Abstract
We solve an optimal control problem governed by an evolution equation using bilinear regular feedback. Using optimization techniques, we show how to approximate the flow of a reaction-diffusion bilinear system by a desired target. For application, we consider the regional flow problem constrained by a bilinear distributed system. The paper ends by an example illustrating the numerical approach of the proposed method.
1. Introduction
Bilinear systems form an important class of dynamic systems for several reasons. Many industrial or natural processes have a bilinear structure. For example, we can cite the transfer of heat by conduction convection, the neutron displacement in a nuclear reactor, and the dynamics of sense organs [1]. Research has shown that bilinear systems are sufficient to approach any nonlinear input-output behavior (see [1, 2]). The control has a double action in the system that allows the adaptation of the model at different levels of input signals. An example is provided by the functioning of sense organ (see [1]).
Optimal control methods continue to provide solutions to many real problems. We cite solutions of smoking models by Mahdy et al. [3] and COVID-19 prediction by Ahmed et al. [4]. Optimal control problems constrained by a distributed bilinear system are initiated by Bradley et al. and Lenhart [5, 6]. In [7], Joshi studies the case of regular velocity terms. Sonawane et al. [8] consider the optimal control for a vibrating string with axial variable. Rao et al. studied plant disease in [9].
Mall et al. propose a uniform method for optimal control problems with control and state constraints (see [10]). Chertovskih et al. in [11] give an indirect method for regular state-constrained optimal control problems in flow fields. Turgut et al. in [12] study an island-based crow search algorithm for solving optimal control problems. Al-Hawasy et al. in [13] consider the optimal control problems for triple elliptic partial differential equations. Bonnet and Frankowska in [14] characterize the necessary optimality conditions for optimal control problems in Wasserstein spaces. Granada and Kovtunenko in [15] consider a shape derivative for optimal control of the nonlinear Brinkman–Forchheimer equation.
For fractional systems, Saidi [16] discusses some results associated to first-order set-valued evolution problems with subdifferentials. Jajarmi and Baleanu [17] consider the fractional optimal control problems with a general derivative operator. Huixian et al. [18] study an averaging result for a class of impulsive fractional neutral stochastic evolution equations. Jafariet al. [19] propose a numerical approach for solving fractional optimal control problems with Mittag–Leffler kernel. Mehandiratta et al. [20] study fractional optimal control problems on a star graph. Heydari et al. [21] propose a numerical solution for an optimal control problems generated by Atangana–Riemann–Liouville fractal-fractional derivative.
The flow problems are one of the most important questions in mathematics. They have applications in several fields such as physics, biology, and engineering. We cite here the problem of controlling the blood flow in a vessel, where we need to calculate the gradient (flow) of the velocity of blood as a rate of change of the blood flow (see [22]).
Recently, many researchers focused on the study of flow problems using optimal control theory. They consider the gradient state of a distributed system and ask if there is an optimal control to reach a desired profile (see [23]). For this approach, one of the most important ideas is called the partial analysis. It has an objective to reach a target on a specific subdomain of the system domain, (see [24, 25]). For partial work on bilinear distributed systems, Ouzehra et al. [26, 27] study the exact and approximate controllability of reaction-diffusion equation using bilinear control. Zerrik and Ould Sidi [28–31] use partial control problems to orient the dynamics of infinite dimensional systems towards the desired state in a specific area. Zine and Ould Sidi in [32–34] deal with partial control problems in the case of hyperbolic systems. Ould Sidi and Beinane [35, 36] treat the partial flow control problems.
The objective of this paper is to control the flow of equation (1) towards a desired target using the penalization problem 3, and with a more regular spatiotemporal control function. In Section 2, we show the existence of a solution to the studied problems. Next, we give the characterization of its solution considering different types of actions. Section 3 is devoted to the study of the partial flow control problems constrained by bilinear distributed systems with regular optimal control time function. The paper ends by an example illustrating the numerical approach of the proposed method.
2. Flow Problem with Regular Control
Let us consider the system described bywith a domain is open bounded, and its regular boundary is . Let and , , where the space of control is .
Let andrepresents the state space (see [5]). The system dynamic is , and system (1) has a unique solution in (see [37]).
We consider the operator :
The flow regular optimal control problem of system (1) iswith , and is the cost penalty defined bywhere the desired flow is .
The main objective is to propose a method to steer the flow of (1) to , using the functional (5) and considering a more regular control space . We characterize the solution of (4) through an extension of the Lagrangian method.
2.1. Existence of Solution
In the next theorem, we study the existence of a solution to the flow problem (4).
Theorem 1. Let us consider be the solution of the system
Then, there exists an optimal control , which is the minimum of (4).
Proof. Let us consider the set , which is a positive nonempty and admits lower bounded. Thus, by choosing a minimizing sequence which verifiesThen, the cost is bounded, and it follows that , with as a positive constant.
We haveBy passing to the limit in the equation , we deduce that , , and . Hence, we obtainFrom the lower semicontinuity of :Therefore, is a solution of (4).
2.2. Characterization of Solution
In this section, the aim is to propose a formulation of the solution of our flow problem. Therefore, we should introduce the so-called optimal equation to find the differential of the functional in (5). The following lemma mentions the differential of with respecting .
Lemma 2. A differential of the mapiswhere verifieswhere , , and is the derivative of with respect .
Proof. We consider the solution of (13), verifyingAlso,Thus,Then, we obtain that is bounded (see [5]).
If we put and , then is the state ofThus,Let which verifies the system; consequently,and we havewhere , and are a constant positive.
In the following, we define a family of optimal equations.The next lemma characterizes the differential of .
Lemma 3. Let be the solution of (4), and we obtain
Proof. The cost (5) can be expressed byIf we put and , using (59), we havethenThe following theorem proposes a solution of the problem (4).
Theorem 4. Let be a solution of (4); then,where is the output of (1), where and is the solution of (22).
Proof. Let and for . The extremal of is realized at ; then,Lemma 9 givesTherefore, using (22), we obtainBy a simple calculus, we haveFrom System(13), we obtainMoreover, if , we deducewhich allow us to introducethat the solution of (4) must satisfy.
Remark 5. According to equation (2),(1)If we consider a spatial control function then the variational formula becomes(2)If we consider a temporal control function , then the variational formula becomes
3. Partial Flow Control Problem
3.1. Problem Statement
We consider the bilinear distributed system (1), with a given . System (1) can be rewritten as follows:and the solution of (37) are often called the mild solution of (1).
The existence of a unique solution in satisfying (37) can be deduced from [37].
We choose , andand ; its adjoint is given by
Definition 6. Equation (1) is called partial flow controllable on , to if there exists a control and such thatwhere is the desired flow in .
Ouzehra in [26], studies the exact and approximate controllability of distributed bilinear systems. The partial flow control problem of (1) iswhere is presented for byThe objective of the presented problem is to command the flow of (1) to a target state , realizing (43), and find , verifying
Remark 7. The existence of solutions for the partial flow control problem can be proved in the same way as in the proof of the previous section.
3.2. Characterization of Solution
Now, we are able to formulate the solution of the flow problem (42).
Lemma 8. A differential of the mapiswhere verifieswhere , , and is the derivative of with respect .
Proof. The output of equation (13) satisfiesAlso,Thus,Then, we obtain that is bounded (see [5]).
If we put and , then is the state ofThus,Let which verifies the system; consequently,and we haveWe introduce the family of optimal systems in the case of partial controllabilitywhere is the adjoint of defined from byThe following lemma mentions the differential of with respecting .
Lemma 9. If is the control realizing (42), is the output of (47), and is the solution of (56), we deduce
Proof. The functional given by (43) can take the formLet and , using (59), we haveConsequently,From (56) and (61), we deduce thatNow, we will deduce the solution of (42), exploiting the family of optimal systems.
Theorem 10. Let be the solution of the partial flow problem, and is its corresponding state of (1), we show thatis a solution of problem (42), where is the unique solution of the adjoint system (56).
Proof. Let with for . The functional get its minimum at , and we deduceUsing Lemma 9, replacing in system (47), we haveConsequently, for an arbitrary control we concludeThen,Consequently,
4. Example
In this section, we propose the numerical approach to computing the solution of our method (68). We consider the one dimensional bilinear equation
The operator admits a set of eigenfunctions associated to the eigenvalues given by
While the operator of system (69) and the perturbation commute, using Pazy [37], the solution of (69) can be written as
The one dimensional adjoint system can be written as
We define the perturbation function
Using (68) and a finite difference schema, the optimal control can be found by solving
By choosing , we define the following sequence of control solution ofwhere and are, respectively, the solution of (71) and (72) perturbed by with .
The penalty cost (43) becomes
The following convergent Algorithm 1 allows the implementation of our results.
Remark 11. (1)The distributed bilinear systems (1), are considered with the feedback map as multiplication of the control by the velocity of the state system. One can consider another different type of perturbation.(2)In the case of partial controllability, we use in general temporal control feedback. This type of control is compatible with real applications.(3)For the simulation point of view, the obtained control formula is easy to calculate numerically. This encourages us to establish numerical approaches and simulations of the proposed problems using Algorithm 1..
5. Conclusion
We consider the flow optimal control problem constrained by a bilinear distributed system. The chosen optimal controls are regular, and the existence of solutions is proved and characterized using optimization techniques. Our method is applied to the partial flow control problem allowing us to control a flow on a specific subdomain of the system domain. Finally, as an example, we present the numerical approach, which makes it possible to concretize the obtained results.
Data Availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the Deanship of Graduate Studies at Jouf University for funding and supporting this research through the initiative of DGS, Graduate Students Research Support (GSR) at Jouf University, Saudi Arabia.