulated first in Reference 18 for continuous Lagrangians of the form with and, in 2003 Reference 16Reference 17 for autonomous and continuous Lagrangians. The growth conditions (H) and (G) will be thoroughly examined below. At this stage we just mention here that if is bounded on bounded sets then superlinearity implies Condition (G) and, in the real valued case, the validity of Condition (G) implies that of Condition (H); Conditions (G) and (H) are satisfied by some Lagrangians with almost linear growth, e.g., satisfies (G) as well as (H), and some Lagrangians of the form –in particular –satisfy (H), but not (G). In the real valued autonomous case these weak growth conditions alone (with no need of convexity or continuity assumptions), instead of superlinearity, ensure the Lipschitz regularity of minimizers, as shown by P. Bettiol and C. Mariconda in Reference 8.

In the nonautonomous case, there are examples of Lagrangians that satisfy Tonelli’s assumptions but whose minimizers are not Lipschitz. Several regularity results appeared on the subject (see Reference 22Reference 24Reference 33), each requiring some extra assumptions on the state or velocity variable, e.g., local Lipschitz conditions on the state variable or Tonelli–Morrey type conditions, mostly motivated by the use of the Weierstrass inequality or Clarke’s Maximum Principle. A Lipschitz regularity result without additional smoothness or convexity requirements on the state and velocity variables was obtained by P. Bettiol–C. Mariconda in Reference 7Reference 8 under the growth condition (H). The price to pay, with respect to Tonelli’s assumptions, is the additional local Lipschitz condition (S) on the time variable, thoroughly examined in § 3, requiring that is locally Lipschitz and that, for all ,

for some and . Moreover, it turns out without need of any growth condition, that is somewhat radially convex on the velocity variable along any given minimizer , in the sense that, for a.e. , the map

has a nonempty subdifferential at , in the sense on convex analysis. The role of radial convexity in Lipschitz regularity was prefigured in Reference 21 by the fact that the velocity constraint is a cone, and was first explicitly formulated for autonomous Lagrangians by C. Mariconda and G. Treu in Reference 31.

The celebrated example by J. M. Ball and V. J. Mizel in Reference 4 of a nonautonomous polynomial Lagrangian that is superlinear and convex in the velocity variable shows that, the violation of Condition (S) may lead not only to minimizers that are not Lipschitz, but even to the Lavrentiev phenomenon, i.e., the fact that

Condition (S) is not new and appeared in several results. It is sufficient for the validity of the Du Bois-Reymond–Erdmann equation (see Reference 19 for smooth Lagrangians, and Reference 8 for a discussion in the general case) and, if one replaces in Tonelli’s assumptions the superlinearity condition with the slower growth (H), it plays an essential role in establishing the Lipschitz continuity of minimizers in F. Clarke–R. Vinter’s Reference 24, Corollary 3. Furthermore, F. Clarke proved in Reference 21 that it provides the existence of a minimizer, which is actually Lipschitz. The Lavrentiev phenomenon has been widely reconsidered in the 1980s, a long time after M. Lavrentiev and B. Manià realized (see Reference 29) that such a pathology could occur. Here again, the autonomous case stands on its own: G. Alberti–F. Serra Cassano proved in Reference 1 that the Lavrentiev phenomenon never occurs if is just Borel, possibly extended valued; we refer to Reference 10, Reference 11, Reference 12 for more insights on Lavrentiev’s gap. More precisely if is an admissible trajectory and , there is no Lavrentiev gap at , i.e., there is a sequence of Lipschitz functions that share the same boundary values with , converging to in and in energy, i.e. . In the nonautonomous case some additional conditions have to be added. To the author’s knowledge, the criteria for the avoidance of the Lavrentiev phenomenon either follow trivially from the fact that minimizers exist and are Lipschitz or, as in Reference 28Reference 35Reference 37, they require that is locally Lipschitz or Hölder continuous in the state variable. One of the reasons is that, as was pointed out by D. Carlson in Reference 15, many of the results on the subject can actually be obtained as a consequence of Property (D) introduced by L. Cesari and T.S. Angell in Reference 20.

Regularity conditions on the state variable or convexity in the velocity variable are not satisfied in several problems arising from real life; discontinuous Lagrangians appear for instance in models arising from combustion in non homogeneous media or light propagation in the presence of layers. Some natural questions arise, and are addressed in the paper:

1.

When a priori existence of a minimizer fails because of the lack of continuity of the Lagrangian with respect to the state or velocity variable, can one at least approach the infimum of the functional through the values of along “nice” minimizing sequences (say equi-Lipschitz)?

2.

May Condition (S) on the time variable replace the customary regularity assumptions in the state variable, in order to prevent the Lavrentiev phenomenon?

Problem 1 was considered by A. Cellina and A. Ferriero in Reference 17, where the authors study autonomous, continuous Lagrangians that are convex or differentiable in the velocity variable and satisfy the growth condition (G). While, in the real valued, convex case, this result may be seen as a consequence of Reference 21, Theorem 2 ((G) implies (H) so that minimizers exist and are Lipschitz), new results arise in the differentiable case or in the extended valued framework, when Condition (G) and Condition (H), in its original formulation (as in Reference 8Reference 9Reference 21), may even not overlap. Most of the present work is based on the intuition that some steps of the proof of the main result in Reference 17 for the basic problem of the calculus of variations could actually be carried on in a more general setting, namely under a weaker growth condition of type (H) instead of (G), no more continuity assumptions on the state and velocity, nor convexity in the velocity variable and in the slightly wider framework of optimal control problems with a controlled-linear dynamics.

In this article we consider the more general Bolza optimal control of minimizing an integral functional

among the absolutely continuous arcs that have a prescribed value at

that are subject to a state constraint

and to a control-linear differential equation

with

and is positive, possibly extended valued, is a cone. If is the identity matrix, is the indicator function of a point and problem (P) is the basic problem of the calculus of variations. The same Bolza problem was considered in Reference 9; the particular form of the dynamics is motivated by the reparametrization techniques used to obtain the results. The results thus apply for instance to the class of problems called of Grushin type (see Reference 30) and to control problems related to subriemaniann metrics (see Reference 3). We take nonautonomous Lagrangians which are Lebesgue–Borel measurable and possibly extended valued. We assume that the Lagrangian is measurable, has at least a linear growth from below and satisfies Condition (S). We admit two different types of Lagrangians: those that are radially convex w.r.t. the control variable or those that are partial differentiable w.r.t the control variable; no kind of lower semicontinuity nor global convexity in the state or control variable are required. The extended valued case needs some extra assumptions. In this situation we impose, moreover, that tends uniformly to at the boundary of its effective domain , together with some structure conditions on that are satisfied if, for instance, where is real valued and is star-shaped.

In Section 4 we study various “slow” growth conditions and describe how they are related. When is smooth Condition (G) imposes that

The interpretation of (G) can be easily understood noticing that is the value of the intersection with the axis of the tangent hyperplane to at . Condition (G) has been considered in the autonomous framework in Reference 16Reference 17Reference 18 and extended to the nonautonomous case in Reference 8. In the smooth setting the original Condition (H), as formulated in Reference 21 for the calculus of variations and in Reference 9Reference 13 for the optimal control problem considered here, requires that once is an admissible pair for (P) with

then there is such that

where

and the are as in Equation 1.1. At a first glance, Condition (H) may appear quite involved since it relies on the essinf of a given admissible pair and on , a function depending on Condition (S). In the autonomous case it appears to be more ductile since in that case (see Figure 1 for the interpretation of Condition (H) in the simple case of a Lagrangian of a positive real control variable). However, as anticipated in Reference 21, Theorem 3 and proved in Reference 9, conditions Equation 1.3Equation 1.4 represent merely a violation of the Du Bois-Reymond–Erdmann equation for high values of the velocity/control.

With respect to Reference 8Reference 9Reference 13Reference 21 we formulate here Condition (H) in a slight different way for several reasons. We take into account that the initial time and value may vary. Furthermore, in the extended valued case the formulation given in § 4.3 widens the class of functions that satisfy Equation 1.4 (see Remark 4.14 and Example 7.2); as a byproduct the validity of (G) implies now that of (H) in any “reasonable” case (Proposition 4.17). At the same time, at least in the real valued and nonautonomous case, our Condition (H) is slightly more restrictive with respect to the original one due to the presence, in Equation 1.4, of a technical factor 2 in front of ; this does not seem, however, to have any consequence in concrete applications. A new growth condition (M) is introduced in § 4.4, so weak that it is in fact satisfied by any Lagrangian that is bounded on the bounded sets “well-inside” the effective domain (in the sense of Definition 4.15) and radially convex in the control variable. In the real valued, smooth case it simply requires that, for a suitable and ,

The main result, formulated in Theorem 5.1, considers the two different types of growth (H) or (M):

If Condition (H) is verified, it states that, whenever is admissible for (P) then there is an admissible pair where is Lipschitz, is bounded, such that

Moreover, the Lipschitz constant of and are uniformly bounded as vary in compact sets.

If the less restrictive Condition (M) holds, given we still get a pair with the above regularity properties, and satisfying

Several examples are provided in § 7 to illustrate the growth conditions involved in the article and the applicability of the results.

In the proof of Theorem 5.1 the Maximum Principle cannot be invoked, due to the lack of Lipschitz continuity of the Lagrangian in the state variable. Instead, we extend the method of Reference 17 to this more general framework in order to build the desired Lipschitz function via a Lipschitz reparametrization of . Without entering into the several technical points of the proof, it may be of interest to briefly illustrate the link between reparametrizations and growth conditions. For simplicity, consider the case of the calculus of variations. Let be a smooth, increasing change of variable on , be an admissible trajectory for , and set . Notice that, by taking high values of , one lowers the norm of the derivative of . The change of variable yields

Supposing that smooth, the derivative of at is

The proof consists on finding a suitable increasing and one-to-one change of variable . By choosing as in Equation 1.4 (resp. Equation 1.5), Conditions of type (H) (resp. (M)) allow to compensate the values of integral in on the sets where with the ones where , up to obtain a lower value than (resp. ). The essential ideas of the multiple step proof of Theorem 5.1 are described at the beginning of Section 9 for the convenience of the reader. Many technical issues are actually related to the fact that the Lagrangian is allowed to take the value ; we invite the reader focused in the real valued case to consult the simplified version in the announcement of the results given in Reference 5. It is worth mentioning that, in the proof of Theorem 5.1, the two growth conditions (H) and (M) share most of the arguments; their difference play a role just in few of the many steps. This fact seems to be a byproduct of the care needed to deal with Condition (H) and was unnoticed in Reference 17, where the authors consider the more restrictive (but easier to handle) growth of type (G).

Theorem 5.1 has several consequences. Under Condition (H), Corollary 5.5 yields “nice” minimizing sequences for (P) formed by equi-Lipschitz trajectories and equi-bounded controls as vary in compact sets. This property, that does not need existence of minimizers, is expected to have a strong impact on the study of the regularity function of the value function

and is investigated in Reference 6. As a further consequences of the main result, the existence of a solution under slow growth conditions to the optimal control problem (P) when, in addition to the conditions of Theorem 5.1, one imposes some standard lower semicontinuity of in , convexity with respect to , closure of the state constraint set , closure and convexity of the control set. Corollary 6.2 almost overlaps Reference 21, Theorem 3 when the problem concerns the calculus of variations, where the major difference relies on the version of Condition (H) mentioned above, but seems to be new in the framework of optimal control problems. Existence for more general controlled differential equations than Equation 1.2 was considered in the autonomous case in Reference 13. However, though the controlled-linear structure of the system Equation 1.2 might appear restrictive, the novelty with respect to the known literature is represented here by the absence of any kind of local Lipschitz condition on the state variable, and by the fact that may be extended valued. Theorem 5.1 does also provide some answers related to Problem 2. The Lavrentiev phenomenon is excluded in Corollary 5.7 for a wide class of Lagrangians, assuming a growth condition of type (M). In particular it is avoided (Corollary 5.9) when is real valued and, moreover:

a)

satisfies Condition (S);

b)

is radially convex in the control variable;

c)

is bounded on bounded sets.

We stress again the fact that, differently from other results on the Lavrentiev phenomenon for optimal control problems (see Reference 13Reference 26Reference 27), we do not assume any kind of Lipschitz continuity of in the state variable, nor we make use of the Maximum Principle.

2. Basic setting and notation

Let and . We consider the Bolza type optimal control problem

Subject to:

with the following basic assumptions.

Basic Assumptions and Notation.

The following conditions hold.

The Lagrangian , is Lebesgue–Borel measurable (i.e., measurable with respect to the -algebra generated by products of Lebesgue measurable subsets of (for ) and Borel measurable subsets of (for ;

(the space of linear functions from to ) is a Borel measurable function such that, for some ,

We refer to as to the controlled differential equation;

The control is measurable;

The state constraint set is a nonempty subset of ;

The control set is a cone, i.e. if then whenever ;

(Linear growth from below) There are and satisfying, for a.e. and every ,

The effective domain of is

We assume that for a.e. and every the set

is strictly star-shaped on the variable w.r.t. the origin, i.e.,