# Optimal Control Problem Governed by an Evolution Equation and Using Bilinear Regular Feedback

Authors: Badriah Shuwaysh Alanazi; Maawiya Ould Sidi
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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, ω⊂D (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 by(1)qmx,m=qxxx,m−vx,mqxx,m,Γ,qx,0=q0x,D,q=qx=0,Π,with a domain D⊂IRnn=1,2,3 is open bounded, and its regular boundary is ∂D. Let M>0 and Γ=D×]0,M[, Π=∂D×]0,M[, where the space of control is v∈L20,M,H01D.Letq0x∈L2D and(2)S=q∈L20,M;H01Dqm∈L20,M;H−2D,represents the state space (see [5]). The system dynamic is qxx=Δq=∑i=1n∂2q/∂xi2, and system (1) has a unique solution qv in S∩L∞0,M;L2D (see [37]).We consider the operator∇:(3)∇:H1D⟶L2Dnq⟶∇q=∂q∂x1,…,∂q∂xn.The flow regular optimal control problem of system (1) is(4)minv∈L20,M,H01DΦεv,with ε>0, and Φε is the cost penalty defined by(5)Φεv=12∇q−qdL20,M,H01Dn2+ε2∫Γvm2x,m+vx2x,mdxdm=12∑i=1n∂q∂xi−qidL20,M,H01D2+ε2∫Γvm2x,m+vx2x,mdxdm,where the desired flow is qd=q1d,…,qnd.The main objective is to propose a method to steer the flow of (1) to qdx, using the functional (5) and considering a more regular control space v∈L20,M;H01D. 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 considerq be the solution of the system(6)qm=qxx−vqx,Γ,qx,0=q0x,D,q=qx=0,Π.Then, there exists an optimal controlv, which is the minimum of (4).Proof.
Let us consider the setΦεv|v∈L20,M,H01D⊂IR, which is a positive nonempty and admits lower bounded. Thus, by choosing a minimizing sequence vnn which verifies(7)Φ∗=limn⟶+∞Φvn=infv∈L20,M,H01DΦεv.
Then, the costΦεvn is bounded, and it follows that vnL20,M,H01D≤B, with B as a positive constant.
We have(8)vn⇀v,L20,M,H01D,qn⇀q,S,qxxn⇀χ,S,qxn⇀Λ,S,qmn⇀Ψ,S.
By passing to the limit in the equationqmnx,m=qxxn−vnqxn, we deduce that qmx,m=Ψ, q↦qxx, qxx=χ and vqx=Λ. Hence, we obtain(9)qm=qxx−vx,mqx.
From the lower semicontinuity ofΦεv:(10)Φεv=infn∑i=1n12∫0M∫D∂qn∂xi−qid2dx+ε2∫Γvm2+vx2ndxdm≤limn⟶0Φεvn=infvΦεv.
Therefore,v 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 functionalΦεv in (5). The following lemma mentions the differential of Φεv with respecting v.Lemma 2.
A differential of the map(11)v∈L20,M,H01D⟶qv∈S,is(12)qv+εl−qlε⇀μ,where μ=μq,l verifies(13)μm=μxx−vμx−lqx,Γ,μx,0=0,D,μ=μx=0,Π,where q=qv, v∈L20,M;H01D, and dqvl is the derivative of v⟶qv with respect v.Proof.
We consider the solution of (13), verifying(14)μS≤k1qL∞0,M;H01DlL20,M,H01D.
Also,(15)μ′S≤k2qL∞0,M;H01DlL20,M,H01D.
Thus,(16)μC0,M;H01D≤k3lL20,M,H01D.
Then, we obtain thatl∈L20,M;L2D⟶μ∈C0,M;H01D is bounded (see [5]).
If we putql=qv+l and ξ=ql−q, then ξ is the state of(17)ξmx,m=ξxx−vx,mξxx,m−lx,mqlx,Γ,ξx,0=0,D,ξ=ξx=0,Π.
Thus,(18)ξL∞0,M;H01D≤k4θL20,M,H01D.
Letγ=ξ−μ which verifies the system(19)γm=γxx+vx,mγxx,m+lx,mξx,Γ,γx,0=0,D,γ=γx=0,Π,γ∈C0,M;H01D; consequently,(20)γC0,M;H01D≤klL20,M,H01D2,and we have(21)qv+l−qv−dqvlC0,M;H01D=γC0,M;H01D≤klL20,M;H01D2,where k1,k2,k3,k4, and k are a constant positive.
In the following, we define a family of optimal equations.(22)−∂pi∂m=∂2pi∂x2+∂vpi∂x+∂q∂xi−qid,Γ,vxx,0=vxx,M=0,D,pix,M=0,D,pi=∂pi∂x=0,Π.
The next lemma characterizes the differential ofΦεv.Lemma 3.
Letv∈L20,M,H01D be the solution of (4), and we obtain(23)limβ⟶0Φεv+βl−Φεvβ=∑i=1n∫D∫0M∂μx,m∂xi∂q∂xi−qiddmdx+ε∫D∫0Mvmlm+vxlxdmdx.Proof.
The costΦεv (5) can be expressed by(24)Φεv=12∑i=1n∫D∫0M∂q∂xi−qid2dmdx+ε2∫D∫0Mvm2+vx2dmdx.
If we putqβ=qv+βl and q=qv, using (59), we have(25)limβ⟶0Φεv+βl−Φεvβ=limβ⟶0∑i=1n12∫D∫0M∂qβ/∂xi−qid2−∂q/∂xi−qid2βdmdx+limβ⟶0ε2β∫D∫0Mvm+βlm2−vm2+vx+βlx2−vx2dmdx,then(26)limβ⟶0Φεv+βl−Φεvβ=limβ⟶0∑i=1n12∫D∫0M∂qβ/∂xi−∂q/∂xiβ∂qβ∂xi+∂q∂xi−2qiddmdx+limβ⟶0ε∫D∫0Mvmlm+vxlxx,mdmdx=∑i=1n∫D∫0M∂μx,m∂xi∂qx,m∂xi−qiddmdx+∫D∫0Mεvmlm+vxlxx,mdmdx.
The following theorem proposes a solution of the problem (4).Theorem 4.
Letv∈L20,M;H01D be a solution of (4); then,(27)vmm+vxx+1εDivpiqx=0,where q=qv is the output of (1), where pi=p1,…,pn and pi∈C0,M;H01D is the solution of (22).Proof.
Letl∈L20,M;H01D and v+βl∈L20,M;H01D for β>0. The extremal of Φε is realized at v; then,(28)0≤limβ⟶0Φεv+βl−Φεvβ.
Lemma9 gives(29)0≤∑i=1n∫D∫0M∂μx,m∂xi∂qx,m∂xi−qiddmdx+∫D∫0Mεvmlm+vxlxx,mdmdx.
Therefore, using (22), we obtain(30)0≤∑i=1n∫D∫0M∂μx,m∂xi−∂pix,m∂m−∂2pix,m∂x2−∂vpix,m∂xdmdx+∫D∫0Mεvmlm+vxlxdmdx.
By a simple calculus, we have(31)0≤∑i=1n∫D∫0M∂μx,m∂xi−∂pix,m∂m−∂2pix,m∂x2+∂vpix,m∂xpix,mdmdx+∫D∫0Mεvmlm+vxlxdmdx.
From System(13), we obtain(32)0≤∑i=1n∫D∫0M∂∂xi−lx,mqxpix,mdmdx+∫D∫0Mεvmlm+vxlxdmdx=∫D∫0M−lx,mqx∑i=1n∂∂xipix,m+εvmlm+εvxlxdmdx.
Moreover, ifl=lt∈L20,M;H01D, we deduce(33)−lqx∑i=1n∂∂xipix,m−εvmml−εvxxl=0,which allow us to introduce(34)vmm+vxx+1εDivpiqx=0,that the solution v of (4) must satisfy.Remark 5.
According to equation (2),(1)
If we consider a spatial control functionv=vx, then the variational formula becomes(35)vxx=−1εDivpiqx,(2)
If we consider a temporal control functionv=vm, then the variational formula becomes(36)vmm=−1εDivpiqx.


## 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 considerq be the solution of the system(6)qm=qxx−vqx,Γ,qx,0=q0x,D,q=qx=0,Π.Then, there exists an optimal controlv, which is the minimum of (4).Proof.
Let us consider the setΦεv|v∈L20,M,H01D⊂IR, which is a positive nonempty and admits lower bounded. Thus, by choosing a minimizing sequence vnn which verifies(7)Φ∗=limn⟶+∞Φvn=infv∈L20,M,H01DΦεv.
Then, the costΦεvn is bounded, and it follows that vnL20,M,H01D≤B, with B as a positive constant.
We have(8)vn⇀v,L20,M,H01D,qn⇀q,S,qxxn⇀χ,S,qxn⇀Λ,S,qmn⇀Ψ,S.
By passing to the limit in the equationqmnx,m=qxxn−vnqxn, we deduce that qmx,m=Ψ, q↦qxx, qxx=χ and vqx=Λ. Hence, we obtain(9)qm=qxx−vx,mqx.
From the lower semicontinuity ofΦεv:(10)Φεv=infn∑i=1n12∫0M∫D∂qn∂xi−qid2dx+ε2∫Γvm2+vx2ndxdm≤limn⟶0Φεvn=infvΦεv.
Therefore,v 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 functionalΦεv in (5). The following lemma mentions the differential of Φεv with respecting v.Lemma 2.
A differential of the map(11)v∈L20,M,H01D⟶qv∈S,is(12)qv+εl−qlε⇀μ,where μ=μq,l verifies(13)μm=μxx−vμx−lqx,Γ,μx,0=0,D,μ=μx=0,Π,where q=qv, v∈L20,M;H01D, and dqvl is the derivative of v⟶qv with respect v.Proof.
We consider the solution of (13), verifying(14)μS≤k1qL∞0,M;H01DlL20,M,H01D.
Also,(15)μ′S≤k2qL∞0,M;H01DlL20,M,H01D.
Thus,(16)μC0,M;H01D≤k3lL20,M,H01D.
Then, we obtain thatl∈L20,M;L2D⟶μ∈C0,M;H01D is bounded (see [5]).
If we putql=qv+l and ξ=ql−q, then ξ is the state of(17)ξmx,m=ξxx−vx,mξxx,m−lx,mqlx,Γ,ξx,0=0,D,ξ=ξx=0,Π.
Thus,(18)ξL∞0,M;H01D≤k4θL20,M,H01D.
Letγ=ξ−μ which verifies the system(19)γm=γxx+vx,mγxx,m+lx,mξx,Γ,γx,0=0,D,γ=γx=0,Π,γ∈C0,M;H01D; consequently,(20)γC0,M;H01D≤klL20,M,H01D2,and we have(21)qv+l−qv−dqvlC0,M;H01D=γC0,M;H01D≤klL20,M;H01D2,where k1,k2,k3,k4, and k are a constant positive.
In the following, we define a family of optimal equations.(22)−∂pi∂m=∂2pi∂x2+∂vpi∂x+∂q∂xi−qid,Γ,vxx,0=vxx,M=0,D,pix,M=0,D,pi=∂pi∂x=0,Π.
The next lemma characterizes the differential ofΦεv.Lemma 3.
Letv∈L20,M,H01D be the solution of (4), and we obtain(23)limβ⟶0Φεv+βl−Φεvβ=∑i=1n∫D∫0M∂μx,m∂xi∂q∂xi−qiddmdx+ε∫D∫0Mvmlm+vxlxdmdx.Proof.
The costΦεv (5) can be expressed by(24)Φεv=12∑i=1n∫D∫0M∂q∂xi−qid2dmdx+ε2∫D∫0Mvm2+vx2dmdx.
If we putqβ=qv+βl and q=qv, using (59), we have(25)limβ⟶0Φεv+βl−Φεvβ=limβ⟶0∑i=1n12∫D∫0M∂qβ/∂xi−qid2−∂q/∂xi−qid2βdmdx+limβ⟶0ε2β∫D∫0Mvm+βlm2−vm2+vx+βlx2−vx2dmdx,then(26)limβ⟶0Φεv+βl−Φεvβ=limβ⟶0∑i=1n12∫D∫0M∂qβ/∂xi−∂q/∂xiβ∂qβ∂xi+∂q∂xi−2qiddmdx+limβ⟶0ε∫D∫0Mvmlm+vxlxx,mdmdx=∑i=1n∫D∫0M∂μx,m∂xi∂qx,m∂xi−qiddmdx+∫D∫0Mεvmlm+vxlxx,mdmdx.
The following theorem proposes a solution of the problem (4).Theorem 4.
Letv∈L20,M;H01D be a solution of (4); then,(27)vmm+vxx+1εDivpiqx=0,where q=qv is the output of (1), where pi=p1,…,pn and pi∈C0,M;H01D is the solution of (22).Proof.
Letl∈L20,M;H01D and v+βl∈L20,M;H01D for β>0. The extremal of Φε is realized at v; then,(28)0≤limβ⟶0Φεv+βl−Φεvβ.
Lemma9 gives(29)0≤∑i=1n∫D∫0M∂μx,m∂xi∂qx,m∂xi−qiddmdx+∫D∫0Mεvmlm+vxlxx,mdmdx.
Therefore, using (22), we obtain(30)0≤∑i=1n∫D∫0M∂μx,m∂xi−∂pix,m∂m−∂2pix,m∂x2−∂vpix,m∂xdmdx+∫D∫0Mεvmlm+vxlxdmdx.
By a simple calculus, we have(31)0≤∑i=1n∫D∫0M∂μx,m∂xi−∂pix,m∂m−∂2pix,m∂x2+∂vpix,m∂xpix,mdmdx+∫D∫0Mεvmlm+vxlxdmdx.
From System(13), we obtain(32)0≤∑i=1n∫D∫0M∂∂xi−lx,mqxpix,mdmdx+∫D∫0Mεvmlm+vxlxdmdx=∫D∫0M−lx,mqx∑i=1n∂∂xipix,m+εvmlm+εvxlxdmdx.
Moreover, ifl=lt∈L20,M;H01D, we deduce(33)−lqx∑i=1n∂∂xipix,m−εvmml−εvxxl=0,which allow us to introduce(34)vmm+vxx+1εDivpiqx=0,that the solution v of (4) must satisfy.Remark 5.
According to equation (2),(1)
If we consider a spatial control functionv=vx, then the variational formula becomes(35)vxx=−1εDivpiqx,(2)
If we consider a temporal control functionv=vm, then the variational formula becomes(36)vmm=−1εDivpiqx.


## 3. Partial Flow Control Problem

### 3.1. Problem Statement

We consider the bilinear distributed system (1), with a given q0∈H1D. System (1) can be rewritten as follows:(37)qm=Smq0+∫0mSm−svsqsds,and the solution of (37) are often called the mild solution of (1).The existence of a unique solutionqvx,m in L20,M;H01D satisfying (37) can be deduced from [37].We chooseω∈D, and(38)χω:L2Dn⟶L2Dnq⟶χωq=q|ω,and χω∗; its adjoint is given by(39)χω∗q=qinD,0∈D∖ω,(40)χ˜ω:L2D⟶L2ωq⟶χ˜ωq=q|ω.Definition 6.
Equation (1) is called partial flow controllable on ω⊂D, to gd∈L2ωn if there exists a control v∈L20,M,H01D and ε>0 such that(41)χω∇qvM−gdL2ωn≤ε,where gd=y1d,…ynd is the desired flow in L2ωn.
Ouzehra in [26], studies the exact and approximate controllability of distributed bilinear systems. The partial flow control problem of (1) is(42)minv∈L20,M,H01DΦεv,where Φε is presented for ε>0 by(43)Φεv=12χω∇qM−gdL2ωn2+ε2∫Γvm2mdm=12∑i=1nχ˜ω∂qM∂xi−yidL2ω2+ε2∫Γvm2mdm.
The objective of the presented problem is to command the flow of (1) to a target state gdx, realizing (43), and find v∗∈L20,M,H01D, verifying(44)Φεv∗=minv∈L20,M,H01DΦεv.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 map(45)v∈L20,M,H01D⟶qv∈S,is(46)qv+εl−qlε⇀μ,where μ=μq,l verifies(47)μm=μxx−vμx−lqx,Γ,μx,0=0,D,μ=μx=0,Π,where q=qv, v∈L20,M;H01D, and dqvl is the derivative of v⟶qv with respect v.Proof.
The output of equation (13) satisfies(48)μS≤k1qL∞0,M;H01DlL20,M,H01D.
Also,(49)μ′S≤k2qL∞0,M;H01DlL20,M,H01D.
Thus,(50)μC0,M;H01D≤k3lL20,M,H01D.
Then, we obtain thatl∈L20,M;L2D⟶μ∈C0,M;H01D is bounded (see [5]).
If we putql=qv+l and ξ=ql−q, then ξ is the state of(51)ξmx,m=ξxx−vx,mξxx,m−lx,mqlx,Γ,ξx,0=0,D,ξ=ξx=0,Π.
Thus,(52)ξL∞[0,M]];H01D≤k4lL20,M,H01D.
Letγ=ξ−μ which verifies the system(53)γm=γxx+vx,mγxx,m+lx,mξx,Γ,γx,0=0,D,γ=γx=0,Π,γ∈C0,M;H01D; consequently,(54)γC0,M;H01D≤klL20,M,H01D2,and we have(55)qv+l−qv−dqvlC0,M;H01D=γC0,M;H01D≤klL20,M,H01D2.
We introduce the family of optimal systems in the case of partial controllability(56)−pimx,m=pixxx,m+vmpixx,m,Γ,pix,M=∂qM∂xi−χ˜ω∗yid,D,pix,m=pixx,m=0,Π,where χ˜ω∗ is the adjoint of χ˜ω defined from L2ω⟶L2D by(57)χ˜ω∗qx=qx,x∈ω,0,x∈D\ω.
The following lemma mentions the differential ofΦεv with respecting v.Lemma 9.
Ifv∈L20,M is the control realizing (42), μ is the output of (47), and pi is the solution of (56), we deduce(58)limβ⟶0Φεv+βl−Φεvβ=∑i=1n∫ωχ˜ω∗χ˜ω∫0M∂pi∂m∂μx,m∂xidm+∫0Mpi∂∂xi∂μ∂mdmdx+∫0M2εlmvmdm.Proof.
The functionalΦεv given by (43) can take the form(59)Φεv=12∑i=1n∫ωχ˜ω∂q∂xi−yid2dx+ε∫0Mvm2mdm.
Letqβ=qv+βl and q=qv, using (59), we have(60)limβ⟶0Φεv+βl−Φεvβ=limβ⟶0∑i=1n12∫ωχ˜ω∂qβ/∂xi−yid2−χ˜ω∂q/∂xi−yid2βdx+limβ⟶0εβ∫0Mvm+βlm2−vm2dm.
Consequently,(61)limβ⟶0Φεv+βl−Φεvβ=limβ⟶0∑i=1n12∫ωχ˜ω∂qβ/∂xi−∂q/∂xiβχ˜ω∂qβ/∂xi+χ˜ω∂q/∂xi−2yiddx+limβ⟶0∫0M2εlmvm+εβlm2dm=∑i=1n∫ωχ˜ω∂μx,M∂xiχ˜ω∂qx,M∂xi−χ˜ω∗yiddx+∫0M2εlmvmdm=∑i=1n∫ωχ˜ω∂μx,M∂xiχ˜ωpix,Mdx+2ε∫0Mlmvmdm.
From (56) and (61), we deduce that(62)limβ⟶0Φεvε+βl−Φεvεβ=∑i=1n∫ωχ˜ω∗χ˜ω∫0M∂pi∂m∂μx,m∂xidm+∫0Mpi∂∂xi∂μ∂mdmdx+∫0M2εlmvmdm.
Now, we will deduce the solution of (42), exploiting the family of optimal systems.Theorem 10.
Letv∈L20,M;H01 be the solution of the partial flow problem, and q=qv is its corresponding state of (1), we show that(63)∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω−2εvmm=0,is a solution of problem (42), where pi∈C0,M;H01D is the unique solution of the adjoint system (56).Proof.
Letl∈L20,M;H01D with v+βl∈L20,M;H01D for β>0. The functional Φε get its minimum at v, and we deduce(64)0≤limβ⟶0Φεv+βl−Φεvβ.
Using Lemma9, replacing ∂μ/∂m in system (47), we have(65)0≤limβ⟶0vεqε+βl−vεqεβ=∑i=1n∫ωχ˜ω∗χ˜ω∫0M∂μ∂xi∂pi∂mdm+∫0M∂∂xiμxx−vμx−lqxpidmdx+∫0M2εlmvmdm0≤∑i=1n∫ωχ˜ω∗χ˜ω∫0M∂μ∂xi∂pi∂m+∂2pi∂x2+vm∂pi∂x+lm∂qx∂xipidmdx+∫0M2εlmvmdm=∑i=1n∫0Mlmχ˜ω∂qx∂xi;χ˜ωpiL2ωdm+∫0M2εlmvmdm=∫0Mlm∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω+2εlmvmdm.
Consequently, for an arbitrary controll=lm we conclude(66)lm∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω−2εvmmlm=0.
Then,(67)∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω−2εvmm=0.
Consequently,(68)vmm=−12ε∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω.


## 3.1. Problem Statement

We consider the bilinear distributed system (1), with a given q0∈H1D. System (1) can be rewritten as follows:(37)qm=Smq0+∫0mSm−svsqsds,and the solution of (37) are often called the mild solution of (1).The existence of a unique solutionqvx,m in L20,M;H01D satisfying (37) can be deduced from [37].We chooseω∈D, and(38)χω:L2Dn⟶L2Dnq⟶χωq=q|ω,and χω∗; its adjoint is given by(39)χω∗q=qinD,0∈D∖ω,(40)χ˜ω:L2D⟶L2ωq⟶χ˜ωq=q|ω.Definition 6.
Equation (1) is called partial flow controllable on ω⊂D, to gd∈L2ωn if there exists a control v∈L20,M,H01D and ε>0 such that(41)χω∇qvM−gdL2ωn≤ε,where gd=y1d,…ynd is the desired flow in L2ωn.
Ouzehra in [26], studies the exact and approximate controllability of distributed bilinear systems. The partial flow control problem of (1) is(42)minv∈L20,M,H01DΦεv,where Φε is presented for ε>0 by(43)Φεv=12χω∇qM−gdL2ωn2+ε2∫Γvm2mdm=12∑i=1nχ˜ω∂qM∂xi−yidL2ω2+ε2∫Γvm2mdm.
The objective of the presented problem is to command the flow of (1) to a target state gdx, realizing (43), and find v∗∈L20,M,H01D, verifying(44)Φεv∗=minv∈L20,M,H01DΦεv.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 map(45)v∈L20,M,H01D⟶qv∈S,is(46)qv+εl−qlε⇀μ,where μ=μq,l verifies(47)μm=μxx−vμx−lqx,Γ,μx,0=0,D,μ=μx=0,Π,where q=qv, v∈L20,M;H01D, and dqvl is the derivative of v⟶qv with respect v.Proof.
The output of equation (13) satisfies(48)μS≤k1qL∞0,M;H01DlL20,M,H01D.
Also,(49)μ′S≤k2qL∞0,M;H01DlL20,M,H01D.
Thus,(50)μC0,M;H01D≤k3lL20,M,H01D.
Then, we obtain thatl∈L20,M;L2D⟶μ∈C0,M;H01D is bounded (see [5]).
If we putql=qv+l and ξ=ql−q, then ξ is the state of(51)ξmx,m=ξxx−vx,mξxx,m−lx,mqlx,Γ,ξx,0=0,D,ξ=ξx=0,Π.
Thus,(52)ξL∞[0,M]];H01D≤k4lL20,M,H01D.
Letγ=ξ−μ which verifies the system(53)γm=γxx+vx,mγxx,m+lx,mξx,Γ,γx,0=0,D,γ=γx=0,Π,γ∈C0,M;H01D; consequently,(54)γC0,M;H01D≤klL20,M,H01D2,and we have(55)qv+l−qv−dqvlC0,M;H01D=γC0,M;H01D≤klL20,M,H01D2.
We introduce the family of optimal systems in the case of partial controllability(56)−pimx,m=pixxx,m+vmpixx,m,Γ,pix,M=∂qM∂xi−χ˜ω∗yid,D,pix,m=pixx,m=0,Π,where χ˜ω∗ is the adjoint of χ˜ω defined from L2ω⟶L2D by(57)χ˜ω∗qx=qx,x∈ω,0,x∈D\ω.
The following lemma mentions the differential ofΦεv with respecting v.Lemma 9.
Ifv∈L20,M is the control realizing (42), μ is the output of (47), and pi is the solution of (56), we deduce(58)limβ⟶0Φεv+βl−Φεvβ=∑i=1n∫ωχ˜ω∗χ˜ω∫0M∂pi∂m∂μx,m∂xidm+∫0Mpi∂∂xi∂μ∂mdmdx+∫0M2εlmvmdm.Proof.
The functionalΦεv given by (43) can take the form(59)Φεv=12∑i=1n∫ωχ˜ω∂q∂xi−yid2dx+ε∫0Mvm2mdm.
Letqβ=qv+βl and q=qv, using (59), we have(60)limβ⟶0Φεv+βl−Φεvβ=limβ⟶0∑i=1n12∫ωχ˜ω∂qβ/∂xi−yid2−χ˜ω∂q/∂xi−yid2βdx+limβ⟶0εβ∫0Mvm+βlm2−vm2dm.
Consequently,(61)limβ⟶0Φεv+βl−Φεvβ=limβ⟶0∑i=1n12∫ωχ˜ω∂qβ/∂xi−∂q/∂xiβχ˜ω∂qβ/∂xi+χ˜ω∂q/∂xi−2yiddx+limβ⟶0∫0M2εlmvm+εβlm2dm=∑i=1n∫ωχ˜ω∂μx,M∂xiχ˜ω∂qx,M∂xi−χ˜ω∗yiddx+∫0M2εlmvmdm=∑i=1n∫ωχ˜ω∂μx,M∂xiχ˜ωpix,Mdx+2ε∫0Mlmvmdm.
From (56) and (61), we deduce that(62)limβ⟶0Φεvε+βl−Φεvεβ=∑i=1n∫ωχ˜ω∗χ˜ω∫0M∂pi∂m∂μx,m∂xidm+∫0Mpi∂∂xi∂μ∂mdmdx+∫0M2εlmvmdm.
Now, we will deduce the solution of (42), exploiting the family of optimal systems.Theorem 10.
Letv∈L20,M;H01 be the solution of the partial flow problem, and q=qv is its corresponding state of (1), we show that(63)∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω−2εvmm=0,is a solution of problem (42), where pi∈C0,M;H01D is the unique solution of the adjoint system (56).Proof.
Letl∈L20,M;H01D with v+βl∈L20,M;H01D for β>0. The functional Φε get its minimum at v, and we deduce(64)0≤limβ⟶0Φεv+βl−Φεvβ.
Using Lemma9, replacing ∂μ/∂m in system (47), we have(65)0≤limβ⟶0vεqε+βl−vεqεβ=∑i=1n∫ωχ˜ω∗χ˜ω∫0M∂μ∂xi∂pi∂mdm+∫0M∂∂xiμxx−vμx−lqxpidmdx+∫0M2εlmvmdm0≤∑i=1n∫ωχ˜ω∗χ˜ω∫0M∂μ∂xi∂pi∂m+∂2pi∂x2+vm∂pi∂x+lm∂qx∂xipidmdx+∫0M2εlmvmdm=∑i=1n∫0Mlmχ˜ω∂qx∂xi;χ˜ωpiL2ωdm+∫0M2εlmvmdm=∫0Mlm∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω+2εlmvmdm.
Consequently, for an arbitrary controll=lm we conclude(66)lm∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω−2εvmmlm=0.
Then,(67)∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω−2εvmm=0.
Consequently,(68)vmm=−12ε∑i=1nχ˜ω∂qx∂xi;χ˜ωpiL2ω.


## 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(69)∂q∂t+α∂2q∂x2=βvx,m∂q∂x,0,1×0,1,qx,0=q0x=2x,0,1,q=0,atx=0,1.The operator−α∂2/∂x2 admits a set of eigenfunctions ϕn⋅ associated to the eigenvalues λn given by(70)ϕnx=2sinnπx;λn=αn2π2,n≥1.While the operator−α∂2/∂x2 of system (69) and the perturbation βvx,m∂q/∂x commute, using Pazy [37], the solution of (69) can be written as(71)qx,m=∑n=1n=Neαn2π2t<eβ∫0m∂v/∂xx,mdmq0,2sinnπx>2sinnπx.The one dimensional adjoint system can be written as(72)−pmx,m=pxxx,m+vmpxx,m,0,1×0,1,px,1=∂qM∂x−χ˜ω∗yd,0,1,p0,m=px0,m=0,0,1.We define the perturbation function(73)fq,p=χ˜ω∂2qm∂x2;χ˜ωpmL2ω.Using (68) and a finite difference schema, the optimal control v can be found by solving(74)vmmx,m=−12εfq,p,0,1×0,1,v0=vm1=0,0,1.By choosingε=1/n, we define the following sequence of control vnn solution of(75)vmmn+1x,m=−n2fqn,pn,0,1×0,1,v0=vm1=0,0,1,where qn and pn are, respectively, the solution of (71) and (72) perturbed by vn with v0=0.The penalty cost (43) becomes(76)Φnvn=12χω∇qn−ydL20,1,H010,12+12n∫01∫01vmnx,m2dxdm.The following convergent Algorithm 1 allows the implementation of our results.Algorithm 1:Algorithm for calculating the solution of the problem (50).
Step 1: ChooseThe desired targyd et.The convergence accuracyζ.The subregionω and time M.Step 2: Untilvn+1−vn≤ζ repeatUsing (71), compute qn associated to vn.Using (72), compute pn associated to vn.Using (74) and (75), compute vn+1.Step 3:vn such that vn+1−vn≤ζ is the minimum of (76).Remark 11.
(1)
The distributed bilinear systems (1), are considered with the feedback map vx,mqxx,m 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.



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*Source: 3521914-2022-04-23.xml*