On some applications of the controllability principle for fixed point equations

Abstract


The aim of this paper is to draw attention to a general principle for solving control problems for operator equations with the help of fixed point techniques. Three distinct applications are presented: a control problem related to the Lotka–Volterra system, the nonlinear Stokes system, and radial solutions of the Neumann problem for φ-Laplace equations. Only for the first application, the control is an explicit external one, while for the next two applications, the control is dependent on the models and arises from the necessity to conform to the actual modeled process or to a certain boundary condition. From the perspective of those interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method thus paving the way for new applications. From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators.

Authors

Radu Precup
Institute of Advanced Studies in Science and Technology, Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Control problem; Fixed point; Boundary value problem; Stokes system; Radial solution; Lotka–Volterra system

Paper coordinates

R. Precup, On some applications of the controllability principle for fixed point equations, Results Appl. Math., 13 (2022), art. no. 100236, https://doi.org/10.1016/j.rinam.2021.100236

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Results in Applied Mathematics

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Elsevier

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25900374

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On some applications of the controllability principle for fixed point equations

On some applications of the controllability principle for fixed point equations

Radu Precup
Institute of Advanced Studies in Science and Technology,
Babeş-Bolyai University, 400084 Cluj-Napoca, Romania &
Tiberiu Popoviciu  Institute of Numerical Analysis, Romanian
Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania
E-mail: r.precup@math.ubbcluj.ro
Abstract

The aim of this paper is to draw attention to a general principle for solving control problems for operator equations with the help of fixed point techniques. Three distinct applications are presented: a control problem related to the Lotka-Volterra system, the nonlinear Stokes system, and radial solutions of the Neumann problem for ϕ-Laplace equations. Only for the first application, the control is an explicit external one, while for the next two applications, the control is dependent on the models and arises from the necessity to conform to the actual modeled process or to a certain boundary condition. From the perspective of those interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method thus paving the way for new applications. From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators.


Keywords: control problem, fixed point, boundary value problem, Stokes system, radial solution, Lotka-Volterra system

Subject Classification: 34H05, 34K35, 35Q30, 37N25

1 Introduction

For the field of differential equations, “the general problem of control theory is that to reconstituting a differential system (as a matter of fact some of its parameters viewed as control variables) from certain properties of solution” ([2, p. 34]). In [7] we have introduced a controllability principle for a general control problem related to operator equations, that we reproduce here for the convenience of the reader. It consists in finding (w,λ), a solution to the following system

{w=N(w,λ)wW,λΛ,(w,λ)D (1.1)

associated to the fixed point equation w=N(w,λ). Here w is the state variable, λ is the control variable, W is the domain of the states, Λ is the domain of controls and D is the controllability domain, usually given by means of some condition/property imposed to w, or to both w and λ. Notice the very general formulation of the control problem, in terms of sets, where W,Λ and DW×Λ are not necessarily structured sets and N is any mapping from W×Λ to W.

In this context, we say that the equation w=N(w,λ) is controllable in W×Λ with respect to D, providing that problem (1.1) has a solution (w,λ). If the solution is unique we say that the equation is uniquely controllable.

Let Σ be the set of all possible solutions (w,λ) of the fixed point equation and Σ1 be the set of all w that are first components of some solutions of the fixed point equation, that is

Σ = {(w,λ)W×Λ:w=N(w,λ)},
Σ1 = {wW: there is λΛ with (w,λ)Σ}.

Clearly, the set of all solutions of the control problem (1.1) is given by ΣD.

Consider the set-valued map F:Σ1Λ  defined as

F(w)={λΛ: (w,λ)ΣD}.

Roughly speaking, F gives the ‘expression’ of the control variable in terms of the state variable.

We have the following general principle for solving the control problem (1.1).

Proposition 1

If for some extension F~:WΛ of F from Σ1 to W, there exists a fixed point wW of the set-valued map

N~(w):=N(w,F~(w)),

i.e.,

w=N(w,λ), (1.2)

for some λF~(w), then the couple (w,λ) is a solution of the control problem (1.1).

Proof. Clearly (w,λ)Σ. Hence wΣ1 and so F~(w)=F(w). Then λF(w) and from the definition of F, it follows that (w,λ)D. Therefore (w,λ) solves (1.1).   

Note that F and F~ can in particular be single-valued maps and in many cases the extension F~ can be done using the expression of F.

Two applications for a system modeling cell dynamics related to leukemia have been included in [7]. The aim of this paper is to present some other problems that can be treated by using this principle, hoping this way to pave the way for other further applications.

2 The applications

2.1 A control problem related to the Lotka-Volterra system

Let us consider the classical Lotka-Volterra system (see [1, 8]) modeling the prey-predator dynamics, with a control parameter λ aimed to modify the rate of attack on the prey population in a such way that after a time period T, the prey population reaches a desired level xT>0:

{x=ax(1λby)y=cy(1dx)x(0)=x0,y(0)=y0,x(T)=xT,x,y>0.
Theorem 2

Let x0<xT<2x0. The system is controllable on a short time interval [0,T] with

T < min{14a,1c,2x0xT4ax0},
12aT > 2cT(12aT+dxT).

Proof. Assume without loss of generality that b=1. Integrating gives

{x(t)=x0+a0tx(s)(1λy(s))𝑑sy(t)=y0c0ty(s)(1dx(s))𝑑s.

This is our fixed point equation with w=(x,y), W=C([0,T];(0,+)2), Λ= and D={(x,y,λ):x(T)=xT}. Using the controllability condition x(T)=xT, we obtain the necessary form of the control parameter λ, namely

λ=F(x,y):=a0Tx(s)𝑑sxT+x0a0Tx(s)y(s)𝑑s.

This expression can be used to define the extension F~ of F from Σ1 to the whole W. Thus we have to find a solution (x,y) of the system

{x(t)=A(x,y):=x0+a0tx(s)(1F~(x,y)y(s))𝑑sy(t)=B(x,y):=y0c0ty(s)(1dx(s))𝑑s.

We shall look for a solution in the closed convex and bounded set

D0:={(x,y)C([0,T];2):mxxMx,myyMy},

where

mx=2x0(12aT)xT12aT,Mx=xT12aT,my=y012cT(1+dMx)1cT(1+dMx),My=y01cT(1+dMx).

One can check that

0<mxx0<xTMx, 0<my<y0<My.

Let N~=(A,B). We have that N~(D0)D0. Indeed,

|A(x,y)(t)x0| aTMx+|a0tx𝑑sxT+x0|
2aTMx+xTx0.

Hence

mx=2x0xT2aTMxA(x,y)(t)2aTMx+xT=Mx.

Also

|B(x,y)(t)y0|cTMy(1+dMx).

Hence

my=y0cTMy(1+dMx)B(x,y)(t)y0+cTMy(1+dMx)=My.

In addition, the Arzelà–Ascoli theorem guarantees that the operator N~ is compact on D0. Thus Schauder’s fixed point theorem applies and the proof is finished.   

2.2 The nonlinear stationary Stokes system

A mathematical model for the steady-state flow of an incompressible fluid in a given domain and with null velocity on its boundary is given by the well-known Stokes system (see [4, 5, 9, 14, 15])

{Δu+p=h(x)in Ωdiv u=0u=0on Ω. (2.3)

Here Ωn is a bounded open set, u=(u1,..,un) and h=(h1,..,hn)H1(Ω;n). Physically, there are relevant the cases n=2 and n=3, when h(x) stands for the external force, the unknown functions u and p are the velocity and pressure, respectively, while the condition div u=0 means that the fluid is incompressible.

It is well-known that the problem has a unique solution

u = PrV(Δ)1h,
q : =p=h+Δu=h+ΔPrV(Δ)1h,
p L2(Ω),

where V={uH01(Ω;n):div u=0}. Note that the representation of q as a gradient of a function p is guaranteed by Rham’s Lemma.

Consider now the case of a reaction force Φ(u), Φ:H01(Ω;n)H1(Ω;n), namely the problem

{Δu+p=Φ(u)in Ωdiv u=0u=0on Ω, (2.4)

or equivalently

{u=(Δ)1(Φ(u)p),uH01(Ω;n)div u=0.

Denoting q:=p, it appears to be a control problem for a fixed point equation, where u is the state variable, q is the control variable and div u=0 is the controllability condition. Notice the specificity of this problem for which the control is of the process itself, in order to keep the fluid incompressibility, and not an external one. Here

W = H01(Ω;n),Λ=H1(Ω;n),D=V×H1(Ω;n),
N(u,q) = (Δ)1(Φ(u)q).

If (u,q) is a solution of the problem, then

u V, u=PrV(Δ)1Φ(u)and
q = q(u)=p(u)=Φ(u)+Δu=Φ(u)+ΔPrV(Δ)1Φ(u).

The last equality gives the necessary expression of the control variable q in terms of the state variable u. Thus, with the notation in Section 1,

F(u)=Φ(u)+ΔPrV(Δ)1Φ(u),uΣ1.

Obviously, we can extend F from Σ1 to the whole space H01(Ω;n) by using the same expression. Thus

F~(u)=Φ(u)+ΔPrV(Δ)1Φ(u),uH01(Ω;n).

Now the map N(u,F~(u)) from the new fixed point equation is in this case

N(u,F~(u)) = (Δ)1(Φ(u)F~(u))=(Δ)1(Φ(u)Φ(u)ΔPrV(Δ)1Φ(u))
= PrV(Δ)1Φ(u).

Thus, the solvability of the nonlinear Stokes system reduces to finding a fixed point u H01(Ω;n) of the operator N(u,F~(u))=PrV(Δ)1Φ(u), that is to solving the equation

u=PrV(Δ)1Φ(u)in H01(Ω;n).

The projection operator PrV being nonexpansive, the fixed point approach for elliptic equations (see [11]) can be easily adapted to this case. Here is an existence and uniqueness result based on Banach’s contraction principle, showing the unique controllability of the above problem.

Theorem 3

If there exists θ[0,1) such that

|Φ(u)Φ(v)|H1θ|uv|H01

for all u,vH01(Ω;n), then problem (2.4) has a unique solution.

Proof. For any u,vH01(Ω;n), one has

|N(u,F~(u))N(v,F~(v))|H01 |(Δ)1(Φ(u)Φ(v))|H01
= |Φ(u)Φ(v)|H1
θ|uv|H01.

Thus the operator N(u,F~(u)) is a contraction on H01(Ω;n).   

For example, the Nemytskii operator Φ(u)=f(x,u,u), where f:Ω×n×n2n is a Carathéodory function such that f(.,0,0)L2(Ω;n) and

|f(x,u,v)f(x,u¯,v¯)|a|uu¯|+b|vv¯|

for all u,u¯n; v,v¯n2 and some a,b0, satisfies the condition of Theorem 3 provided that θ:=a/λ1+b/λ1<1. Here λ1 is the first eigenvalue of the Dirichlet problem for Δ. Indeed, using two times Poincaré’s inequality, one has

|Φ(u)Φ(v)|H1 = |f(.,u,u)f(.,v,v)|H1
1λ1|f(.,u,u)f(.,v,v)|L2
1λ1(a|uv|L2+b|(uv)|L2)
= 1λ1(a|uv|L2+b|uv|H01)
(aλ1+bλ1)|uv|H01.

Some other existence results for problem (2.4) can be obtained via Schauder’s and Leray-Schauder’s fixed point theorems, as in [11], if the global Lipschitz condition on Φ is relaxed.

2.3 Radial solutions of the Neumann problem for ϕ-Laplace equations

In paper [12], it is discussed the existence and localization of radial solutions of the Neumann problem for general ϕ-Laplace equations with a state-dependent source term. The starting assumption was that for each  hC[R0,R], there exists at least one solution to the problem

{L(v)(r):=(rn1ϕ(v))+εrn1v=rn1h(r)in (R0,R)v(R0)=v(R)=0. (2.5)

Here ϕ:(a,a)(0<a+) is an increasing homeomorphism with ϕ(0)=0, ε>0 and 0R0<R<+. Notice that this happens for singular homeomorphisms, i.e., for a<+, as shows Corollary 2.4 in [3].

A double integration leads (see [12, Lemma 2.1]) to the conclusion that a function v is a solution of (2.5) if and only if it satisfies the following two conditions:

v(r)=λ+R0rϕ1(s1nsRτn1(hεv)𝑑τ)𝑑s(r[R0,R]) (2.6)
R0Rrn1(hεv)𝑑r=0, (2.7)

where λ=v(R0). Relation (2.6) is a fixed point equation in the unknown vC[R0,R], λ appears as a control parameter and (2.7) can be seen as the controllability condition. Hence, under the notations in Section 1, we have

W = C[R0,R],Λ=,D=D0×,
D0 = {vC[R0,R]:v satisfies (2.7)},
N(v,λ)(r) = λ+R0rϕ1(s1nsRτn1(hεv)𝑑τ)𝑑s.

Assuming that (v,λ) is a solution and replacing it in (2.7) we have

R0Rrn1(hελεR0rϕ1(s1nsRτn1(hεv)𝑑τ)𝑑s)𝑑r=0,

whence the expression of λ, namely

λ = F(v)=R0Rrn1(hεR0rϕ1(s1nsRτn1(hεv)𝑑τ)𝑑s)𝑑r/(εR0Rrn1𝑑r)
= n(RnR0n)εR0Rrn1(hεR0rϕ1(s1nsRτn1(hεv)𝑑τ)𝑑s)𝑑r.

We use the same expression for the extension F~ of F to the whole space C[R0,R] and we consider the operator

N~(v) = N(v,F~(v))=F~(v)+R0rϕ1(s1nsRτn1(hεv)𝑑τ)𝑑s
= F~(v)+R0rϕ1(s1nR0sτn1(hεv)𝑑τ)𝑑s.

Clearly, N~(D0)D0 and N~ is completely continuous. We think that the existence of a fixed point of N~ could be obtained via Schauder’s fixed point theorem. The set D0 is closed and convex but unbounded. Hence, in order that Schauder’s theorem apply we need to find a ball of a sufficiently large radius which is invariated by N~ . Such a ball exists if we consider that the function ϕ1 has only a linear growth, i.e., |ϕ1(s)|θ|s|+C for all s and 2ε(RR0)2θ<1. Indeed, one has

|R0rϕ1(s1nR0sτn1(hεv)𝑑τ)𝑑s| R0R(θs1nR0sτn1|hεv|𝑑τ+C)𝑑s
(RR0)2θ(|h|+ε|v|)+(RR0)C

and

|F~(v)||h|ε+(RR0)2θ(|h|+ε|v|)+(RR0)C.

Hence

|N~(v)(r)||h|ε+2(RR0)2θ(|h|+ε|v|)+2(RR0)C.

Since 2ε(RR0)2θ<1, we can choose the radius ρ of the ball sufficiently large such that

|h|ε+2(RR0)2θ(|h|+ερ)+2(RR0)C ρ,
ρ > |h|ε,

which guarantee that the intersection Dρ of D0 with the closed ball centered at the origin and of radius ρ is nonempty (h/ε belongs to both of them) and N~(Dρ)Dρ. Hence Schauder’s fixed point applies. Thus we have the following result.

Theorem 4

If ϕ is an unbounded increasing homeomorphism (i.e., b=+) and ϕ1 is such that

|ϕ1(s)|θ|s|+C (2.8)

for every s and some nonnegative constants θ,C with

2ε(RR0)2θ<1, (2.9)

then for each hC[R0,R], problem (2.5) has at least one solution.

Notice that for a singular homeomorphism, i.e., when a<+, condition (2.8) holds with θ=0 and C=a. Thus Theorem 4 applies for every ε>0.

Also note that if ϕ1 is θ-Lipschitz on and (2.9) holds, then for any h, problem (2.5) has a unique solution. In this case, the operator N~ is a contraction on D0 and the result follows from Banach’s contraction principle.

Finally, note that if instead of h(r) we consider more generaly f(r,v(r)), and we assume that |f(r,s)|μ|s|+γ for all s, then

|N~(v)(r)| |f(.,v)|ε+2(RR0)ϕ1((RR0)|f(.,v)εv|)
1ε(μ|v|+γ)+2(RR0)ϕ1((RR0)((μ+ε)|v|+γ)).

Hence, if there exists ρ>0 with

1ε(μρ+γ)+2(RR0)ϕ1((RR0)((μ+ε)ρ+γ))ρ,

then Schauder’s theorem applies in the ball of radius ρ. This sufficient condition holds if μ<ε and ϕ is singular or more general with lims+ϕ1(s)/s=0. The case μ=0 and ϕ singular is covered by Corollary 2.4 in [3]. More general, the invariance of a large ball is obtained if μ/ε+2(RR0)2(μ+ε)l<1, where l=lims+ϕ1(s)/s.

3 Conclusion

Through this paper we wanted to highlight the advantages of a general method of solving control problems, inside the theory of operator equations, benefiting from fixed point techniques. Three types of apparently distinct applications were considered.

The first one illustrates those problems with an external explicit control through which it intervenes so that the evolution of the investigated process follows the desired path or leads to the desired result. Such kind of controls are frequently imposed in engineering, economics, ecology and medicine. For many control problems associated to various classes of equations, additional techniques of investigation, and related topics, the interested reader can see the excellent Coron’s monograph [6]; some examples from medicine can be found in paper [7].

Surprisingly, a number of seemingly uncontrolled problems can still be treated as control problems. In such situations, the control is a hidden one and acts from inside the model so that it fits the problem under investigation. Such is the case of the Stokes system - the second application of this paper - for which the control is given by the pressure and comes from the necessity to adjust the flow rate of the incompressible fluid through the porous medium. Another example is of a boundary value problem - as in our third application - where the unknown value of the solution at some point takes over the function of a control variable in order for the solution to satisfy a boundary condition.

From the perspective of those readers interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method, thus paving the way for new applications.

From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators, especially for operators of the decomposable type, as in works [10] and [13].

References

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  • [3] C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces, Math. Nachr. 283 (2010) 379–391.
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  • [7] I.Ş. Haplea, L.G. Parajdi and R. Precup, On the controllability of a system modeling cell dynamics related to leukemia, Symmetry 2021,13, 1867.
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2022

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