Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 586130, 16 pages

http://dx.doi.org/10.1155/2013/586130

Research Article

Consistency of the Duhem Model with Hysteresis

Mohammad Fuad Mohammad Naser and Fayçal Ikhouane

Universitat Politècnica de Catalunya, Escola Universitària d'Enginyeria Tècnica Industrial de Barcelona, Departament de Matemàtica Aplicada III, Comte d'Urgell 187, 08036 Barcelona, Spain

Received 26 March 2013; Accepted 27 July 2013

Academic Editor: Piermarco Cannarsa

Copyright © 2013 Mohammad Fuad Mohammad Naser and Fayçal Ikhouane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The Duhem model, widely used in structural, electrical, and mechanical engineering, gives an analytical description of a smooth hysteretic behavior. In practice, the Duhem model is mostly used within the following black-box approach: given a set of experimental input-output data, how to tune the model so that its output matches the experimental data. It may happen that a Duhem model presents a good match with the experimental real data for a specific input but does not necessarily keep significant physical properties which are inherent to the real data, independent of the exciting input. This paper presents a characterization of different classes of Duhem models in terms of their consistency with the hysteresis behavior.

1. Introduction

Hysteresis is a nonlinear behavior encountered in a wide variety of processes including biology, optics, electronics, ferroelectricity, magnetism, mechanics, structures, among other areas. The detailed modeling of hysteresis systems using the laws of Physics is an arduous task, and the obtained models are often too complex to be used in applications. For this reason, alternative models of these complex systems have been proposed [1–5]. These models do not come, in general, from the detailed analysis of the physical behavior of the systems with hysteresis. Instead, they combine some physical understanding of the system along with some kind of black-box modeling.

One of the popular models for hysteresis is the Duhem model proposed in [6]. The generalized form of the Duhem model consists of an ordinary differential equation of the form , where is the input and is the state or the output [7]. Other special forms of the model have been used, like the form [8] or the semilinear form [9]. Other important special cases of the Duhem model are the LuGre model of friction [10], the Dahl model of friction [11], and the Bouc-Wen model of hysteresis [12, 13]. The Duhem model has been used to represent friction [7], electromagnetic behavior [14, 15], or hysteresis in magnetorheological dampers [16].

In the current literature, the Duhem model is mostly used within the following black-box approach: given a set of experimental input-output data, how to adjust the Duhem model so that the output of the model matches the experimental data? The use of system identification techniques is one practical way to perform this task. Once an identification method has been applied to tune the Duhem model, the resulting model is considered as a “good” approximation of the true hysteresis when the error between the experimental data and the output of the model is small enough. Then, this model is used to study the behavior of the true hysteresis under different excitations. By doing this, it is important to consider the following remark. It may happen that a Duhem model presents a good match with the experimental real data for a specific input but does not necessarily keep significant physical properties which are inherent to the real data, independent of the exciting input. In the current literature, this issue has been considered in [17, 18] regarding the passivity/dissipativity of Duhem model.

In this paper, we investigate the conditions under which the Duhem model is consistent with the hysteresis behavior. The concept of consistency is formalized in [19] where a general class of hysteresis operators is considered. The class of operators that are considered in [19] are the causal ones, with the additional condition that a constant input leads to a constant output. For these classes of systems, consistency has been defined formally. This property is useful in system modeling and identification as it limits the search for the system's parameters to those regions where consistency holds. From the results of [19], it can be concluded that to check consistency one has to consider the sequence of inputs ,   and the corresponding sequence of outputs with . For the Duhem model to represent a hysteresis system, it is necessary that the sequence of functions converges uniformly when . In this paper, we seek necessary conditions and sufficient ones for this uniform convergence to hold.

This paper is organized as follows. Section 2 presents the needed mathematical background. The problem statement is formalized in Section 3. A classification of functions that is used throughout the paper, is introduced in Section 4. Sections 5 and 6 present necessary conditions and sufficient ones for the Duhem model to be consistent with the hysteresis behavior. Conclusions are given in Section 7.

2. Background Results

This section summarizes the results obtained in [19].

A function is said to be increasing (resp., decreasing) if (resp. ), and it is said to be nondecreasing (resp. nonincreasing) if (resp., ).

The Lebesgue measure on is denoted . A subset of is said to be measurable when it is Lebesgue measurable. Consider a function for some interval ; the function is said to be measurable when is -measurable where is the class of Borel sets of and is the class of measurable sets of . For a measurable function , denotes the essential supremum of the function on where is the Euclidean norm on . When , and it is denoted simply .

Consider the Sobolev space of absolutely continuous functions , where is a positive integer. For this class of functions, the derivative is defined almost every where with and . Endowed with the norm , is a Banach space [20].

For , let be the total variation of on ; that is, . The function is well defined as . ( is the space of locally integrable functions ). It is nondecreasing and absolutely continuous. Denote . If for some , let (in this case is necessarily finite). On the other hand, if for all , let (in this case may be finite or infinite).

Lemma 1. Let be nonconstant. Then, there exists a unique function that satisfies .

Consider the linear time scale change , for any and . Let be a set of initial conditions. Let be an operator that maps the input function and initial condition to an output in , where is a positive integer. That is . We consider causal operators such that   for all , if   in  , then in , [1, page 60].

Let and let . In the rest of this work, only causal operators are considered. Additionally, we consider that the following holds.

Assumption 2. Let and ; if such that is constant on ; then is constant on .

Lemma 3. There exists a unique function that satisfies . Moreover, one has . If is continuous on , then is continuous on and one has .

Definition 4. Let and initial condition be given. The operator is said to be consistent with respect to input and initial condition if and only if the sequence of functions converges in as .

It is shown in [19] that for hysteresis process, the sequence of functions converges in as . This fact shows that consistency is a mathematical property that any model of hysteresis should satisfy.

3. Problem Statement

The generalized Duhem model is defined for almost all by [7]

where and state take values in for some positive integer , input , function is continuous, where and are positive integers. Finally, is continuous and satisfies . Observe that if is constant; then ,  for all  . For this reason, we consider only nonconstant inputs in this paper.

Since is continuous and , we have . The differential equation (1) satisfies Carathéodory conditions, thus, for each initial state , (1) has an absolutely continuous solution that is defined on an interval of the form [21, page 4].

Consider the time scale change . When the input is used instead of , the system (1)-(2) becomes

where is the maximal solution of (3). When , system (3)-(4) reduces to (1)-(2). For any , define as . System (3)-(4) can be rewritten as

for all and for almost all , where is the maximal interval of existence of the solution .

Observe that Lemma 3 implies that for any there exists a unique function such that (when , we get ). The latter equality is equivalent to . According to Definition 4, the system (1)-(2) is consistent with respect to if and only if the sequence of functions converges in .

Proposition 5. The system (1)-(2) is consistent with respect to in the sense of Definition 4 if and only if the sequence of function converges in as .

Proof. To prove the if part, define the causal operator that maps to , where is given in (1)-(2). Assume that there exists such that . We know from (5), that is a sequence of continuous functions. Thus, the function is continuous as a uniform limit of continuous function. Lemma 3 implies that there exists a unique continuous function such that . Let . Since is continuous, there exists some such that . We get from the relation that for all : . This implies that so that , which means that the system (1)-(2) is consistent with respect to .

To prove the only if part, assume that , then the relation implies that for almost all

Thus, we have so that .

Proposition 5 implies that the consistency of the system (1)-(2) can be investigated by studying the uniform convergence of the sequence of functions instead of . Thus, we know from Section 2 that the system (1)-(2) is a hysteresis only if converges uniformly as .

Problem. In this paper, our objective is to derive necessary conditions and sufficient ones for the uniform convergence of the sequence of functions as .

4. Classification of Function

This section introduces a classification for the function that is used throughout the paper.

Definition 6. Let  such that is continuous on . The right and left local fractional derivatives of at with respect to order are defined respectively as follows [22]:

where is the gamma function.

The local fractional derivative of a vector-valued function is the vector of local fractional derivatives of its components.

Definition 7. The function is said to be of class   if   and the quantities   and   exist, are finite, and at least one of them is nonzero.

Proposition 8. The function is of class if and only if

where is defined as

Proof. Immediate using of the change of variables .

Proposition 9. If the function is of class ; then , where is defined in (9).

Proof. The result is trivial when is constant. Assume that is nonconstant. Given . Since is of class , there exists some ; that depend solely on , such that

The boundedness of implies that there exists a positive constant such that ,  for  all  . Thus, we have for all ,

Thus, we get from (9) that

which completes the proof.

Proposition 10. If the function is of class for some ; then it cannot be of any class different than .

Proof. Assume that the function is of class and with . Then,

which contradicts the fact that is of class .

Proposition 11. If the function is of class ; then there exists , such that

Proof. see Appendix A.

5. Necessary Conditions

The objective of this section is to derive necessary conditions for the uniform convergence of the sequence of functions as .

The standard way to ensure that the system (1)-(2) admits a unique solution is to prove that the right-hand side of (1)-(2) is Lipschitz with respect to . A function is Lipschitz with respect to if there exists a summable function such that , for almost all and for all that satisfy [21].

Lemma 12. Assume that the system (1)-(2) has a unique global solution for each input and initial condition . Assume that the function is of class . Suppose that there exists a continuous function such that

for each initial state and each input . Assume that the system (1)-(2) is consistent with respect to ; that is, there exists such that (see Proposition 5); then

 If , one has

(i).(ii)one has for all that

  where is given in (9).

 If , one has

(i).(ii). (iii)one has for almost all that

 where is defined in (9).

 If , one has , for all .

Proof. By (15), the fact that , the continuity of the function , and the relation , for all , there exists some independent of , and such that

where is given in (3)-(4). Thus,

On the other hand, we conclude from Lemma 3 that and , for all . Hence, the continuity of and (19) imply that

Thus, the continuity of and , the boundedness of , and Proposition 11, imply that there exists a constant independent of such that , for almost all , for all . Thus, we can apply the Dominated Lebesgue Theorem [23] to get

On the other hand, since is continuous as a uniform limit of continuous sequence of functions, we have and (note that , for all ).

When , we obtain from (21) and (5) that , for all . Thus, the continuity of the functions and along with the boundedness of the functions , , and implies that the function is bounded. Therefore, and (16) is satisfied.

When , we get from inequality (20) that as . Moreover, (5) can be written for all as

The fact that , along with (21) and (22) implies that

which proves (17).

Finally, when , (5) implies for all that

and thus, we get from (21) that , for all . Therefore, the uniqueness of limits and the continuity of imply that , for all .

Remark 13. Observe that for , the fact that ,  for  all  , means that system  (1)-(2)  does not represent a hysteresis behavior [24].

Remark 14. For the case , (17) and the fact that imply that , whenever exists.

Example 15. Consider the Following LuGre model [10]:

where parameter is the stiffness, is the average deflection of the bristles and is the output of the system, is the initial condition, is the relative displacement, and is the input of the system. The function is defined as

where is the Coulomb friction force, is the stiction force, and is the Stribeck velocity.

System (25)-(26) has a unique global solution [21, page 5]. On the other hand, the sequence of function is given by (see (5))

The following facts are proved in Example 29.

(i)There exist such that ,  for all  , where is the output when we use input instead of (see system (3)-(4)).(ii)  as , where the function is defined for all as

Thus, all conditions of Lemma 12 are satisfied.

Now, we have to find the value of and the function . We have

Thus, the function in (25) is of class (see Definition 7) and the function in (9) is defined as

Therefore, by applying Lemma 12, it follows that the system (29) satisfies (16).

Simulations. Take N/m, m/s,  N,  N, N, and  m, for all (values taken from [7]). Figure 1(a) shows that the graphs converge to the hysteresis loop as . This is the main characteristic of a hysteresis system. Also, observe that are different for different values of . This is what is called “rate-dependent” property of the model (25)-(26). Figure 1(b) presents the graph of versus ; we observe that converges uniformly to the zero function as which means that converges uniformly to when . The graph of versus is presented in Figure 1(c).

(a)

(b)

(c)

Figure 1:  (a) versus for different values of . (b) versus for different values of . (c) versus . The plots are for the LuGre model (25)-(26), where the solid lines are for .

6. Sufficient Conditions

This section presents sufficient conditions for the uniform convergence of the sequence of functions as (and hence for consistency of the system (1)-(2) with respect to ). The main results of this section are given in Lemmas 20, 23,  and 27.

6.1. Class Functions

In this subsection, sufficient conditions for the uniform convergence of as , are derived when the function is of class .

Definition 16 (see [25]). A continuous function is said to belong to class if it is increasing,   satisfies ,  and .

The following lemma generalizes Theorem in [25, page 172]. Indeed, in [25], continuous differentiability is needed, while in Lemma 17, we only need absolute continuity. Also, in [25], the inequality on the derivative of the Lyapunov function is needed everywhere, while in Lemma 17 it is needed only almost everywhere.

Lemma 17. Consider a function , where is finite or infinite. Assume the following.

(1) The function is absolutely continuous on each compact interval of . (2) There exist and such that , and

Then, for all  .

Proof. see Appendix B.

Example 18. We want to study the stability of the following system

where and state take values in , and input . System (33) has an absolutely continuous solution that is defined on an interval of the form [21, page 4].

Let be such that ,  for  all  . The function is absolutely continuous on each compact subset of because is absolutely continuous. Thus, condition in Lemma 17 is satisfied.

We have for almost all that

Thus,

Therefore, condition in Lemma 17 is satisfied with and can be any positive real number such that . Thus, we deduce from Lemma 17 that for all, and hence , for all  .

Corollary 19. Consider a function , where may be infinite. Assume the following.

(1) The function is absolutely continuous on each compact subset of .(2) There exist a class   function and constants , , and such that , and

Then, for all  .

Proof. We have from (36) that

and hence the result follows directly from Lemma 17.

Although the latter corollary follows immediately from Lemma 17, it is useful in many situations [25].

Lemma 20. Suppose that the system (1)-(2) has a unique solution and that the function is of class . Assume that there exists such that for almost all

For all , define as

for all , where is the maximal interval of existence of solution in (5). Suppose that we can find a continuously differentiable function such that

(1) there exists a function that satisfies

(2) there exist constants , continuous functions and class functions , satisfying

Then,

(i)there exist such that for all :   and , where is given in (3)-(4).(ii).

Proof. From (5) and (39), we get for all and almost all that and

For any , define as for all . Note that function is absolutely continuous on each compact subset of as a composition of a continuously differentiable function and an absolutely continuous function . The derivative of along with trajectories (44) is given for almost all and all by

By (40), there exists some , such that , for all . Let . By (41), we have for any , for almost all that

Thus, we deduce from (42), (43), (45), and (46) that

Therefore, (46) and the continuity of the functions   and   imply that there exists a constant that does not depend on , such that

Thus, we deduce from (41) that

where is defined as .

On the other hand, since , there exists such that for  all. Hence, Corollary 19 and the fact that forall, imply that foralland  for  all. Therefore, (41) implies for all and for all that

Thus, for all. Furthermore, (40), (50), and the fact that imply that as . This proves the consistency with respect to because of Proposition 5.

Moreover, by (50), there exist some , such that

On the other hand, let . Since is continuous, Lemma 3 ensures that . Let . Due to the continuity of , there exists some such that and thus (51) and the continuity of lead to ,  for  all. Therefore, for  all, which completes the proof.

Remark 21. For ,  if the function in Lemma 20, such that for some , then the graphs converge to the curve as . Hence, (1)-(2) is not a hysteresis because the hysteresis loop cannot be a function [24]. This fact is illustrated in Example 22.

Example 22. Consider the following semilinear Duhem model:

where is a Hurwitz matrix (i.e., every eigenvalue of has a negative real part), vector and state take values in . The right-hand side of (52) is Lipschitz and thus the system has a unique solution [21]. Take an input such that and that for almost all and for some . Assume that the function is of class and that . Thus, there exists such that , for all , where the function is defined in (9). On the other hand, Proposition 9 states that . This means that there exists such that we get for almost all , and all that

Thus, the facts that and imply that

The function which is defined as satisfies (38) because .

Since is Hurwitz, there exists a positive-definite matrix such that [25, page 136]

where is the identity matrix. Consider the continuously differentiable quadratic Lyapunov function candidate such that , for all . Since is symmetric, we have for all that

where and are, respectively, the maximum and minimum eigenvalues of the matrix . This shows that (41) is satisfied with and for all . Since is symmetric, we have the following matrices derivation:

Thus, we get

where is the induced 2-norm for the matrix and hence (43) is satisfied with ,  for all  . From (57), we have  for  all   that

Therefore, (54) implies that for almost all and that

where is defined in (39). Thus, (42) is satisfied with , for all and , for all .

Let be the zero function. Then (40) is verified. Take arbitrary in (say , ). Hence, all conditions of Lemma 20 are satisfied. Thus, it follows from Lemma 20 that there exist some such that for all , the solution of (52) is global with , for all . Moreover, the operator which maps to is consistent. In particular, we have as .

As a conclusion, the graphs converge to the graph of the linear function , which is defined as , for all . This means that for , the model (52) does not represent a hysteresis (see Remark 21).

Simulations. Take , , , and . Let , for all , then . Let be the function of period such that , for all , and , for all . Then, we have , for almost all . We also have . Figure 2(a) shows that the graph collapses into the identity function when . This happens because of the fact that and Remark 21. Figure 2(b) shows that the sequence of functions converges uniformly to as .

(a) versus for system  (52)

(b) versus for system  (52)

Figure 2:  (a) versus for different values of . (b) versus for different values of . In each plot, the function is the solid line.

6.2. Class Functions

In this subsection, we consider class functions. (A function is of class if and the limits and exist, are finite, and at least one of them is nonzero (see Section 5)). The main results of this subsection are given in Lemmas 23 and 27.

Lemma 23. Assume the following.

(1) The system (1)-(2) has a unique global solution.(2) For the function in system (1)-(2), there exist such that

Then, the sequence of functions of (5) is independent of and the operator which maps to is consistent.

Proof. By condition , the right-hand side of (5) is independent of . Thus, the solution of (5) is independent of . Since , the function is also independent of (this is the so-called “rate-independent hysteresis”) and hence consistency holds.

Example 24. Consider Bouc's hysteresis model [13] as follows:

where , , input , and .

The right-hand side of (62) is Lipschitz with respect to . Thus, the system has a unique solution. Furthermore, we have

Thus, , for each input and each initial state . Since is bounded and is continuous, the solution of (62) is bounded and hence global. Hence, condition in Lemma 23 is satisfied. Equation (62) can be written as

Clearly, the function is of class and satisfies condition in Lemma 23. This fact implies that the operator which maps to is consistent and is independent of .

Simulations. Let , , for  all  ,  , and input . The function is independent of and is plotted in Figure 3(a). Furthermore, Figure 3(b) shows a rate-independent hysteresis behavior; that is graphs are the same for different values of .

(a) versus

(b) versus

Figure 3:  (a) versus . (b) versus for Boucs hysteresis model (62).

Proposition 25. Let be nonconstant. There exists a unique function that is defined by . Moreover, . Assume that is nonzero on a set that satisfies . Then, is nonzero almost everywhere.

Proof. Consider the left-derivative operator defined on by . The operator is causal as depends only on values of for , and we have almost every where as so that . The operator satisfies Assumption 2. The first part of Proposition 25 follows immediately from Lemma 3. Now, let , then which implies that .

Remark 26. Observe that if    is nonzero almost everywhere, then so that by [26] we have as is absolutely continuous. An example in which does not need to be nonzero almost everywhere, is when is constant on some interval, or on a finite number of intervals, or an infinite number of intervals such that this infinite number has measure zero (e.g., countable).

Lemma 27. Let be such that is nonzero on a set that satisfies . Consider the semilinear Duhem model with and   as follows:

where , , and are row vectors, state and function and of class , and nonconstant input . Denote

For any , assume that

Suppose that there exists some , such that

Then,

(i)there exist such that ,  for all    ( is the output of the system (65) when we use the input instead of the input (see system (3)-(4))).(ii)There exists a function such that , as where is the norm of the Banach space (and hence the consistency of the system (65) is guaranteed with respect to input and initial condition ).(iii). Furthermore, for almost all , one has

 where the function is defined as in (9), that is;

Proof. The semilinear Duhem model has a unique global solution due to the Lipschitz property of the right-hand-side [21, page 5].

For any and any , let be the th component of the function . From (66), there exists some constant , such that for all  

Let . We get from (67) and (71) that

For the case , take . Let . From (67) and (68), we get . Thus, we get from (72) that

By Proposition 25, a function can be defined almost every where as with almost every where The boundedness of implies that there exists such that for almost all and all . Thus, we deduce from (73) that for all ,

Let be a function such that

For any , define as

Since , relations (66) imply that

This result can be easily checked using the same techniques used in the proof of Proposition 9. Now, consider the system

where state . The differential equation (78) verifies Carathéodory conditions with Lipschitz property with respect to . Thus, system (78)-(79) has a unique and absolutely continuous local solution [21, page 4]. Consider the Lyapunov function . We deduce from (78)-(79) that and that

We get from (67) and (68) that for almost all . Thus, the boundedness of the function (see Lemma 1) along with (80) imply that , for some . This leads to , whenever . Therefore, Lemma 17 and the fact that imply that which means that the solution of the system (78)-(79) is bounded and hence is global (i.e., is defined on ). On the other hand, the relation implies that . Thus, we obtain from systems (65) and (5) and the relations , and , that and that

for all , for almost all , where is the maximal interval of existence [21, page 4]. For any , let be defined as . Since , the system (81) can be written for all , for almost all as

For any , consider the Lyapunov function with ,  for all  . By (82) and the boundedness of both and the solution of (78)-(79), there exists some such that for almost all and all ,

Thus, we get from (74) that

Therefore, Lemma 17 and the fact that imply that for all and almost all , and hence we obtain, for almost all that

which implies that ,  for  all   and (77) implies that

On the other hand, the continuity of implies that ,  for all   (see Lemma 3). Thus, there exists some and with

Moreover, we get from (78) and (81) for all that

Thus, by the boundedness of functions and , and the relation (88), there exist positive constants and independent of , such that

which means that converges to in as because of (77) and (87). Define as . Since for all , relations (78)-(79) imply for all that

Moreover, using the relation , it can be easily verified that .

Remark 28. In Lemma 27, the sequence of functions converges in as . This result is stronger than the one obtained in Lemma 20, where the convergence is only in . An application to this stronger convergence is given in the following example.

Example 29. The LuGre model is described by [10] as follows:

where parameters , , and are, respectively, the stiffness, damping, and viscous friction coefficients, is the average deflection of the bristles, is the initial state, is the relative displacement and is the input of the system, and is the friction force and is the output of the system. The function is defined as

where is the Coulomb friction force, is the stiction force, and is the Stribeck velocity.

The LuGre model can be written in the form of the system (65) with , , and . We have

Clearly, conditions (67) and (68) are satisfied. Thus, Lemma 27 implies that , as , where the functions and are defined for all as

Also, there exist some , such that for all , the solution of (92) is global with .

Now, the following analysis is not a part of Lemma 27, but it follows straightforwardly from it.

Let be the output of the system when we use the input instead of . We obtain from (93) for almost all that

which leads to

where is defined as , for all and for all  .

Since , as , we have and . Thus, we get from the boundedness of and (99) that

which means that the operator which maps input and initial state to output is consistent.

The conclusion of the analysis is that the hysteresis loop of the LuGre model is , where is given in (97). Observe that this conclusion has been obtained due to the convergence of in (see Remark 28).

Simulations. Take  N/m, m/s, N, N, ,    and N. Let be a function of period that is measured in meters such that , for all  s, and , for all s. Figure 4(a) shows the uniform convergence of to as . Figure 4(b) shows that the graphs converge to the hysteresis loop . Figures 4(c) and 4(d) show that the sequence of functions and converges uniformly to and as . Figures 4(e) and 4(f) give the graphs of the functions and .

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4:  (a) versus . (b) versus . (c) versus . (d) versus . (e) versus . (f) versus . The plots are for system (92) of Example 29. In each plot, and are the solid black lines.

7. Conclusion

This paper presented a classification of the possible Duhem models in terms of their consistency with the hysteresis behavior. Three classes of models have been considered in relation with the range of a parameter . For , it has been shown that the corresponding generalized Duhem model does not represent a hysteresis behavior. For , it has been shown that the semilinear Duhem model is not compatible with a hysteresis behavior. In all other cases, necessary conditions and sufficient ones have been derived to insure the consistency of the Duhem model with the hysteresis property. Numerical simulations have been carried out to illustrate the obtained results.

Appendices

A. Proof of Proposition 11

Without loss of generality, assume that . Definition 7 implies that there exists some , such that

Therefore, there exists some with

Thus, the substitution implies that

Define , as

The function is continuous because is continuous. Moreover, we have ,  for all . The function is nondecreasing on and nonincreasing on . This implies that

Thus, we get from (A.3) for all and all that

which completes the proof.

B. Proof of Lemma 17

We discuss two cases, Case  1   and Case  2  .

Case 1 (). The objective of what follows is to prove that we have . To this end, assume that , such that . Put . The set is nonempty because . Define ; then there exists a real sequence such that . By the continuity of , we have . This fact implies that leading to . Also, there exists a real sequence such that , for all and . Since is continuous, we get which leads to . Let . The set is nonempty since . Define ; then using a similar argument as above we get which implies that and .

Claim 1. .

Proof. Assume that such that . By definition of , we get . Since is continuous and , we can use the intermediate value theorem to find such that which implies that and which is a contradiction.

Let . We consider the following subcases and . If , then and hence Claim 1 and (32) imply that

Thus, the absolute continuity of the function implies that which contradicts the fact that . Now, if , let = Inf. It can be shown that and . Thus, Claim 1 and (32) give

which also contradicts the facts that is absolutely continuous on each compact interval and .

We have, thus, proved that in Case 1,

Case 2 (). Assume that such that . Let be the smallest real number such that (it exists due to the continuity of ). Then, seeing as an initial time, and as an initial condition, it follows from Case 1 that , we have . So, we have to just analyze what happens in the interval and discuss the case . The analysis of both situations is the same so that we focus on the case .

Assume that such that . Let be the smallest real number such that (it exists due to the continuity of ). Then, , we have which implies that for almost all we have . Since is absolutely continuous, it follows that . This contradicts the fact that which means that . Since , it follows that so that .

As a conclusion, we have proved that in Case 2, .

Acknowledgment

Supported by Grant DPI2011-25822 of the Spanish Ministry of Economy and Competitiveness.

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