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Optimal control for cooperative systems involving fractional Laplace operators
Journal of Inequalities and Applications volume 2021, Article number: 196 (2021)
Abstract
In this work, the elliptic \(2\times 2\) cooperative systems involving fractional Laplace operators are studied. Due to the nonlocality of the fractional Laplace operator, we reformulate the problem into a local problem by an extension problem. Then, the existence and uniqueness of the weak solution for these systems are proved. Hence, the existence and optimality conditions are deduced.
1 Introduction
Nonlocal operators have been a useful area of investigation in different branches of mathematics such as operator theory and harmonic analysis. Also, they have gained vital attention because of their strong connection with real-world problems since they form a fundamental part of the modeling and simulation of complex phenomena that span vastly different length scales.
Nonlocal operators appear in several applications such as image processing, boundary-control problems, electromagnetic fluids, materials science, porous-media flow, turbulence, optimization, nonlocal continuum field theories, and others. Consequently, the domain of definition Ω may be in its general form.
In this paper, we discuss the elliptic \(2\times 2\) cooperative system containing one of the nonlocal operators namely the fractional Laplace operator \((- \Delta )^{s}\).
Let \(\Omega \subset \mathbb{R}^{N}\), \(N\geq 2s\), be an open, bounded and connected domain with Lipschitz boundary ∂Ω. Then, we shall study the following system:
where \(y=\{y_{1},y_{2}\}\) are the states of the system, \(f=\{f_{1},f_{2} \}\) are the external sources. The fractional Laplace operator \((- \Delta )^{s}\) is defined by Fourier transform as follows:
From (1.2), it becomes clear that the fractional laplace operator \((- \Delta )^{s}\) is a nonlocal operator.
Definition 1.1
For given numbers a, b, c and d the system
is called cooperative if \(b,c>0\); otherwise, the system (1.3) is said to be noncooperative.
Optimal control for partial differential equations (PDEs) has been widely studied in many fields such as biology, ecology, economics, engineering, and finance [5–10, 18, 22, 24, 25, 30, 34, 37]. These results have been expanded in [12, 14, 15, 29, 31–33] to cooperative and noncooperative systems. The fractional optimal control problems are the generalization of standard optimal control problems. Hence, it allows treatment of more general applications in physics, chemistry, and engineering [13, 17, 20, 21, 23, 39]. Several papers discuss time-fractional optimal control. In [27, 28], the distributed optimal control problem for a time-fractional diffusion equation is discussed. Moreover, the optimality conditions are derived. In [19], the distributed control for a time-fractional differential system involving a Schrödinger operator is studied, and the optimality conditions are derived. Furthermore, space-fractional optimal control is introduced. In [1, 2, 11], the nonlocal system is reformulated to a local system by an extension problem. Hence, the optimality conditions are achieved.
Henceforth, to treat with the nonlocality of the fractional Laplace operator, in [4], Caffarelli and Silvestre proved that the fractional Laplace operator can be characterized as an operator that maps a Dirichlet boundary condition to a Neumann-type condition via an extension problem as follows:
Let \({D}^{+}\subset \mathbb{R}^{N+1}\) be a semi-infinite cylinder as
where t is a new extended variable. Therefore, the nonlocal problem (1.1) is reformulated locally as follows:
where \(\frac{\partial Y}{\partial \nu }=-\lim_{t\to 0^{+}}t^{1-2s} \frac{\partial Y(x,t)}{\partial t}\), ν is the unit outer normal to \({D}^{+}\) at \(\Omega \times \{0\}\), \(\lim_{t \to \infty } Y(x, t) = 0\) and \(k_{s}=2^{1-2s} \frac{\Gamma (1-s)}{\Gamma (s)}>0\).
In this paper, we generalize some previous results obtained for the classical cooperative systems. Indeed, we consider the elliptic \(2 \times 2\) cooperative system involving one of the nonlocal operators called the fractional Laplace operator. The nonlocality of the fractional Laplace operator creates some difficulties. To overcome these, we transform the nonlocal system into a local system via an extension problem. Hence, via the Lax–Milgram lemma, we are able to prove the existence and uniqueness of the weak solution for the local system. Moreover, for both local and nonlocal systems, the optimality conditions are derived via the Lions technique. The results obtained tend to the classical results if \(s\rightarrow 1\). This article is organized as follows. In Sect. 2, we introduce some functional spaces to represent the fractional cooperative systems and their extension, and also furnish the existence results. In Sect. 3, the weak solution and the optimality condition are established for the scalar case. In Sect. 4, we can generalize our results to a \(2\times 2\) cooperative system involving the fractional Laplace operator. Section 5 is devoted to a summary and discussion.
2 Preliminaries
In our work, the optimal control of the cooperative system depends on the variational formulation. Hence, we introduce the Sobolev spaces, which are the solution spaces for our problem. This section includes three subsections. Section 2.1 provides a short overview of classical fractional Sobolev spaces. In Sect. 2.2, we recollect the idea of the weighted Sobolev spaces and their embedding properties. The characterization of the principal eigenvalue problem is presented in Sect. 2.3.
2.1 Fractional Sobolev spaces
For \(0< s<1\), define the fractional-order Sobolev space [3, 35]
which is a Hilbert space endowed with the norm
Also, define the space \(H_{0}^{s}(\Omega )\) as:
which can be endowed with norm
Moreover, the Lions–Magenes space is given by [26]
where \(d(x, \partial \Omega )\) is the distance from x to ∂Ω. Combining (2.1), (2.3) and (2.5) we have for any \(s\in (0,1)\), the following fractional Sobolev space:
Moreover, we denote by \(\mathcal{H}^{-s}(\Omega )\) the dual space of \(\mathcal{H}^{s}(\Omega )\) such that
Also, we have the following embedding
By a Cartesian product, we have the following chain of Sobolev spaces
2.2 Weighted Sobolev spaces
To set the weak solution for problem (1.5), it is useful to establish the following weighted space [16]
equipped with the norm
Hence, the space of all functions in \(X^{s} ({D}^{+} )\), whose trace over \(\mathbb{R}^{N}\) vanishes outside of Ω, is given by
which furnishes a precise meaning of the solutions to problem (1.5) in a bounded domain Ω. It is clear that \(\mathcal{H}^{s}(\Omega )= \{ Y | _{\Omega \times \{0\}}: Y \in X_{\Omega }^{s} (D^{+} ) \} \). In addition, we have the following compact embedding.
Lemma 2.1
Let \(1 \leq p<2_{s}^{\#}=\frac{2 N}{N-2s} \). Then, \(\operatorname{Tr}_{ \Omega } (X_{\Omega }^{s} (D^{+} ) )\) is compactly embedded in \(L^{p}(\Omega )\).
Remark 2.1
For a function \(Y\in X_{\Omega }^{s} (D^{+} )\), the operator \(\operatorname{Tr}_{\Omega }: X_{\Omega }^{s} (D^{+} ) \rightarrow \mathcal{H}^{s}(\Omega )\) is called the trace operator and satisfies
Furthermore, \({\operatorname{Tr}_{\Omega }}Y=Y(x,0)=y(x)\) is the trace of Y onto \(\Omega \times \{0\}\).
2.3 Eigenvalue problem
In this subsection, we state some results given in [36] concerning the eigenvalue problem for the following fractional elliptic equation
Theorem 2.1
([36])
The first eigenvalue of problem (2.14) is positive and can be characterized as follows:
or equivalently,
3 Scalar case
For \(a>0\), consider the following system:
Using the extension problem, the nonlocal problem (3.1) is reformulated in a local way as follows:
3.1 Weak solution
Multiplying the first equation in (3.2) by a test function \(\phi (x,t) \in X_{\Omega }^{s} ({D}^{+} )\) and integrating over \({D}^{+}\) we obtain
Applying Green’s formula, we have
Take the bilinear form \(a(Y,\phi )\) as follows:
Also, take the linear form \(F(\phi )\) as follows:
Lemma 3.1
If \(\lambda >a\ k_{s}\), the bilinear form \(a(Y,\phi )\) defined in (3.6) is coercive.
Proof
Replacing \(\phi (x,t)\) by \(Y(x,t)\) in (3.6) we obtain
Hence, using (2.16) we obtain
Then, for \(\lambda >a\ k_{s}\) the coerciveness condition is satisfied, i.e.,
□
Remark 3.1
([38])
If \(Y(x,t)\) is a solution of the extended problem (3.2), then the trace function \(y(x)=\operatorname{Tr}_{\Omega }Y(x,t)=Y(x,0)\) will be called a weak solution to problem (3.1).
3.2 The optimality condition
Consider \(L^{2}(\Omega )\) as the space of controls. For a control \(u \in L^{2}(\Omega )\), the state \(Y(u)\) solves the systems
For a given \(z_{d}\in L^{2}(\Omega )\) and \(v\in L^{2}(\Omega )\), the cost-functional subject to the systems (3.11) is given by
where \(\operatorname{\mathbf{N}} \in \mathcal{L}(L^{2}(\Omega )) \)Footnote 1 is a positive-definite Hermitian operator satisfying
Let v belong to a subset \(\mathcal{U}_{ad}\) of \(L^{2}(\Omega )\) (the set of admissable controls); we assume \(\mathcal{U}_{ad}\) is a closed nonempty subset of \(L^{2}(\Omega )\). Then, the optimal control problem is now
Theorem 3.1
If the cost functional is given by (3.12) and the condition (3.13) is satisfied, then there exists a unique optimal control \(u\in \mathcal{U}_{ad}\). Moreover, this control is characterized by the following equations:
together with
where \(P\in X_{\Omega }^{s} (D^{+} )\) is the adjoint state.
Proof
The control \(u\in \mathcal{U}_{ad}\) is optimal, if and only if
and hence, via an explicit computation of \(J^{\prime }(u)\), (3.17) is equivalent to [24]
In order to transform (3.18) into a more convenient form, we introduce the adjoint state P defined by \((\operatorname{\mathbf{A}}Y,P)=(Y, \operatorname{\mathbf{A}}^{*}P)\), where \(A^{*}\) is the adjoint of A. Now
By applying Green’s formula, (3.19) is transformed to
and hence (3.15) is satisfied.
Take the adjoint systems as follows:
Remark 3.2 The variational form of (3.21) is
where we define
Then, (3.18) is equivalent to
Hence, using the last condition in (3.15), we obtain
By using (3.11), we have
and hence, the optimality condition becomes
Thereby, the proof is completed. □
4 \(2\times 2\) cooperative system
In this section, we generalize the results obtained in the previous section to a \(2\times 2\) cooperative system. This section is divided into two subsections. In Sect. 4.1, we prove the existence and uniqueness of the weak solution by using the Lax–Miligram Lemma. The optimality condition is obtained in Sect. 4.2.
4.1 The weak solution
To obtain a weak solution of the systems (1.5), we first transform (1.5) into a weak form. Indeed, multiplying the first and second equations in (1.5) by a test function \(\phi (x,t)=\{\phi _{1}(x,t),\phi _{2}(x,t)\} \in (X_{\Omega }^{s}(D^{+}) )^{2}\) and integrating over \(D^{+}\) we obtain
Applying Green’s formula, we have
Then, we obtain
By using the systems (1.5), Eqs. (4.5) and (4.6) are equivalent to
To this end, we can define a bilinear form on \((X_{\Omega }^{s}(D^{+}) )^{2}\) as follows:
Also, we can define a linear form as follows:
Lemma 4.1
The bilinear form (4.9) is coercive and bounded.
Proof
Replacing \(\phi =\{\phi _{1},\phi _{2}\}\) by \(Y=\{Y_{1},Y_{2} \}\) in (3.9) yields
By using the Cauchy–Schwarz inequality, we have
from (2.15), we deduce
Take
and
Then, if \(C_{2},C_{3}\geq 0\),we have
or
Hence, the bilinear form \(a(Y,Y)\) is coercive, if and only if the following conditions are satisfied
□
4.2 The optimality conditions
The control-problem formulation is the main target of this work. For the control problem, we construct the adjoint state. Furthermore, we originate the conditions of optimality via the Lions technique [24, 25]. This subsection consists of two parts. Section 4.2.1 contains the derivation of the necessary and sufficient conditions for fractional optimal control. Meanwhile, the equivalence extended optimal control is obtained in Sect. 4.2.2.
4.2.1 Fractional optimal control
Consider \((L^{2}(\Omega ))^{2}\) as the space of controls. For a control \(u= \{u_{1},u_{2}\} \in (L^{2}(\Omega ))^{2}\), the state \(y(u)=\{y_{1}(u),y_{2}(u)\}\) solves the following system:
The observation equations are given by
For a given \(z_{d}=\{z_{1d},z_{2d}\}\in (L^{2}(\Omega ))^{2}\) and \(v=\{v_{1},v_{2} \}\in (L^{2}(\Omega ))^{2}\), the cost-functional subject to the systems (4.1) is given by
where \(\operatorname{\mathbf{N}} \in \mathcal{L}((L^{2}(\Omega ))^{2})\) is a positive-definite Hermitian operator satisfying
Let \(\mathcal{U}_{ad}\) be a closed and convex subset of \(L^{2}(\Omega ) \). Then, the control problem is given as follows:
Theorem 4.1
If the cost functional is given by (4.3), and the condition (4.4) is satisfied, then there exists a unique optimal control \(u\in (\mathcal{U}_{ad})^{2}\). Moreover, this control is characterized by the following equations:
In addition,
where \(p=\{p_{1},p_{2}\}\in (\mathcal{H}^{s}(\Omega ))^{2}\) is the adjoint state.
Proof
Since \(\operatorname{\mathbf{N}}>0\), then the cost functional (4.3) is strictly convex. Furthermore, the set \(\mathcal{ U}_{ad}\) is nonempty, closed, bounded and convex in \(L^{2}(\Omega )\). Therefore, the existence and uniqueness of the optimal control is proved.
The control \(u=\{u_{1},u_{2}\}\in (\mathcal{U}_{ad})^{2}\) is optimal, if and only if
which is equivalent to [24]
Now, since \((\operatorname{\mathbf{A}}y,p)=(y, \operatorname{\mathbf{A}}^{*}p)\), where
then
where A∗ is the adjoint operator of A, p is the adjoint state.
Take the adjoint system as follows:
By using Eqs. (4.1) and (4.12), we deduce
Thus, the proof is completed. □
4.2.2 Extended optimal control
If \(y(v)\in (\mathcal{H}^{s}(\Omega ))^{2}\) is a solution of (4.1) with \(v= \{v_{1},v_{2}\}\in (\mathcal{H}^{-s}(\Omega ))^{2}\) and \(Y(v)\in (X_{\Omega }^{s}(D^{+}) )^{2}\) solves the following systems:
then, we have
Hence, the equivalence extended optimal control problem is given by
Theorem 4.2
If the cost functional is given by (4.16) and the condition (4.4) is satisfied, then there exists a unique optimal control \(u=\{u_{1},u_{2}\}\in ( \mathcal{U}_{ad})^{2}\). Moreover, this control is characterized by the following equations:
In addition,
where \(P=\{P_{1},P_{2}\}\in (X_{\Omega }^{s}(D^{+}) )^{2}\) is the adjoint state.
Proof
The control \(u\in (\mathcal{U}_{ad})^{2}\) is optimal, if and only if
which is again equivalent to [24]
Now, since \((\operatorname{\mathbf{A}}Y,P)=(Y, \operatorname{\mathbf{A}}^{*}P)\), then
and hence, (4.17) is satisfied.
Take the adjoint systems as follows:
Then, (4.20) is equivalent to
Hence, using the last two conditions in (4.17), the optimality condition becomes
which completes the proof. □
5 Summary and conclusion
In the present work, we investigate the optimal control problem for \(2 \times 2\) cooperative systems involving the fractional Laplace operator, wherein these systems are subject to the zero Dirichlet condition. Due to the difficulty arising from the nonlocality of the fractional Laplace operator, we follow the Caffarelli and Silvestre technique to extend our problem to local cooperative systems. With the aid of the Lax–Milgram lemma, the existence and uniqueness of the solution to the extended problem are proved. Moreover, the conditions of optimality are proved by the Lions technique for both fractional and extended optimal control. If \(s\rightarrow 1\), the obtained results are similar to the classical results.
Availability of data and materials
The data that support the findings of this study are available from the authors upon request.
Notes
\(\mathcal{L}(X)\) is the space of all bounded and linear operators from X into itself.
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Acknowledgements
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Groups Program under grant RGP.1/68/42.
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This research was funded by King Khalid University, grant RGP.1/68/42.
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Serag, H.M., Hyder, AA. & El-Badawy, M. Optimal control for cooperative systems involving fractional Laplace operators. J Inequal Appl 2021, 196 (2021). https://doi.org/10.1186/s13660-021-02727-3
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DOI: https://doi.org/10.1186/s13660-021-02727-3
MSC
- Optimal control
- Weak solution
- Lax–Milgram lemma
- Fractional Laplace operator
- Nonlocal operator
- Sobolev spaces
- Cooperative systems