Bilinear Minimax Optimal Control Problems of the Velocity Term in an Extensible Beam Equation with Rotational Inertia

Hwang, Jinsoo

doi:https://doi.org/10.1155/2023/9903316

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 9903316 | https://doi.org/10.1155/2023/9903316

Bilinear Minimax Optimal Control Problems of the Velocity Term in an Extensible Beam Equation with Rotational Inertia

Jinsoo Hwang¹

Academic Editor: Xiaolong Qin

Received18 Jul 2022

Revised10 Sept 2022

Accepted19 Nov 2022

Published05 Jan 2023

Abstract

The author studies the minimax optimal control problem for an extensible beam equation that takes into account rotational inertia effects. A velocity term multiplied by the bilinear control and disturbance functions to construct a minimax optimal control strategy is added. The existence of an optimal pair, meaning optimal control and perturbation, is proved by assuming some conditions on the considered quadratic cost function. The optimal conditions for optimal pairs are provided by the adjoint systems that correspond to some physically meaningful observation cases.

1. Introduction

This study is devoted to the study of the minimax optimal control problem for the following model of an extensible beam with rotational force:where is a bounded domain in with a sufficiently smooth boundary , where (mainly ), the positive constant represents the effect of the rotational inertia of the beam, is a nonlinear term that is explicitly explained later, is a control input function, and is a forcing term. We consider either hinged boundary conditionor clamped boundary conditionwhere is the outward unit normal vector tailing on . And we consider the initial condition

Equation (1) with has been extensively studied in many articles. Initially, Woinowsky [1] proposed a one-dimensional version with the goal of describing the transverse deflection of an extensible beam. From a more physical point of view, including mathematical analysis of the extensible beam model, we can refer to Ball [2], Dickey [3], and Eisley [4].

We can also find many pioneering studies on the same equation, such as the well posedness of the equation with an additional damping term, or the stability of a solution to the equation (see [5–9]).

By the way, the case where in Equation (1) is called a Rayleigh beam that is introduced to consider the effect of rotational inertia on a large deflection beam model. For the introduction and studies on Rayleigh beam models, we can refer to Chueshov and Lasiecka [10, 11].

As a contribution to control theory to nonlinear beam equations without rotational inertia , we studied in [12] the optimal control problem by the framework of Lions [13] (cf. [14], [15]) using distributed forced control variables. In [12], the optimal control problems were studied for Equation (1) in which and without and under the framework of Lions [13]. And quite recently, we studied in [16] that the nonlinear solution map of equation (1) with is Fréchet differentiable and applied our results to a bilinear robust control problem in which the control variable is taken as a multiplier of the displacement term rather than the velocity term. That is to say, is replaced by in Equation (1). By the minimax optimal control strategy (cf. [17]), the existence of an optimal pair was shown and the necessary optimality condition of the optimal pair was studied, which satisfieswhere is a quadratic cost, and are distributed control and disturbance, respectively, and and are admissible sets.

As is noted in [18], [19], in a hyperbolic control system, a control strategy with a velocity term rather than a displacement term seems preferable. The author in this study has already pointed this out and tried to apply it to Equation (1) with . However, we found that this requires more regularity of the control variable or solution, which is not favorable in control theory. With this in mind, in this study, the author is motivated to study the minimax optimal control problems for the bilinear control input of the velocity term of Equation (1) with .

A description of the minimax optimal control strategy is given in [16]. To set up the control problem, the multiplier function in Equation (1) is replaced with . Here, and are control and disturbance variables, respectively, and and belong to the set of admissible control and disturbance sets and , respectively. The following quadratic cost function is considered:where is a solution satisfying Equation (1), is an observation operator, are the Hilbert spaces of observation, control, and disturbance (or noise) variables, respectively, is the aiming value, and and are the positive constants related to the weights of the second and third terms of (6).

In this study, the author tries to find and characterize control and noise variables to minimize and maximize the quadratic cost (6) within and , respectively. In other words, it is a problem of finding and characterizing a saddle point that satisfies (5). As stated in [16], without any confusion, the term optimal pair is still used to denote these saddle points in (5). For the study of the existence of an optimal pair satisfying (5), the minimax theorem in infinite dimensions proposed by Barbu and Precupanu [20] (cf. [17]) is used. To do this, we need to ensure that the solution map is differentiable and that its derivative is continuous in the Hilbert norm topology. Therefore, the Fréchet differentiability of the solution map was verified from in Equation (1) to the solution of Equation (1), and the local Lipschitz continuity of the Fréchet derivative was proved by imposing some conditions on the nonlinear function in Equation (1).

Next, the necessary optimality conditions for the optimal pairs are derived, corresponding to physically meaningful observation cases. In this paper, the author mainly considers two observation cases: The first observation case is the distribution of velocities, and the other is the distributive and terminal value observation case. In order to derive the optimal conditions for the optimal pair, we need to find and use the relevant adjoint equation corresponding to the observed case. The optimal pair can then be given explicitly through the adjoint system.

The novelty of this paper is summarized as follows: In the case of velocity distribution observation, the author is faced with the difficulties of regularity in the process of deriving the optimal condition for the optimal pair through the adjoint equation, but this problem is overcome by the double regularization method. For the distributive and terminal value observation case, due to the lack of regularity of the control and noise variables, it is not possible to explicitly construct the related adjoint system. In general, control can be irregular; instead of assuming regularity in admissible control and noise sets, the transposition method is used to derive the necessary optimality condition for the optimal pair.

2. Notations and Preliminaries

Given a Banach space , its topological dual is denoted by , and the duality pairing between and by . For simplicity, the following abbreviations are used:where and represent the scalar product and norm on (or ). As is known, it is denoted by that the Sobolev space of order on with the -based Sobolev norm. is the completion of for the norm.

It is denoted asand the operator is defined by

Since is self-adjoint from into , it is strictly positive on due to (9), and the injection of in is compact. Thus, from the spectral theory of self-adjoint compact operators in the Hilbert space as given in ([27], Theorem 7.7) (cf. [21]), a complete orthonormal basis of , , can be found,which consists of eigenvectors of , such that

Thus, the powers of can be defined for any , such that for ,

For , is the completion of for the following norm:

For , the scalar product and norm of can be written alternatively as

Embedding is as follows:

Furthermore, for ,

For simplicity, throughout the paper, the author denotes . Then, for , are Hilbert spaces with the following scalar products and norms:

For all . Thus,

For all .

From the well-known embedding theorem, as given by Adams [22], the following embeddingis compact when .

The following assumptions for in Equation (1) are given as follows. in Equation (1) is a function with : (A1) Let be the function given by , and we make the following assumption: (A2) There exists a constant such that

It is deduced from (20) that, for every , there exists a constant such that

We can convert Equation (1) with the boundary condition (2) or (3) and the initial condition (4) to an absolute evolution equation, which is given bywhere .

Remark 1. From (A1), (A2), and (19), the following conditions are deduced which are suitable for the nonlinear operator in Equation (23):(i)It follows from (21) and (19) that the nonlinear operator in Equation (23) is a bounded operator from into , a Fréchet differentiable with the differential , and Lipschitzian from the bounded sets of into . Indeed, for every , there exists such that where .(ii)As a consequence of (A1) and (22), it is deduced that there exists , such that and that for every , there exists a constant such thatThe Hilbert space is defined byequipped with the normwhere and represent the distributional derivatives of with respect to time variables.
In the following, for simplicity, is frequently used to denote the generic constant and the integral variable is omitted from all definite integrals.

Definition 1. The author calls a weak solution of Equation (23) if , and the following holdsThe following existence theorem can be given by referring to Chueshov and Lasiecka [10, 11].

Theorem 1. Let (A1) and (A2) be fulfilled and and . Then, there exists a weak solution of Equation (23), satisfying

To prove the regularity and uniqueness of a weak solution of Equation (23), the steps mentioned by Lions and Magenes are followed ([23], pp. 275–278). First, the following lemma provided by Lions and Magenes is exploited [23].

Lemma 1. Let be two Banach spaces, with dense and being reflexive.and then,

Then, the following improved regularity for the weak solution of Equation (23) can be proved.

Corollary 1. Let be a weak solution of Equation (23). Then, it can be seen that

Proof. From Dautray and Lions ([26], p. 480), it is clear that ↪ . Therefore, since , the proof is the immediate consequence of Lemma 1 obtained by setting to have and by setting to have .
Hence, the proof is completed.
The following lemma is frequently used to obtain many estimates throughout the study.

Lemma 2. If is the weak solution of Equation (23), then the equation is obtained as follows:

Proof. Based on (32), we follow the proof given in Lions and Magenes ([23], pp. 276–279). By regarding in ([23], pp. 276–279) as , the following equation is obtained through the double regularization, as given in ([23], pp. 276–279):Since(33) can be obtained by combining (34) with (35).
Hence, the proof is completed.
Now, it paves the way to state the following theorem.

Theorem 2. Let be the weak solution of Equation (23). Then, . Moreover, the mapping of into is locally Lipschitz continuous. Let . Then, the equation is obtained as follows:

Proof. Using the energy equality method by Dautray and Lions ([26], pp. 578–581) combined with (32) and (34); then, the equation is obtained as follows:Therefore, the author focuses on showing (36). For each , the author denotes by . Then, it is deduced from Equation (23) that satisfiesin the weak sense whereBy analogy with (34), the energy equality for the weak solution of Equation (38) can be deduced as follows:From (24) and the fact that ↪ ,where depends only on . Thanks to (41),Since ↪ when ,where depends only on . Then, for the other terms to the right of (40), the author refers to [16] and uses Gronwall’s lemma to obtainSince is isomorphism, it is inferred from Equations (38) and (44) thatwhere is also isomorphism. Thus, finally from (44) and (45), the following is obtained:Hence, the proof is completed.

3. Statement of the Main Results

The goal of this section is to describe the minimax optimal control problem for the proposed equation and to state the main results. As stated before, the control problem is set up as follows: in equation (23) is replaced by , where and belong to the admissible control set and the admissible disturbance (or noise) set , respectively.

The assumptions for and are given as follows: (A3) The control set of is given by Here, . (A4) The set of (noises) is given by Here, .

For simplicity, the author denotes Thanks to Theorem 1, for fixed , there exists a unique weak solution satisfyingwhere Thus, it is deduced that the solution map is well defined. The quadratic cost function is considered as follows:where is the aiming state value and and denote the Hilbert space of observation variables and the observation operator, respectively.

As stated before, the control strategy is to find and characterize optimal control even at the worst disturbance (or noise) , satisfyingwhere is called the saddle point of the function in (50). Then, using the methods given in [16] (cf. [18]), the author characterizes the optimal pair by stating the necessary optimal conditions through the adjoint equations related to Equation (49) and the cost (50).

The main results of this paper are as follows.

3.1. Case of and Observe

If we take and observe in (50), then the optimal pair can be characterized as follows.

Theorem 3. Let (A1)– (A5) be fulfilled. For sufficiently large and in the cost (50), there exists an optimal pair satisfying (50) such that it can be given bywhere is the weak solution of Equation (49), corresponding to , and is the weak solution ofwhere

is an aiming observation value.

3.2. Case of and Observe

If we take and observe in (50), then the optimal pair can be characterized as follows.

Theorem 4. Let (A1)–(A5) be fulfilled. For sufficiently large and in the cost (50), there exists an optimal pair satisfying (50) such that it can be given bywhere is the weak solution of Equation (49), corresponding to , and is the weak solution satisfyingwhereis defined in (54), and is an aiming observation value.

Remark 2. (i) Assumption (A5) is specified in the next section. (ii) The weak solution satisfying (56) is usually called the transposition solution (cf. ([13], pp. 291–295)) and formally satisfies the following equation:The reason for using the transposition solution in the necessary optimality conditions is explained later.

4. Differentiability of the Control-to-State Map

The goal of this section is to show the Fréchet differentiability of the nonlinear solution map from to , where satisfieswhere is fixed. Then, from Theorem 2, it is deduced that the control-to-state map from the term of Equation (59) to is uniquely defined and continuous.

The following definitions of functional differentiation are presented.

Definition 2. The control-to-state map of into is said to be Fréchet differentiable at if there exists an operator and a mapping with the following properties. For any , there existsas , where and are the weak solutions of Equation (59), corresponding to and , respectively.

Definition 3. Let be a weak solution of Equation (59), corresponding to . Suppose that the first variation at in the direction exists, and there exists a continuous linear operator such thatThen, the control-to-state map is said to be Gâteaux differentiable at in the direction , and we write .
As is well known, Fréchet differentiability also implies Gâteaux differentiability, and the Gâteaux derivative then coincides with the Fréchet derivative. Therefore, the author gives priority to the study on the Fréchet differentiability of the control-to-state map.

Theorem 5. Let be the weak solution of Equation (59). Then, the Fréchet derivative of at in the direction of , denoted by , is given by the weak solution

Proof. LetThen, from Theorem 2 and Theorem 3 in [16], the following equation is obtained:Due to (A2) with ↪ and (64), it can be seen from Theorem 2 that there exists a unique weak solution to linearized system Equation (62) and satisfiesHence, from (65), the mapping is linear and bounded. Therefore, it is verified that the existence of an operator with for any .
For convenience of calculation, the author denotes by . Then, from Theorem 3 given in [16],whereIt is known that satisfiesIn the weak sense,Regarding in Equation (38) as and estimating the weak solution of Equation (68) through the energy equation for , the following inequality is obtained:First, since when , the following inequality is obtained with (65) thatFor all . Thus,By similar arguments in [16] together with Theorem 2 and (65), it can be deduced thatwhere . And also,As and is a - class nonlinear operator, the author infers with (A2) and the Lebesgue-dominated convergence theorem thatTherefore, the author deduces with (74), (75), and (76) thatHence, from (70)–(77), the following equation can be obtained:Hence, the proof is concluded.
The following results are used to prove the existence of optimal pairs in the next section. Another assumption for is needed to prove the following result. In addition to (A1) and (A2), we give another assumption for in equation (1) as follows: (A5) in Equation (1) is a function, and there exists a constant such that

Proposition 1. Let (A1), (A2), and (A5) hold. Given , the Fréchet derivative is locally Lipschitz continuous on . Indeed, it is satisfied that

Proof. Let be the weak solutions of Equation (62), corresponding to . The author sets . Then, it is deduced that satisfiesIn the weak sense,By analogy with (70), it is deduced that the weak solution of Equation (81) can be estimated asFrom (65), the following equation can be obtained:Hence, by Theorem 2 and (84), can be estimated asSimilarly, can be estimated as follows:Thanks to (A5) and (84) with , can be estimated byBy analogy with (71) and (72), the following equation can be obtained:Finally, by (83)–(88), the following equation can be obtained:Hence, the proof is concluded.

5. Proof of the Existence

The goal of this section is to study the existence of an optimal pair satisfying (5) under some reasonable conditions for the constants and in the quadratic cost. First, the following result is needed.

Proposition 2. The mapping is sequentially continuous from , endowed with weak- topology, to .

Proof. Let , and let be a sequence such thatrespectively, as . From now on, is denoted by , which is the solution of Equation (49), in which and are replaced by and , respectively. From Theorem 2, the following equation can be obtained:Hence, by Rellich’s extraction theorem, a subsequence can be extracted from , say again , and can be found such thatFrom the fact that embedding is compact, we can apply Simon’s compact embedding theorem [24] to (93) and (94) to verify thatHence, if necessary, a subsequence can be found such thatTherefore, from (21), (92), and (96),Since embedding is compact, we can apply again Simon’s compact embedding theorem [24] to (92) and (94) to verify thatHence, a subsequence can also be found if necessary such thatFor all , the following equation can be obtained:By (99) and (90) with , it is evident that (100) converges to 0 as . Thus,as. Finally, in (28) is replaced by , in (28) is replaced by , and is taken. Then, the limit satisfiesFrom Theorem 2, it is finally deduced that the unique solution of Equation (102) equals to in . Thus,Hence, the proof is concluded.

Remark 3. Since we are considering observation spaces in which is compactly embedded, the author infers from (103) thatwhere is the observation operator in (50).
From Theorem 5, it is known that the Fréchet derivative of at in the direction , which is denoted by , is given as the weak solution of

Proposition 3. Let (A1)– (A5) be satisfied. For sufficiently large and in (50),where are the Gâteaux derivatives of at in the directions of and , respectively.

Indeed, the maps and are convex for all and concave for all , respectively.

Proof. From the Fréchet differentiability of the solution map where is fixed, (106) can be written again bywhere is the weak solution of Equation (105), in which is replaced by . It can easily be verified that (107) equals toBy Theorem 2, (65), and Proposition 1, the left hand side of (108) can be estimated as follows:From (108)–(110), sufficiently large can be found, depending on , , and so that (106) is satisfied for any . Thus, we know the map , where is fixed is convex for such .
Similarly, sufficiently large can also be found, depending on , , and so that (106) is satisfied for any . This implies the concavity of the map , where is fixed.
Hence, the proof is concluded.
Finally, the following existence theorem of optimal pairs is presented (cf. [17] and [20]).

Theorem 6. Assume that (A1)–(A5) is fulfilled. For sufficiently large and in (50), there exists such that satisfies (3.3).

Proof. By Proposition 3, for sufficiently large and , we know that the maps with fixed and with fixed are convex and concave, respectively. Together with (104), this theorem can be proved by referring to Theorem 5 of [16].
Hence, the proof is concluded.

6. Proofs of Necessary Optimality Conditions

This section is dedicated to deriving the optimal conditions required for each of the optimal pairs of minimax optimal control problems along with the costs from Theorem 3 and Theorem 4.

6.1. Proof for the Case of and

For this purpose, adjoint Equation (53)is introduced. The well posedness of Equation (53) is clarified by the following proposition.

Proposition 4. Equation (53) admits a unique weak solution .

Proof. By reversing the direction of time in Equation (53), it can be seen that Equation (53) is changed as follows:where is defined in (54) and .
Estimating the weak solution of Equation (111) through energy equality, as given in the proof of Theorem 2, the following equation is obtained:We noteBy (21) with (19), it is deduced thatTherefore, if we combine (112) and (114) and apply Gronwall’s lemma to the combined inequality, we obtainBased on this, using Faedo–Galerkin’s approximation procedure, it can be verified that Equation (111) has a unique weak solution .
Hence, the proof is concluded.
Before discussing the first-order optimality condition for the minimax optimal control problem (51) for the observation case of Theorem 3, the following remark is noted.

Remark 4. In deriving the optimality condition for the observation case of Theorem 3, the weak form of Equation (53) needs to be multiplied by . However, as is often encountered in hyperbolic problems, it is the only formal procedure because and . In [25], this difficulty can be eliminated by employing the regularization method proposed by Lions ([13], pp. 286–288).

Proof of Theorem 1. Thanks to the assumptions of Theorem 3, it is evident from Theorem 6 that there exists an optimal pair in (51) for the observation case of Theorem 3.
Let be an optimal pair in (51) with the cost (50) through the observation case of Theorem 3 and be the corresponding weak solution of Equation (49).
The Gâteaux derivative of the cost (50) through the observation case of Theorem 3 at in the direction , with for sufficiently small , yieldswhere is a solution of Equation (105).
To eliminate the difficulty caused by the regularity of weak solutions, Lions’ regularization method is employed [13] (cf. [25]). For that, the author extends the time domains of Equation (105) and Equation (53) to as follows:AndIn fact,andFor convenience of representation, the scalar product on or antiduality is denoted as . Let be a mollifying sequence on . Then, the right hand side of (120) equals may be integrated by parts to obtainUsing(122) can be given again bywhereWe immediately know by integration by parts. Sinceas . It can be deduced that as . In a similar manner, and as . This justifiesSince is an optimal pair in (51), the following equation is obtained:From (116), (120), and (127), (128) implieswhere . By referring to the proof of ([16], Theorem 4.2), it can be deduced from (129) that the optimal pair can be given byHence, the proof is concluded.

6.2. Proof for the Case of and

Before proceeding the proof of Theorem 4, the following remark is noted.

Remark 5. In the case of observation in Theorem 4, adjoint equations cannot be constructed directly, because they must involve the term , as shown in (58), where is an adjoint state. However, the regularity of the admissible set is insufficient to guarantee the existence of adjoint equations. This difficulty is overcome by employing the modified transposition method.
The transposition method is explained. For any given , the following equation is considered:where , , andFrom Theorem 2, it is evident that Equation (131) has a unique weak solution satisfyingThus, the space Equation (131) as ranges over .
Then,Endowed with the norm , is the Hilbert space, and the map defined byis an isomorphism from onto . Therefore, for any continuous linear functional on , there exists unique such thatWe call such a to be a solution by the transposition method associated with the bounded linear functional on for the transposed equation with respect to .
To discuss the existence of a solution to (formal) adjoint system Equation (58), in (136) is taken as follows:It is easily verified thatHence, from (137)–(139), the following equation is obtained:As a consequence, the functional is linear and bounded on . Also, it is deduced that the map is continuous from to . Hence, the following proposition is obtained.

Proposition 5. For given and , there exists a unique solution of Equation (58) such thatwhere is defined in (135) and is defined in (137). Moreover, the solution of (141) satisfies

Proof of Theorem 2. Let be an optimal pair in (51) with the cost (50) through the observation case of Theorem 4 and be the corresponding weak solution of (49).
The Gâteaux derivative of the cost (50) through the observation case of Theorem 4 at in the direction , with for sufficiently small , yieldswhere is a weak solution of Equation (105). Hence, if the author takes in (141), then it is inferred from (142) thatSince is an optimal pair in (50), the following equation is obtained:Thus, from (144) and (145), the following equation is obtained:where . By analogy with the case of (130), it can be inferred that from (145), the optimal pair can be given as (possibly not unique)Hence, the proof is concluded.

7. Conclusion

In this study, the author investigated the bilinear minimax optimal control problems for an extensible beam equation with rotational inertia effects. The well posedness of the solution to the Cauchy problem and the improved regularity theorem were given for the underlying equation. The author formulated the bilinear minimax optimal control problem for the state equation by adding the bilinear control and noise inputs of the velocity term to the state equation. Using and analyzing the properties of the Fréchet derivative of the nonlinear solution map from the bilinear control and noise inputs to the solution of the equation, the author proved the existence of an optimal pair and found its necessary optimality conditions corresponding to the following physically meaningful observation cases. The main novelties of this study can be summarized as follows: In the case of velocity distribution observation, the double regularization method was used to derive the optimality condition for the optimal pair through the adjoint equation. For the distributive and terminal value observations, instead of assuming regularity for the data and control sets required to construct an explicit adjoint equation, the transposition method was used to derive the necessary optimality condition for the optimal pair.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This research was supported by the Daegu University Research Grant 2019.

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Copyright © 2023 Jinsoo Hwang. 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.

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