Quadratic Cost Minimax Optimal Control Problems for a Semilinear Viscoelastic Equation with Long Memory

Hwang, Jin-soo

doi:https://doi.org/10.1155/2021/6689119

Journal of Mathematics

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

Research Article | Open Access

Volume 2021 | Article ID 6689119 | https://doi.org/10.1155/2021/6689119

Quadratic Cost Minimax Optimal Control Problems for a Semilinear Viscoelastic Equation with Long Memory

Jin-soo Hwang¹

Academic Editor: Dimitri Mugnai

Received25 Dec 2020

Revised05 May 2021

Accepted11 May 2021

Published24 May 2021

Abstract

In this paper, we study the quadratic cost minimax optimal control problems for a semilinear viscoelastic equation with long memory. A global well-posedness theorem regarding the solutions to its Cauchy problem is given. We formulate the minimax control problem with bilinear control inputs and corresponding disturbances. Under some assumptions, we prove the existence of optimal pairs and give necessary optimality conditions for optimal pairs in some observation cases.

1. Introduction

We study the minimax optimal control problems for the following viscoelastic equation with long memory:

As is widely known, equation (1) is considered a typical model of viscoelastic waves with a long-memory damping. By analyzing the above equation with various boundary conditions or additional linear or nonlinear terms, one can study the eventual properties of the viscoelastic dynamics, such as exponential decay or blowup. We can find many research articles in this direction, to name just a few, refer to Cavalcanti et al. [1, 2], Xiao and Liang [3], and Messaoudi [4] and references therein. However, the research on other applications including optimal control or identification problems for the above state equation is few.

In this paper, we study the quadratic cost bilinear minimax control problems for equation (1) with the control or disturbance function . For the bilinear control problems, we refer to Bradley and Lenhart [5–7] in which they studied the bilinear control problems on linear hyperbolic state equations. Recently, Belmiloudi [8] employed the bilinear minimax control framework to study robust control in a linear parabolic state equation. In [9], we extended the results in [8] to a quasilinear beam equation by proving the Fréchet differentiability of the nonlinear solution map.

The minimax control methods have been employed by many mathematicians for various control problems (see Arada and Raymond [10], Lasiecka and Triggiani [11], and Li and Yong [12]). As explained in [8, 9], in this paper, the minimax control framework is employed to consider the effects of disturbance (or noise) in control inputs such that a cost function achieves its optimum (minimum) even in the case of the worst disturbances of the system. For the purpose, we replace the bilinear multiplier in equation (1) by , where is a control variable that belongs to the admissible control set and is a disturbance (or noise) that belongs to the admissible disturbance set . We also introduce the following cost function to be minimized within and maximized within :where is the solution of equation (1) with , is a Hilbert space of observation variables, is a continuous observation operator, is a desired value, and the positive constants and are the relative weights of the second and third terms on the right-hand side of (2).

In this paper, our goal is to find and characterize the optimal controls of the cost function (2) even in the worst disturbances through the control input in equation (1). This leads to the problem of proving the existence of an admissible control and disturbance (or noise) such that is a saddle point of the functional of (2), that is,

Then, we derive optimality conditions for such in (3) corresponding to some observation cases. As in [9], we use the term optimal pair to indicate such a saddle point in (3).

As a main tool to prove the existence of an optimal pair satisfying (3), we use the minimax theorem in infinite dimensions given in Barbu and Precupanu [13]. For this, we prove the Fréchet differentiability of the nonlinear solution map and its local Lipschitz continuity.

Next, we derive the necessary optimal conditions for some observation cases that should be satisfied by the optimal pairs in these observation cases. To derive these conditions, we refer to the previous results in [5–7] dealing with linear problems. Especially, we deduce the necessary optimality condition for the velocity distribution observation case which is physically meaningful. We propose an appropriate adjoint system for the velocity observation case. To author’s knowledge, this is a newly developed adjoint system to deal with the second-order Volterra-type state equation. Finally, when deducing the optimality condition for the velocity observation case, we use the regularization method proposed by Lions [14] (cf. [15]) to overcome some difficulties. This is another novelty of the paper.

2. Preliminaries

2.1. Notations and Assumptions

Let be an open, connected, and bounded set of with the smooth boundary . We set and for . We consider the following viscoelastic equation with long memory with the Dirichlet boundary condition:where and are given initial values, is a given forcing function, is a memory kernel function, and is a bilinear forcing control function applied to the system together with displacement .

We denote , where We endow the space with the usual scalar product and norm:where means the inner product on . Let us denote the topological dual space of by and the dual pairing between and by . Then, operator defined byis a nonnegative self-adjoint operator with domain . Also, it is obvious that each natural topological imbeddingis continuous and compact.

According to Adams [16], we know the following embeddings:

Throughout this paper, the following assumptions will be effective: For the exponent in equation (4), we assume that For the above , we define the nonlinear function by . Then, it is easily verified that and . For given , we can deduce from (8) and (11) that the nonlinear operator , defined by , is well defined. For the memory kernel function in equation (4), we assume that

Now, equation (4) can be rewritten as the abstract initial value problem described by the following second-order semilinear Volterra integrodifferential system:

2.2. Well-Posedness and Continuity of the Solution Map

The Hilbert space of the weak solutions of equation (13) is defined byequipped with the normwhere and denote the first- and second-order time derivatives of in the sense of distribution.

Definition 1. A function is said to be a weak solution of equation (13) if and satisfieswhere is the space of distributions in .

Remark 1. For the existence and uniqueness of weak or strong solutions of equation (13) without the integral term, one can refer to the results of an abstract semilinear wave equation given in Teman ([17], pp. 212–214). In addition, by referring to the results in [18] dealing with the Volterra-type semilinear evolution equations, one can verify the well-posedness of equation (13).
Let be a Banach space. Set

Lemma 1. Assume thatand is a corresponding weak solution of equation (13). Then, we can verify that

Proof. By regarding in equation (13) as in [18] and noting (18), we can use the result of [18] to obtainThen, as shown in [18] (cf. Lions and Magenes [19], pp. 275–278), we can obtain (19) from (20). This completes the proof.
The following lemma is used frequently and importantly in this paper.

Lemma 2. Let be a weak solution of equation (13) with condition (18). Then, for each , we have the following energy equality:

Proof. By (19) and the uniform boundedness theorem, and for each . Thus, we know that all terms in (21) are meaningful. From (18) and (20), we know that . Thus, by applying the result in Proposition 2.1 of [18] to equation (13), we obtainSincewe can combine (22) and (23) to obtain (21).
This completes the proof.
With the help of (21) or (22), we can address the following theorem.
Throughout this paper, we use as a generic constant and omit the integral variables in any definite integrals without confusion.

Theorem 1. For given condition (18), let be the corresponding weak solution of equation (13). Then, we can assure that . Moreover, the solution mapping of into is locally Lipschitz continuous. Indeed, for each and , the following is satisfied:where is a constant depending on the data.

Proof. Referring to Dautray and Lions ([20], pp. 578–581), we can make use of (22) to show that is continuous, which implies the following:Thus, every weak solution of equation (13) with data condition (18) exists in . From this result, we show inequality (24): we denote by . Then, from equation (13), we can know that satisfies the following in the weak sense:whereBy using , (8), (11), and the Hölder inequality, we note thatTo obtain the estimation for in equation (13), we apply energy equality (22) to equation (26). Then, the energy equality for can be given byBy (28), we can verify thatBy estimating other terms in the right-hand side of (29) as in [18], we can getHere, we note thatThus, plugging (32) to (31) and applying Gronwall’s lemma, we have the following estimation:which immediately impliesSince is an isomorphism, by conducting similar estimations in equation (26), we can obtain from (34) thatHence, by (34) and (35), we can prove (24).
This completes the proof.

Remark 2. From Theorem 1, for fixed , we can define the continuous solution map from the term to satisfying equation (13). Indeed, for each , we haveThus, from now on, to emphasize the fact that the only varying variable of the weak solution of equation (37) is the bilinear multiplier , we use the notation where satisfiesFurthermore, we present the following weakly continuous results to study the existence of the optimal pair in the subsequent section. For this, we need the following lemma.

Lemma 3. (see Simon [21]). Let , and be Banach spaces such that each embedding is continuous and the embedding is compact. Then, a bounded set of is relatively compact in .

Proposition 1. For fixed , the solution map from to of equation (37) is weakly continuous.

Proof. Let , and let be a bounded sequence such thatFrom now on, each state is a solution ofFrom Theorem 1, we havewhich implies that remains in a bounded set of . Therefore, we can find a subsequence of , say again , and find such thatSince the embedding is compact, we can apply Lemma 3 to (42) and (43) with to verify thatThus, we can find a subsequence of , if necessary, still denoted by , such thatFor any given , using notation (27), we can deduceTherefore, from (45) and (46), we can readily find a subsequence of , if necessary, still denoted by itself, such thatFrom (38) and (45), we may extract a subsequence of , denoted again by , such thatReplacing by in equation (39), if necessary, and letting , we can conclude by the standard arguments as in Dautray and Lions ([20], pp.561–565) that the limit is a weak solution ofMoreover, by the uniqueness of the weak solutions, we can conclude that in , which implies that weakly in .
This completes the proof.

3. Quadratic Cost Minimax Control Problems

In this section, we study the quadratic cost minimax optimal control problems for equation (52). Let the following be the set of the admissible controls:where and are lower and upper bounds of the control variables, respectively. Let the following be the set of the admissible disturbance or noises:where and are lower and upper bounds of the disturbance (noise) variables, respectively. For simplicity, let be a product space defined by .

Using Theorem 1, for fixed , we can uniquely define the solution map , which maps from via to the weak solution , where satisfies

The weak solution is called to be the state of the control system.

The quadratic cost function associated with control system (52) is given bywhere is a Hilbert space of observation variables, is a continuous linear operator, that is to say, observer, is a desired value, and the positive constants and are the relative weights of the second and third terms on the right-hand side of (53).

As indicated in the introduction, the minimax optimal control problem can be summarized as follows:(i)Find an admissible control and a disturbance (or noise) such that is a saddle point of the functional of (53), that is, In this paper, we call such a pair in (54) as the optimal pair for the minimax optimal control problem with cost (53). To show the existence of the above saddle point (optimal pair), we need to show the solution map is differentiable in some sense. Then, with the assumptions on the weight constants and the exponent , we utilize the arguments in Barbu and Precupanu [13] to show the existence of the optimal pairs.(ii)Characterize (optimality condition): in characterizing these optimal pairs, we introduce an appropriate adjoint system corresponding to the observed case and deduce the necessary optimal conditions through the variational inequality.

3.1. Differentiability of the Nonlinear Solution Map

In this section, we address the Fréchet differentiability of the nonlinear solution map, which is desirable for many applications.

For our study, we define the Fréchet differentiability of the nonlinear solution map as follows.

Definition 2. The solution map of into is said to be Fréchet differentiable on if for any , there exists such that, for any ,The operator is called the Fréchet derivative of at , which we denote by . is called the Fréchet derivative of at in the direction of .

Theorem 2. The solution map is Fréchet differentiable on , and the Fréchet derivative of at in the direction , that is to say , is given by the solution of

Proof. LetThen, by (11) and Theorem 1, we can verifyHence, from (58), we see thatSince and , we know that . By (59), we can use the results in [18] to verify that equation (56) admits a unique weak solution . And by Theorem 1, we can know that the weak solution of equation (56) satisfiesHence, from (60), the mapping is linear and bounded.
We set . Then, using notation (27) and notingwe know that satisfiesin the weak sense, whereArguing as in the proof of Theorem 1, we can know that the weak solution of equation (62) exists, and the following is fulfilled:By (60), we can deduceBy Theorem 1, we can see thatThus, if , then by (66), we can extract a subsequence, still denoted by , such thatAnd from (11), we know thatThus, by the Lebesgue dominated convergence theorem through (67) and (68), we can obtainHence, from (64) to (69), we can obtainwhich immediately implies that as .
This completes the proof.
As before, for simplicity, we denote and by and , respectively. From Theorem 2, we know that the map of to is Fréchet differentiable at , and the Fréchet derivative of at in the direction , say , is a unique weak solution of the following problem:The following results will be used in showing the existence of optimal pairs.

Proposition 2. Assume that the exponent . For given , the Fréchet derivative is locally Lipschitz continuous on . Indeed, it is satisfied thatwhere is a constant depending on the data.

Proof. Let be the weak solutions of equation (56) corresponding to . If we set , then we can know that satisfiesin the weak sense, whereAs before, arguing as in the proof of Theorem 1, we can deduce thatIn estimating , the case of will be clarified as we will see later. Thus, we restrict to the case of to estimate . By (11), Theorem 1, and (60), we can obtainFrom (75) to (77), we can get the following:This completes the proof.

3.2. Existence of the Optimal Pairs

By the minimax theorem in infinite dimensions in Barbu and Precupanu [13], we study the existence of the optimal pairs in the underlying control system with quadratic cost function (53).

Theorem 3. Assume that the exponent in control system equation (52). Then, for sufficiently large and in (53), there exists satisfying (54).
In order to prove this theorem, we mainly illustrate the following as in [8] (cf. [9]):(i)For sufficiently large and in (53), the maps and are convex and concave for all and for all , respectively(ii)The maps and are lower and upper semicontinuous for all and for all , respectively

Proof. (i)Since the map where is fixed is Fréchet differentiable, the map is also Fréchet differentiable. Let and be the Fréchet derivatives of the map at and in the direction , respectively, where holds fixed. To verify the convexity of the map , it is sufficient to show that for any fixed . (79) means where are weak solutions of equation (71), in which is replaced by , respectively. We can deduce that (80) is equivalent to By Theorem 1, (60), and Proposition 2, we can have the following estimations: From (81) to (83), we can verify that there exists sufficiently large such that, for any , inequality (79) holds. Thus, the map is convex. Similarly, we can show that there exists sufficiently large such that the following inequality is satisfied for any fixed and . This also indicates the concavity of the map .(ii)Since is bounded in , we can extract a subsequence such thatThen, since is a continuous linear operator on , by Proposition 1, we obtain. Since the norm is weakly lower semicontinuous, we can see from (85) and (86) that the map is lower semicontinuous for all . By similar arguments, we can prove that the map is upper semicontinuous for all .
Next, we prove the existence of an optimal pair : let be a minimizing sequence of where is fixed. Thus,Then, by (i) and (ii), we know thatbut since , we haveSimilarly, we also know that there exists such thatFrom (89) and (90), we can conclude that is an optimal pair for cost (53).
This completes the proof.

3.3. Necessary Conditions of Optimal Pairs

In this section, we study the necessary optimality conditions that have to be satisfied by each optimal pair of the minimax optimal control problem with cost (53) in which the following two types of observations are considered:(1)We take and and observe (2)We take and and observe

Since , the above observations are meaningful.

3.3.1. Case of Distributive and Terminal Values’ Observations

We consider the cost functional expressed bywhere and are desired values. Let be an optimal pair subject to equation (52) and quadratic cost (91). Now, we are about to formulate the following adjoint equation corresponding to cost (91) and equation (52) in which with is replaced by with :

Proposition 3. Equation (92) admits a unique weak solution .

Proof. From the observation conditions and and (59), we can refer to the results in [18] (cf. Dautray and Lions [20], pp.655–659) to ensure that equation (92), after reversing the direction of time admits a unique weak solution . This completes the proof.
We now discuss the first-order optimality conditions for minimax optimal control problem (54) for quadratic cost function (91).

Theorem 4. If and in cost (91) are large enough and the exponent , then an optimal control and a disturbance , namely, an optimal pair satisfying (54), can be given bywhere is the weak solution of equation (92).

Proof. By the assumptions of Theorem 4, we can verify through Theorem 3 that there exists an optimal pair in (54) with cost (91). Let be an optimal pair in (54) with cost (91) and be the corresponding weak solution of equation (52).
Due to Theorem 2, we know that the map is Fréchet differentiable at in the direction which satisfies for sufficiently small that Thus, the map is also (strongly) Gâteaux differentiable at in the direction . Indeed, we haveas , where is a unique solution of equation (71). Therefore, we can obtain from (94) the Gâteaux derivative of cost (91) at in the direction as follows:where is a solution of equation (71).
We multiply both sides of the weak form of equation (92) by which is a solution of equation (71) and integrate it over . Then, we obtainBy integration by parts, the terminal value of the weak solution of equation (92), and Fubini’s theorem, (96) can be written again asSince is the solution of equation (71), by (97), we can deduce the following:Therefore, from (95) and (98), we can obtain thatSince is an optimal pair in (54), we knowTherefore, we can obtain the following from (99) and (100):where . By considering the signs of the variations and in (101), which depend on and , respectively, we can deduce from (101) thatThis completes the proof.

3.3.2. Case of Velocity’s Distributive Value Observation

In this case, we consider the following cost functional:where is the desired value. Let be an optimal pair subject to equation (52) and quadratic cost (103). In this case, we introduce the following adjoint equation corresponding to cost (103) and equation (52) in which is replaced by :

Proposition 4. Equation (104) admits a unique weak solution .

Proof. By the reversion in time and the reversion in equation (104), we can obtain the following:where . By changing in equation (105) and then changing every function in the reversed equation like this , we can have the following initial value problem:Now, we apply the well-known Faedo–Galerkin procedure to equation (106): since is separable, there exists a basis in such that is a complete orthonormal system in and free and total in . For each , we define an approximate solution of equation (106) bywhere satisfiesWe multiply both sides of equation (108) by and sum over to obtainIntegrating (109) from 0 to and by integration by parts, we obtainBy (59), we can obtain the following estimations:And we also have the following estimations:By (111)–(114) with a properly determined and estimating other terms in (110) as in the proof of Theorem 1, we can arrive atThus, by Gronwall’s lemma, we see thatThen, following similar arguments in [18], we can ensure that equation (104) admits a weak solution . Uniqueness will be proved straightforwardly for .
This completes the proof.
Now, we investigate the first-order optimality conditions for minimax optimal control problem (54) for quadratic cost function (103).

Theorem 5. If and in cost (103) are large enough and the exponent , then an optimal control and a disturbance , namely, an optimal pair satisfying (54), can be given bywhere is the weak solution of equation (104).

Remark 3. To get the optimality condition in Theorem 5, we need to multiply the weak form of equation (104) by . Since we just know that , it would be a just formal procedure. To overcome this difficulty, we employ the regularization method of Lions ([14], pp. 286–288) which was used to deal with linear hyperbolic problems (cf. [15]).

Proof of Theorem 5. By the assumptions of Theorem 5, we can verify through Theorem 3 that there exists an optimal pair in (54) with cost (103). Let be an optimal pair in (54) with cost (103) and be the corresponding weak solution of equation (52).
By similar arguments in Theorem 4, we can know that cost (103) is also (strongly) Gâteaux differentiable at in the direction which satisfies for sufficiently small that . Thus, we deduce the following:where is a solution of equation (71).
As explained above, we prove this theorem by the regularization method. For this purpose, we extend the time domains of equations (71) and (104) to by introducing and as the solutions ofwhereIn fact, we haveFor the simplicity, we shall denote the scalar product in or the antiduality by . Let be a regularizing sequence on . Then, the right-hand side of (124) becomesNow, we integrate by parts of (125). Then, we putUsing equation (119), that is, (126) can be given again bywhereWe immediately know by integration by parts. By integration by parts, we can see thatThus,as . From (121) and Fubini’s theorem, we know that the right-hand side of (130) is 0. Therefore, we can obtain as . And we can easily verify that as . Consequently, we can obtainSince is an optimal pair in (54), we knowFrom (118), (124), and (131), we know that (132) implieswhere . By similar arguments in the proof of Theorem 4, we can deduce from (133) thatThis completes the proof.

4. Conclusion

In this paper, we study the bilinear minimax optimal control problems for a semilinear viscoelastic equation with long memory. A global well-posedness theorem regarding the solutions to its Cauchy problem is given. We show the Fréchet differentiability of the nonlinear solution map from the bilinear control input to the solution of the state equation given by the above equation. We formulate the minimax optimal control problem for the state equation. By using and analyzing the properties of the Fréchet derivative of the nonlinear solution map, we show the existence of optimal pairs and find their necessary optimality conditions corresponding to the practical observation cases.

Data Availability

All the data sets generated for this study are included within this manuscript.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

The author contributed solely to the writing of this paper and read and approved the manuscript.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIT) (NRF-2020R1F1A1A01074403).

References

M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, “Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron,” Journal Differential Equations, vol. 44, pp. 1–14, 2002.
View at: Google Scholar
M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, “Global existence and asymptotic stability for viscoelastic problems,” Differential Integral Equations, vol. 15, pp. 731–748, 2002.
View at: Google Scholar
T.-J. Xiao and J. Liang, “Coupled second order semilinear evolution equations indirectly damped via memory effects,” Journal of Differential Equations, vol. 254, no. 5, pp. 2128–2157, 2013.
View at: Publisher Site | Google Scholar
S. A. Messaoudi, “Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 902–915, 2006.
View at: Publisher Site | Google Scholar
M. E. Bradley and S. Lenhart, “Bilinear optimal control of a Kirchhoff plate,” Systems & Control Letters, vol. 22, no. 1, pp. 27–38, 1994.
View at: Publisher Site | Google Scholar
M. E. Bradley and S. Lenhart, “Bilinear spatial control of the velocity term in a Kirchhoff plate equation,” Electron Journal Differential Equations, vol. 27, pp. 1–15, 2001.
View at: Google Scholar
M. E. Bradley, S. Lenhart, and J. Yong, “Bilinear optimal control of the velocity term in a Kirchhoff plate equation,” Journal of Mathematical Analysis and Applications, vol. 238, no. 2, pp. 451–467, 1999.
View at: Publisher Site | Google Scholar
A. Belmiloudi, “Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 620–642, 2007.
View at: Publisher Site | Google Scholar
J.-s. Hwang, “Fréchet differentiability for an extensible beam equation and its application to bilinear minimax control problems,” Journal of Mathematical Analysis and Applications, vol. 474, no. 1, pp. 157–178, 2019.
View at: Publisher Site | Google Scholar
N. Arada and J.-P. Raymond, “Minimax control of parabolic systems with state constraints,” SIAM Journal on Control and Optimization, vol. 38, no. 1, pp. 254–271, 1999.
View at: Publisher Site | Google Scholar
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, I, Cambridge University Press, Cambridge, UK, 2000.
X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, MA, USA, 1995.
V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Reidel, Dordrecht, Germany, 1986.
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 1971.
J.-s. Hwang, “Optimal control problems for an extensible beam equation,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 436–448, 2009.
View at: Publisher Site | Google Scholar
R. A. Adams, Sobolev Spaces, Academic Press, Cambridge, MA, USA, 1975.
R. Teman, “Infinite-dimensional dynamical systems in mechanics and physics,” Applied Mathematics Science, vol. 68, 1997.
View at: Google Scholar
J. Hwang and S. Nakagiri, “On semi-linear second order Volterra integro-differential equations in Hilbert space,” Taiwanese Journal of Mathematics, vol. 12, pp. 679–701, 2008.
View at: Publisher Site | Google Scholar
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, II, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 1972.
R. Dautray and J. L. Lions, “Mathematical analysis and numerical methods for science and technology,” Evolution Problems I, Springer-Verlag, Berlin, Germany, 1992.
View at: Google Scholar
J. Simon, “Compact sets in the space ,” Annali di Matematica Pura ed Applicata, vol. 146, no. 4, pp. 65–96, 1987.
View at: Google Scholar

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Copyright © 2021 Jin-soo 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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