On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

Kogut, Peter I.; Kupenko, Olha P.; Manzo, Rosanna

doi:https://doi.org/10.1155/2020/7418707

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2020 | Article ID 7418707 | https://doi.org/10.1155/2020/7418707

On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

Peter I. Kogut,¹Olha P. Kupenko,^2,3and Rosanna Manzo⁴

Academic Editor: Fasma Diele

Received11 Jul 2020

Revised24 Sept 2020

Accepted26 Oct 2020

Published17 Nov 2020

Abstract

We discuss the existence issue to an optimal control problem for one class of nonlinear elliptic equations with an exponential type of nonlinearity. We deal with the control object when we cannot expect to have a solution of the corresponding boundary value problem in the standard functional space for all admissible controls. To overcome this difficulty, we make use of a variant of the classical Tikhonov regularization scheme. In particular, we eliminate the PDE constraints between control and state and allow such pairs run freely by introducing an additional variable which plays the role of “compensator” that appears in the original state equation. We show that this fictitious variable can be determined in a unique way. In order to provide an approximation of the original optimal control problem, we define a special family of regularized optimization problems. We show that each of these problems is consistent, well-posed, and their solutions allow to attain an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation.

1. Introduction

The main object of our study is the following optimal control problem for a nonlinear elliptic equation: subject to constrains where is a bounded open domain in , , the boundary is assumed to be Lipschitz, , where is a given nonlinear function, is a given distribution, is a nonempty closed convex subset of , , and is a given weight coefficient.

Optimal control governed by PDEs has been examined thoroughly since the pioneering work of J.L. Lions (see [1, 2], for instance). Other important references that also deal with the numerical approximation and in addition to those already mentioned above without any attempt to be exhaustive, are [3–9]. However, as for the optimal control problem (OCP) (1)–(4) and the corresponding Dirichlet boundary value problem (BVP) (2)–(3), it is well known that they are ill-posed, in general, and it is unknown whether the set of optimal pairs to the problem (1)–(4) is nonempty. In particular, there is no reason to assert the existence of weak solutions to (2)–(3) for a given or to suppose that such solution, even if it exists, is unique (see, for instance, I.M. Gelfand [10], M.G. Crandall and P.H. Rabinowitz [11], F. Mignot and J.P. Puel [12], T. Gallouët, F. Mignot and J.P. Puel [13], H. Fujita [14], R.G. Pinsky [15], R. Ferreira, A. De Pablo, J.L. Vazquez [16], J. Dolbeault and R. Stańczy [17]).

The novelty of this paper is that we discuss the existence of optimal pairs to OCP (1)–(4) using an indirect approach based on the classical Tikhonov regularization technique in its special implementation. The idea to involve the Tikhonov regularization is inspired by the following reason: the main characteristic feature of BVP (2)–(3) is the fact that because of the specificity of nonlinearity (in many particular implementations of the model (2)–(3), , [18, 19]), we have no a priori estimate for the weak solutions in the standard Sobolev space . As a result, the consistency of OCP (1)–(4) and existence of optimal pairs can be established only if we impose rather strict assumptions on the original data. In particular, it was shown in [20] that the set of optimal solutions of (1)–(4) is nonempty provided , , the domain is star-shaped with respect to some interior point , and the set of feasible pairs contains at least one pair such that .

Therefore, our main intention is to show that these assumptions can be essentially weakened or even eliminated. With that in mind, in the framework of Tikhonov regularization technique, we introduce the additional variable (the so-called “defect” in the state equation) into the regularized problem in order to let the pairs “ control-state” run freely in the feasible space so that there is no dependence of on . At the same time, there is a principle difference between the standard implementation of the Tikhonov regularization of OCPs (see, for instance, [21–23]) and the proposed scheme. This difference lies in the exploitation of the terms and in the perturbed cost functional . We show that the boundedness of these terms on the set of feasible solutions to the original problem plays a crucial role in the study of asymptotic behaviour of global solutions to regularized OCPs. Having introduced a special family of optimization problems, we also show that there exists an optimal solution to the original OCP that can be attained with a prescribed level of accuracy by the sequence of optimal solutions for the regularized minimization problems (for benefit of this approach and its comparison with other ones, we refer to the recent papers [24–31]).

The paper is organized as follows. In Section 2 we give some preliminaries and describe in details the characteristic features of OCP (1)–(4). The Tikhonov regularization of the original OCP is discussed in Section 3. The key result of this section is Theorem 8, where we announce the sufficient conditions of the existence of optimal solutions to the regularized problems. In Section 4, we focus on deriving and substantiation of optimality conditions for regularized optimal control problem. The details of the indirect approach to the study of the original optimal control problem are discussed in last section. The key points of such approach are summarized in Theorem 13.

2. Preliminaries

Let be a bounded open subset of (). Let be a mapping such that . We specify this mapping as follows: there exists a constant such that

Then, it is easy to deduce that

Following the standard notation, by , we denote the Sobolev space as the closure of with respect to the norm . Let be the dual space to .

In order to make a precise meaning of the weak solution to BVP (2)–(3) in the sense of distributions (or shortly, distributional solution), we begin with the following concept.

Definition 1. Let be a given control function. We say that is a weak solution to the boundary value problem (2)–(3) in the sense of distributions, if it belongs to the class of functions and the integral identity holds for every test function .
Since for each test function , there exists a compact set such that it follows that the second term in (8) is well defined, namely,

At the same time, it is unknown whether the original BVP admits at least one weak solution in the sense of Definition 1 for each admissible control . Moreover, as follows from (8), the continuity of form on the set is not evident. For the details related with this issue, we refer to the classical paper Casas, Kavian, and Puel [20].

Before proceeding further, we make use of the following observation. Assume that for a given , we have , and the pair is related by integral identity (8). Then, for each test function , the following estimate holds true. Hence, the mapping can be extended by continuity onto the set of all using (11) and the standard rule where and strongly in as . In particular, if , then we can define the value , and this one is finite for every . As a consequence, we deduce: if is a weak solution to boundary value problem (2)–(3), then satisfies the energy equality

However, it is unknown whether the value preserves a constant sign for all . Therefore, we cannot make use of the energy equality (13) in order to derive a priori estimate in -norm for the weak solutions.

In particular, to specify the term , we have the following result (we refer to [20], Lemma 2.1) where this result was proven for a particular nonlinearity (see also [27, 28, 32] for the more general cases).

Lemma 2. Let be a weak solution to BVP (2)–(3) for a given . Then, , and, therefore, for every .

Proof. Taking into account the Friedrich’s inequality and following the definition of the weak solution, we have (see (8)) Hence, by Definition 1.
Let be an arbitrary element. Since , it follows that the term is well defined. Let be a sequence such that in . Moreover, in this case, we can suppose that Hence, due to the fact that , we get Thus, we arrive at relation (15) for each .
Let us take now such that almost everywhere in . For every , let be the truncation operator defined by The following property of is well known (see [33]): if , then Hence, almost everywhere in . Since it follows that is a pointwise nondecreasing sequence, and also, for almost all . Therefore, by monotone convergence theorem, is a measurable function on , and Thus, (15) holds true for each such that .
As for a general case, i.e., , it is enough to note that with and in , where , .
To complete the proof, it remains to observe that holds true for an arbitrary element . As a result, we have and

Remark 3. As follows from Lemma 2, whenever is related by integral identity (8) and , then , but for a general , it is not necessarily true that the duality action is given by an integral , hence the need for a rigorous definition of .

As a direct consequence of Lemma 2 and relation (13), we can specify the energy equality (13) as follows.

Corollary 4. Let be a given control and let be a weak solution to BVP (2)–(3) in the sense of Definition 1. Then, the energy equality for takes the form

Since it is unknown whether there exists a weak solution to BVP (2)–(3) for a given , or to suppose that such solution, even if it exists, is unique, it motivates us to introduce the following set.

Definition 5. We say that a pair is a feasible solution for optimal control problem (1)–(4) if , , , and the pair is related by integral identity (8). By , we denote the set of all feasible solutions.
As for the optimal control problem (1)–(4), it was mentioned in [20] that its study is a nontrivial matter because of the specific of nonlinearity (in [20], the authors consider the case ). The main troubles in this case are strongly related with the following circumstances: (i)The set of feasible solutions can be empty, in general(ii)Even if , we have no a priori estimate for the weak solutions of (2)–(3) with arbitrary (iii)Some a priori estimates can be established if only , the domain has a sufficiently smooth boundary, and it is star-shaped with respect to some interior point , i.e, where denotes the outward unit normal vector to at the point , and the considered weak solutions of (2)–(3) satisfies the extra property (iv)Since we have no estimates for the states (especially without the above mentioned extra property ), it follows that we cannot deduce the boundedness in of a minimizing sequence to the problem (1)–(4)(v)Even if a minimizing sequence is weakly compact in with , it does not allow to pass to the limit in integral identity (8) as , and, therefore, we are not able to prove the existence of an optimal pair to the problem (1)–(4).

Although this list can be extended by many other options, we can summarize this issue by the following existence result (in order to prove this assertion, it is enough to closely follow the arguments of the proofs of Theorems 3.5 and 3.6 in [20]).

Theorem 6. Let us assume that the following conditions hold true: , , the domain has a boundary, this domain is star-shaped with respect to some interior point , and the set of feasible pairs contains at least one pair such that . Then, there exists a unique pair such that where In spite of the fact that not every pair of needs to be a feasible pair to (1)–(4) with the extra property , and constrained minimization problems are distinguished from a formal point of view, we can deduce the following result (for the proof we refer to [20], Proposition 3.2).

Proposition 7. Assume that and is star-shaped with respect to some interior point . Assume also that boundary value problem (2)–(3) has a weak solution for some control with . Then, there exists a solution of (2)–(3) corresponding to the same control and such that

In the next section, we will show that the main restrictions coming from Theorem 6 and Proposition 7 can be eliminated by introducing a new additional variable into the problem which lets pairs run freely in the feasible space so that there is no dependence of on .

3. On the Tikhonov Regularization of the Original OCP

Let us introduce the Tikhonov regularized optimal control problem associated to the original OCP (1)–(4). Let be a given small parameter. Then, the regularized problem reads as follows (for comparison, we refer to [22, 23]). subject to constraints

To begin with, let us stress again that the main reason to introduce the additional variable into the regularized problem is to let pairs run freely in the feasible space so that there is no dependence of on . On the other hand, there is a principle difference between the standard scheme of the Tikhonov regularization of OCPs (see, for instance, [22, 23]) and the proposed regularization in the form (32)–(35). This difference lies in the exploitation of the terms and in the perturbed cost functional . As it will be shown later on, the boundedness of these terms on the set of feasible solutions to the original problem (see Lemma 2) plays a crucial role in the study of asymptotic behaviour of global solutions as tends to zero.

Our main assumptions are: (a) is a bounded open domain in , (b) is a nonempty closed convex subset of , (c) is a given monotonically increasing mapping such that .

We say that a tuple is a feasible solution to regularized problem (32)–(35) (in symbols, ), if , , and the following variational equality holds true for all , where and denotes the bilinear form

Let us show that, for each , the set of feasible solutions to regularized problem (32)–(35) is nonempty. Indeed, let be an arbitrary pair in such that

Then, the right hand side of (36) is well defined for each test function and satisfies the following estimate (see (12)).

Since, the bilinear form is continuous and uniformly coercive on , it follows from Lax-Milgram theorem that the variational problem (36) has a unique solution . Hence, and, therefore, for a given . Thus, and this implies that regularized optimal control problem (32)–(35) is consistent for all .

Our next intention is to discuss the issue related to the existence of optimal solutions of the regularized problems (32)–(35).

Theorem 8. Assume that conditions (a)–(c) indicated before are valid. Then, for each , there is a triplet such that

Proof. Let be a given value. Since the cost functional is nonnegative on , it follows that there exist a and a sequence such that Then, we can immediately deduce from (43) and definition of the set that for each and the sequences are uniformly bounded in , , , and , respectively. In particular, Hence, without loss of generality, we can suppose that there exist elements , , , and such that as .
Let us show that, in fact, in . Indeed, from (36), using as a test function, we find that for all . Then, utilizing the Poincaré’s inequality, we obtain From this and estimates (45)–(46), we deduce that Thus, without loss of generality, we can suppose that (up to a subsequence) Utilizing the pointwise convergence (55)₃ and assumption (c), we see that Let us show that this fact together with (47) implies the strong convergence To begin with, let us show that the sequence is bounded in . With that in mind, for each , we make use of the decomposition with and set Then, for all such that . Using as a test function in (36), we find that Since in as , the limit passage in (60) as leads to the relation From this and the fact that , we deduce the estimate Arguing in a similar manner, it can be shown that Since , it follows from (62)–(63) that there exist positive constants , , independent of and such that In order to prove the strong convergence (57), we make use of Vitali’s theorem. To do so, we fix an arbitrary and take and such that , , where Then, for every measurable set with Lebesgue measure , we have As a result, we see that the sequence is equi-integrable and, hence, the desired convergence (57) is a direct consequence of the pointwise convergence (56) and Vitali’s convergence theorem. From this and (51), we obtain Since the set of admissible controls is convex and closed in , it follows from Mazur’s theorem that it is sequentially closed with respect to the weak topology of . Hence, . Thus, in order to decide that is a feasible solution to the regularized problems (32)–(35), it remains to show that this tuple is related by the variational equality (36). To do so, we utilize the following integral identity which holds true for each , , and . Taking into account properties (48)–(51), (67), the limit passage in (68) as becomes trivial. As a result, we arrive at the following relation Thus, is a feasible solution to the regularized problems (32)–(35).
To conclude the proof, let us show that, in fact, the triplet is optimal to the problem (32)–(35). Indeed, in view of the strong convergence (57) and lower semicontinuity of norms in reflexive Banach spaces , , and with respect to the weak convergence, passing to the limit in (42), we obtain Thus, the equality (41) holds true with and, therefore, the tuple is optimal for regularized problems (21)–(24).

4. Optimality Conditions for Regularized Problem

In this section, we focus on deriving of optimality conditions for regularized optimal control problem (32)–(35) corresponding to the case .

We begin with the following observations.

Remark 9. In spite of the natural expectations, the mapping , where is the solution of (34)–(35) associated to and , is not of class from . Indeed, if we assume this property, then the mapping should be of class from into on the subset defined by Apart from the fact that, under the assumptions on the function , it is not clear whether the set has a nonempty interior . Even if it is so, then the assumption that the mapping is of class at some point would imply for all . However, when , even for function such as , this result does not hold.
Indeed, let us consider the case , , and is the unit ball . Then, the function , for , satisfies . Now, if we set for and a function such that and for , we have and . However, when .

Remark 10. In practice, the numerical simulation of -term in the cost functional is quite specific and a delicate matter. Usually, it is associated with providing a very precise numerical analysis. In order to avoid these difficulties, it makes sense to substitute the term in the cost functional by some equivalent norm. For instance, let be an element of such that where in is a given a vector-function.
It is clear that On the other hand, due to the Lax-Milgram theorem, the Dirichlet boundary value problem has a unique solution for each . Moreover, in view of the energy equality which holds true for the weak solution of Dirichlet problems (74), we can deduce the following a priori estimate Combining this result with (73), we obtain the following chain of inequalities for the dual norm in : Hence, in this case, the standard norm in is equivalent to the following one Taking this fact into account, in this section, we specify the cost functional (32) as follows

As a result, to derive optimality conditions for regularized optimal control problem (79), (33)–(35), we apply the following reasoning. Let be a fixed value. In addition to (5), we assume that and are a convex function for which there exists a constant such that

Note that this property does not come into conflict with relation (6), and as a particular case of satisfying (80) is .

Let be the following subset of

In view of the properties of function , it is unknown whether this set has a nonempty interior. However, taking into account that and , it follows from Sobolev embedding theorem that for all .

We know that boundary value problem (34)–(35) has a unique solution for every and . Let be an optimal solution to the problem (79), (33)–(35) with and .

Let and be arbitrary chosen functions. Then, property (80) immediately implies that

By convexity of , we have

Then, where and belong to (see (82)). From this, we deduce that

For every , , we set

Then, property (80) implies that and in . So, is a feasible point for the problem (32)–(35). As follows from (84)–(85), there exists an element such that and as . Let us show that, for a given , the following extra properties hold true

Indeed, due to property (80), we deduce from (84) and (88) that

Since , the above inequalities imply that

Hence,

In order to deduce the asymptotic property as , we utilize property of . Then, for a given and sufficiently small, we have

As a result, we see that

From this and definition of the directional derivative, we finally deduce

Hence, as .

In the functional , we will distinguish three terms where

Now, using Lebesgue’s convergence theorem and the fact that is an optimal triplet, we get where

As for the term , we notice that where

Here, we have utilized the following obvious equality and the fact that for the nonnegative function .

Since it follows from (89) that

Then, (102) implies that

Since for any element , we have , it follows that

Hence,

As a result, utilizing relations (101), (107), and (109), we obtain

From the linearity of , , and with respect to and , we deduce from (98), (99), (100), and (110) that for every and .

Let us set . Then, (111) implies that

From the last equality, we immediately deduce that , where is a weakly harmonic function (it satisfies Laplace’s equation in the sense of distributions). As a consequence, we have .

Taking into account that is dense in , we see that relations (112)–(113) can be rewritten as follows. for every and .

Thus, we can summarize the obtained result as follows.

Theorem 11. Let be a bounded open domain in with . For a given , let be the subset defined as in (81). Assume that is a monotonically increasing function such that and this function is convex and satisfies property (80). If is an optimal solution to the problem (79), (33)–(35), then and setting , one has a.e. in , and

5. Asymptotic Analysis of Regularized Optimal Control Problem

Our main aim in this section is find out whether the original optimal control problem (1)–(4) is solvable under assumptions (a)–(c) and its optimal solutions can be attained (in some sense) by optimal solutions to the regularized problem (32)–(35).

The key point of our consideration is that, in contrast to the well-known approaches (see, for instance, [20, 27, 28]), we do not assume here the fulfillment of the “standard” extra properties such that the domain is an open subset of with , this domain should be star-shaped and exists a weak solution of Dirichlet problem (2)–(3) satisfying . Because of this, the existence of at least one optimal pair to the problem (1)–(4) is an open question provided we restrict our consideration only by assumptions (a)–(c).

In what follows, in order to guarantee the consistency of the original problem (1)–(4), we accept the following hypothesis.

Hypothesis 12. The set of feasible solutions to the problem (1)–(4) is nonempty.

It is worth to notice that the verification of Hypothesis 12 is not too restrictive from practical implementation point of view. Indeed, let be an arbitrary function. Then, it is clear that and , that is, . Let us define the control as follows in . Then, is a feasible pair to the problem (1)–(4) if only . So, this hypothesis is obviously true if we do not impose any additional restrictions on the class of admissible controls, i.e., .

The following result is crucial in this paper and it shows that solvability of the original OCP (1)–(4) in some sense is equivalent to its consistency, i.e., OCP (1)–(4) admits at least one solution if and only if Hypothesis 12 is fulfilled. However, in order to establish this fact, we apply an indirect approach based on the variant of Tikhonov regularization which is described in Section 4.

Theorem 13. Let be a bounded open domain in with , let be a nonempty closed convex subset of , , and let be a monotonically increasing function such that . Let be a sequence of optimal solutions to regularized problems (32)–(35) when the parameter varies in a strictly decreasing sequence of positive numbers converging to . Assume that Hypothesis 12 holds true and the sequence is bounded in . Then, there is a subsequence of , still denoted by the suffix , such that

Proof. Since , it follows that, for a given , there exists such that is a feasible solution to optimal control problem (1)–(4), and Hence, in view of Definition 5 and Lemma 2, , , , and the pair is related by integral identity (8). Therefore, for each , the triplet is a feasible solution for regularized problem (32)–(35), i,e, for all . Taking this fact into account, we see that where and (see (118)) Since this relation holds true for each varying in a given interval and each , it follows that In addition, the sequence is assumed to be bounded in . Taking this fact into account, we deduce that Hence, the sequences , , and are weakly compact in , , and , respectively, whereas estimate (122) implies that the sequence is strongly convergent to in . So, we can suppose that there exist elements , , , and a sequence monotonically converging to zero as such that Let us show that, in fact, we have the weak convergence in . Indeed, arguing as in the proof of Theorem 8, we utilize the integral identity which holds true for each and reflects the fact that the triplets are feasible to the problem (32)–(35) for each . Then, we deduce from (129) that Hence, estimates (122)–(124) imply that Thus, without loss of generality, we can suppose that (up to a subsequence) Utilizing the pointwise convergence (132)₃ and (c)-property, we see that almost everywhere in as . Let us show that, in fact, we have the strong convergence and, as a consequence of (127), and .
To this end, we make use of some arguments of the proof of Theorem 8. With that in mind, for each , we make use of the decomposition with and set Then, for all such that . Using as a test function in (25), we find that Since in as , the limit passage in (135) as leads to the relation From this and the fact that , we deduce the estimate Arguing in a similar manner, it can be shown that Since , it follows from (137)–(138) that there exists a positive constant , independent of and such that In order to prove the strong convergence (133), it remains to make use of Vitali’s theorem. We fix an arbitrary and take and such that , . Then, for every measurable set with Lebesgue measure , we have As a result, we see that the sequence is equi-integrable and, hence, the desired convergence (133) is a direct consequence of the pointwise convergence almost everywhere in and Vitali’s convergence theorem.
We are now in a position to show that is a feasible solution to the original OCP (1)–(4). Indeed, in view of the initial assumptions, the set is sequentially closed with respect to the weak topology of . Hence, . It remains to show that the pair is related by the integral identity (8). To this end, we note that for all . Hence, the equality holds true for each test function . As a result, the limit passage in (141) as becomes trivial, and it immediately leads us to the expected integral identity (8). Thus, combining all properties of the pair established here and before, we finally deduce that .
To conclude the proof, we have to show that in an optimal pair to the problem (1)–(4). To do so, we assume the converse, namely, there is a pair such that . Then, the triplet is feasible to the regularized problem for each . Hence, Therefore, passing in (142) to the limit as and using the properties we obtain As a result, it leads us to a contradiction. Thus, in an optimal pair to the problem (1)–(4).
To the end of proof, we note if the original OCP admits a unique solution, then the asymptotic analysis given before remains valid for each subsequence of the sequence of optimal solutions . Therefore, the limits in (125)–(128) do not depend on the choice of a subsequence, and, hence, is a unique limit pair for the entire sequence of optimal triplets .

Data Availability

All the results in this manuscript are produced by the authors.

Conflicts of Interest

No potential conflict of interest was reported by the authors.

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Copyright © 2020 Peter I. Kogut et al. 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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