A mutual control problem for semilinear systems via fixed point approach

Radu Precup R. Precup, Faculty of Mathematics and Computer Science and 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 r.precup@ictp.acad.ro and Andrei Stan A. Stan, Department of Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania andrei.stan@ubbcluj.ro

Abstract.

In this paper, we introduce and discuss the concept of a mutual control problem. Our analysis relies on a vector fixed-point approach based on the fixed-point theorems of Perov, Schauder, and Avramescu. In our analysis we employ a novel technique utilizing Bielecki equivalent norms.

Key words and phrases: control; fixed point; nonlinear operator; differential system

Mathematics Subject Classification (2010): 34B15, 34K35, 47H10

1. Introduction and preliminaries

The study of systems of abstract or concrete equations has been the subject of research for a long time, especially from the perspective of the existence and uniqueness of solutions. In the present paper, our objective goes beyond merely establishing the existence of solutions; we strive to identify solutions whose components maintain a level of control over each other.

Let $D_{1},D_{2},C$ be sets with $C\subset D_{1}\times D_{2}$ and let $E:D_{1}\times D_{2}\rightarrow Z$ be a mapping, where $Z$ is a linear space. Consider the equation $E(x,\lambda)=0_{Z}$ and the problem of finding $\lambda\in D_{2}$ such that the equation has a solution $x\in D_{1}$ with $(x,\lambda)\in C$ . We say that $x$ is the state variable, $\lambda$ is the control variable, and $C$ is the controllability domain. One way to solve the problem (see, e.g, [12]) is to use the controllability condition to express $\lambda$ as a function of $x$ , $\lambda=N(x)$ , and then find a solution to the equation $E(x,N(x))=0_{Z}$ . Alternatively, we can express $x$ as $x=N(\lambda)$ and then solve the equation $E(N(\lambda),\lambda)=0_{Z}$ . In the first case, we solve the problem

\left\{\begin{array}[]{l}\lambda=N\left(x\right)\\[5.0pt] E\left(x,N\left(x\right)\right)=0_{Z},\end{array}\right.

while in the second case, the problem

\left\{\begin{array}[]{l}x=N\left(\lambda\right)\\[5.0pt] E\left(N\left(\lambda\right),\lambda\right)=0_{Z}.\end{array}\right.

For a system of two equations, we view each variable as a control over the other, and we aim to find solutions that achieve equilibrium under certain controllability conditions. We call this a mutual control problem. Specifically, we consider four sets $D_{1},D_{2},C_{1},C_{2}$ with $C_{1}\subset D_{1}\times D_{2}$ and $C_{2}\subset D_{2}\times D_{1}$ , $E_{1},E_{2}:D_{1}\times D_{2}\rightarrow Z$ two mappings and the problem

\begin{cases}E_{1}(x,y)=0_{Z}\\[5.0pt] E_{2}(x,y)=0_{Z}\\[5.0pt] (x,y)\in C_{1},\text{ }(y,x)\in C_{2}.\end{cases}

The interpretation of this problem is as follows: the first equation describes the state $x$ controlled by $y$ , and the second equation describes the state $y$ controlled by $x$ . The restriction of $(x,y)$ to $C_{1}$ and $(y,x)$ to $C_{2}$ represents the controllability conditions. One way to solve such a problem, is to incorporate the controllability conditions into the equations and give the problem a fixed point formulation

\left(x,y\right)\in\left(N_{1}(x,y),\ N_{2}(x,y)\right),

where $N_{1},N_{2}$ are set-valued mappings $N_{1}:D_{1}\times D_{2}\rightarrow D_{1},$ $N_{2}:D_{1}\times D_{2}\rightarrow D_{2}.$ A solution $\left(x,y\right)$ of this fixed point equation is said to be a solution of the mutual control problem, while the problem is said to be mutually controllable if such a solution exists.

In some cases, we can do even more, namely to express $x$ and $y$ as state variables in terms of the controls $y$ and $x$ , respectively. In such situations, one has $x\in N_{1}(y)\$ and $y\in N_{2}(x)$ , where

	$\displaystyle N_{1}:D_{2}\rightarrow D_{1},\ N_{1}(y):=\left\{x:\text{ }E_{1}(x,y)=0_{{}_{Z}}\text{ and }(x,y)\in C_{1}\right\},$
	$\displaystyle N_{2}:D_{1}\rightarrow D_{2},\ N_{2}(x):=\left\{y:\text{ }E_{2}(x,y)=0_{{}_{Z}}\text{ and }(y,x)\in C_{2}\right\}.$

Note that $N_{1}$ indicates how the state $x$ is controlled by $y$ , and $N_{2}$ shows how the state $y$ is controlled by $x$ . The equilibrium is achieved if there exists $(x,y)\in D_{1}\times D_{2}$ such that

(x,y)\in\left(N_{1}(y),N_{2}(x)\right).

It is interesting to note the similarity between the mutual control problem and the Nash equilibrium problem. In case of the last one, we have two functionals $J_{1}\left(x,y\right)$ and $\ J_{2}\left(x,y\right).$ Minimizing $J_{1}\left(.,y\right)$ on $D_{1}$ for each $y\in D_{2}$ yields to the set of minimum points denoted $s_{1}\left(y\right),$ while minimizing $J_{2}\left(x,.\right)$ on $D_{2}\$ for each $x\in D_{1}$ yields to the set of minimum points denoted $s_{2}\left(x\right).$ A point $\left(x,y\right)$ is a Nash equilibrium with respect the two functionals if

\left(x,y\right)\in\left(s_{1}\left(y\right),\ s_{2}\left(x\right)\right).

In situations when the two functionals are differentiable, a Nash equilibrium $\left(x,y\right)$ solves the system

\left\{\begin{array}[]{l}J_{11}\left(x,y\right)=0\\ J_{22}\left(x,y\right)=0,\end{array}\right.\

where $J_{11},J_{22}$ are the derivatives of $J_{1},J_{2}$ in the first and second variable, respectively. These equations replace the ones associated to $E_{1},E_{2},$ while the controllability conditions are such that $x,y$ minimize $J_{1},J_{2}$ in the first and second variable, respectively, that is

	$\displaystyle C_{1}$	$\displaystyle=$	$\displaystyle\left\{\left(x,y\right)\in D_{1}\times D_{2}:\ x\text{ minimizes }J_{1}\left(.,y\right)\right\},$
	$\displaystyle C_{2}$	$\displaystyle=$	$\displaystyle\left\{\left(y,x\right)\in D_{2}\times D_{1}:\ y\text{ minimizes }J_{2}\left(x,.\right)\right\}.$

Therefore, in the differential case, the Nash equilibrium problem appears as a particular case of the mutual control problem as specified above. For further details on such results, we refer the reader to [1, 8, 10, 11, 14, 15, 16, 17, 18].

Compared to classical control theory, where the problems involve explicit control variables (see, e.g., [3, 20]), mutual control problems feature controls that are implicit by the very form of the coupling terms. In this way, the problem of mutual control appears to be more general.

In this paper, we discuss the solvability of a mutual control problem for a system of semilinear first-order differential equations, namely

(1.1)

\left\{\begin{array}[]{l}x^{\prime}+Ax=f\left(x,y\right)\\[5.0pt] y^{\prime}+By=g\left(x,y\right),\text{ on }\left[0,T\right].\end{array}\right.

Here, $T>0$ ; $x,y\in\mathbb{R}^{n};$ $A,B\in M_{n\times n}\left(\mathbb{R}\right)$ and $f,g:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ are continuous mappings. Each of the two states, $x$ and $y$ , controls the other with the aim of achieving the equilibrium specified by the controllability condition

(1.2)

x(T)=ky(T),

for some given $k>0$ . Thus,

	$\displaystyle C_{1}$	$\displaystyle=$	$\displaystyle\left\{(x,y)\,:\,x(T)=ky(T)\right\},$
	$\displaystyle C_{2}$	$\displaystyle=$	$\displaystyle\left\{(y,x)\,:\,y(T)=\frac{1}{k}x(T)\right\}.$

Related to our system $\left(\ref{se}\right)$ , we assume that only one initial state is know, say $y(0).$ Then, based on this and the controllability condition $(\ref{cc0}),$ we determine both the solution and the other initial state $x\left(0\right).$ Note that, in this case, we are dealing a semi-observability problem since, given a control and some partial initial information, we must determine both the output and the necessary initial condition $x\left(0\right)$ that leads to this output satisfying the control. We send to [20] for more details on the observability problems.

Such a control condition is of interest in dynamics of populations when it expresses the requirement that at a certain moment $T$ the ratio between two populations, for example prey and predators, should be the desired $k.$ Similarly, for the control of epidemics, it expresses the requirement that at some time one reach a certain ratio between the infected population and that susceptible to infection. Analogous interpretations can be given in the case of some chemical or medical reaction models.

Our analysis relies on the basic fixed point theorems of Perov, Schauder, and Avramescu. We present both the advantages and limitations of each theorem. Using a vector approach based on matrices instead of constants allows us to obtain results independent of the norm of the spaces we are working in.

1.1. Bielecki-type norms

For each number $\theta\geq 0$ , on the space $C([0,T];\mathbb{R}^{n})$ , we define the Bielecki norm

\left|\left|u\right|\right|_{\theta}=\max_{t\in\left[0,T\right]\ }e^{-\theta t}\left|u\left(t\right)\right|.

Note that if $\theta=0$ , one has $\|\cdot\|_{0}=\|\cdot\|,$ where $\|\cdot\|$ is the uniform norm. We mention that, with a suitable choice of $\theta$ , the Bielecki-type norms allow us to relax the restrictions on constants required by the Lipschitz or growth conditions when using fixed point theorems.

1.2. Matrices convergent to zero

A square matrix $M\in M_{n\times n}\left(\mathbb{R}_{+}\right)$ is said to be convergent to zero if its power $M^{k}$ tends to the zero matrix as $k\rightarrow\infty$ . The next lemma provides equivalent conditions for a square matrix to be convergent to zero (see, e.g., [2, 9, 13]).

Lemma 1.1.

Let $A\in M_{n\times n}\left(\mathbb{R}_{+}\right)$ be a square matrix. The following statements are equivalent:

(a):

The matrix $A$ is convergent to zero.

(b):

The spectral radius of $A$ is less than $1$ , i.e., $\rho(A)<1$ .

(c):

The matrix $I-A,$ where $I$ is the unit matrix of the same size, is invertible and its inverse has nonnegative entries, i.e., $\left(I-A\right)^{-1}\in M_{n\times n}\left(\mathbb{R}_{+}\right).$

(d):

In case $n=2$ , a matrix $A=[a_{ij}]_{1\leq i,j\leq 2}\in M_{2\times 2}\left(\mathbb{R}_{+}\right)$ is convergent to zero if and only if

a_{11},\ a_{22}<1\ \ \text{and\ \ }a_{11}+a_{22}<1+a_{11}a_{22}-a_{12}a_{21}.

In what follows, the vectors in $\mathbb{R}^{n}$ are treated as columns. By $\left|x\right|$ , we mean the Euclidean norm of the vector $x$ ; $\left\langle x,y\right\rangle$ represents the inner product in $\mathbb{R}^{n}$ ; and $\left\|x\right\|$ stands for the uniform norm in $C\left([0,T];\mathbb{R}^{n}\right)$ . For a square matrix $A$ , we denote $S_{A}(t)=e^{-tA}$ ( $t\in\mathbb{R}$ ). Additionally, let $C_{A}$ be a upper bound of the norms of the linear operators $e^{-tA}$ on $[-T,T]$ , i.e.,

\max_{t\in[-T,T]}\left|e^{-tA}\right|\leq C_{A}.

We conclude this introduction by recalling two less known fixed point theorems which are applied in order to establish our results.

Theorem 1.2 (Perov).

Let $\left(X_{i},d_{i}\right),$ $i=1,2$ be complete metric spaces and $N_{i}:X_{1}\times X_{2}\rightarrow X_{i}$ be two mappings for which there exists a square matrix $M$ of size two with nonnegative entries and the spectral radius $\rho\left(M\right)<1$ such that the following vector inequality

\left(\begin{array}[]{c}d_{1}\left(N_{1}\left(x,y\right),N_{1}\left(u,v\right)\right)\\[5.0pt] d_{2}\left(N_{2}\left(x,y\right),N_{2}\left(u,v\right)\right)\end{array}\right)\leq M\left(\begin{array}[]{c}d_{1}\left(x,u\right)\\[5.0pt] d_{2}\left(y,v\right)\end{array}\right)

holds for all $\left(x,y\right),\left(u,v\right)\in X_{1}\times X_{2}.$ Then, there exists a unique point $\left(x,y\right)\in X_{1}\times X_{2}$ with

(x,y)=\left(N_{1}\left(x,y\right),\ N_{2}\left(x,y\right)\right).

Theorem 1.3 (Avramescu).

Let $D_{1}$ be a closed convex subset of a normed space $\ Y$ , $(D_{2},d)$ a complete metric space, and $N_{i}:D_{1}\times D_{2}\rightarrow D_{i}$ , $i=1,2$ be continuous mappings. Assume that the following conditions are satisfied:

(i)

$N_{1}(D_{1}\times D_{2})$ is a relatively compact subset of $Y$ ;
(ii)

There is a constant $L\in[0,1)$ such that

$d(N_{2}(x,y),N_{2}(x,\bar{y}))\leq L\,d(y,\overline{y}),$

for all $y,\overline{y}\in D_{2}$ ;

Then, there exists $(x,y)\in D_{1}\times D_{2}$ such that

N_{1}(x,y)=x,\quad N_{2}(x,y)=y.

Some reference works in fixed point theory are the books [4] and [5].

2. Controllability of the mutual control problem

In the first part of this section, we present some properties of a particular type of matrix that will be used throughout this paper.

2.1. On a type of matrices convergent to zero

In the subsequent analysis based on the fixed point arguments, it is advantageous to use the matrix

M(\theta)=\begin{bmatrix}a_{11}&a_{12}\frac{e^{\theta T}-1}{\theta}\\[3.0pt] a_{21}&a_{22}\frac{1-e^{-\theta T}}{\theta}\end{bmatrix},

where $\theta\geq 0,$ and we need to find $\theta$ such that $M(\theta)$ is convergent to zero. Here, $a_{ij}\,(i,j=1,2)$ are nonnegative numbers with

a_{11}<1\ \ \ \text{and\ \ \ }a_{22}<\frac{1}{T}.

Notice that the last inequality guarantees $a_{22}\frac{1-e^{-\theta T}}{\theta}<1$ since $\frac{1-e^{-\theta T}}{\theta}\leq T$ for every $\theta\geq 0.$

From Lemma 1.1, the matrix $M(\theta)$ is convergent to zero if and only if $h(\theta)<0$ , where

	$\displaystyle h(\theta)$	$\displaystyle=\text{tr}(M(\theta))-1-\text{det}(M(\theta))$
		$\displaystyle=a_{11}+a_{22}\frac{1-e^{-\theta T}}{\theta}-1-a_{11}a_{22}\frac{1-e^{-\theta T}}{\theta}+a_{12}a_{21}\frac{e^{\theta T}-1}{\theta}\ \ \left(\theta\geq 0\right).$

Denoting $\tau=a_{22}(1-a_{11})-a_{12}a_{21},$ one has

	$\displaystyle h^{\prime}(\theta)=\frac{1}{\theta^{2}}\left(a_{22}e^{-\theta T}(1-a_{11})(1+\theta T)+a_{12}a_{21}(\theta T-1)e^{\theta T}-\tau\right),$
	$\displaystyle h(0)=-\left(1-a_{11}\right)\left(1-a_{22}T\right)+a_{12}a_{21}T,$
	$\displaystyle h^{\prime}\left(0\right)=-a_{22}(1-a_{11})\left(\frac{T^{2}}{2}+1\right)<0,\text{ and }$
	$\displaystyle\lim_{\theta\rightarrow\infty}h\left(\theta\right)=\left\{\begin{array}[]{l}a_{11}-1\ \ \text{if }a_{12}a_{21}=0\\[5.0pt] +\infty\ \ \text{if }a_{12}a_{21}>0.\end{array}\right.$

In the following lemma, we discuss conditions to guarantee the existence of a $\theta$ such that $M(\theta)$ is convergent to zero.

Lemma 2.1.

Assume $0\leq a_{11}<1$ and $0<a_{22}<\frac{1}{T}$ .

(i)

If $h(0)<0$ , then $M(0)$ converges to zero.
(ii)

If $h(0)\geq 0$ , then there exists $\theta_{1}>0$ with $h^{\prime}(\theta_{1})=0$ and
- (a)
  
  if $h(\theta_{1})<0$ , then the matrix $M(\theta)$ converges to zero for every $\theta$ between the zeroes of $h$ and does not converge to zero otherwise;
- (b)
  
  if $h(\theta_{1})\geq 0$ , then there are no $\theta$ such that $M(\theta)$ converges to zero.

Proof.

(i) is obvious. The next assertions are based on the convexity of the function $h.$ To prove this, we show its second derivative is positive everywhere on $(0,\infty)$ . Indeed, one has

h^{\prime\prime}(\theta)=\frac{1}{\theta^{3}}\left(2\left(a_{22}-a_{22}a_{11}-a_{12}a_{21}\right)+a_{12}a_{21}\varphi(\theta)-a_{22}\left(1-a_{11}\right)\varphi(-\theta)\right),

where

\varphi(\theta)=e^{\theta T}\left(\left(\theta T-1\right)^{2}+1\right).

Since

\varphi(\theta)\geq 2\,\text{ \ and }\ \varphi(-\theta)\leq 2,\text{ for all $\theta\geq 0,$ }

we find

\theta^{3}h^{\prime\prime}(\theta)\geq 2\left(a_{22}-a_{22}a_{11}-a_{12}a_{21}\right)+2a_{12}a_{21}-2a_{22}(1-a_{11})=0,

which gives our conclusion.

(ii) Assume that $h(0)\geq 0$ . Since $h^{\prime}(0)<0$ and $\lim_{\theta\rightarrow\infty}h(\theta)=+\infty$ , the function $h$ has a minimum at some $\theta_{1}\in(0,+\infty)$ , whence $h^{\prime}(\theta_{1})=0$ . Clearly, if $h(\theta_{1})<0$ , then $h$ has two positive zeros and is negative between them and nonnegative otherwise. If $h(\theta_{1})\geq 0$ , then $h(\theta)\geq 0$ for all $\theta\geq 0$ , and thus there are no $\theta$ such that $M(\theta)$ converges to zero. ∎

Remark 1.

In case $a_{22}=0,$ $M\left(\theta\right)$ is convergent to zero if and only if

a_{11}<1\text{ \ \ and\ \ \ }a_{12}a_{21}\frac{e^{\theta T}-1}{\theta}<1-a_{11}.

Clearly, solving the equation $h^{\prime}(\theta)=0$ analytically is challenging. Thus, instead of $M(\theta)$ , we may consider an approximate variant, larger than $M\left(\theta\right),$ namely

\widetilde{M}(\theta)=\begin{bmatrix}a_{11}&a_{12}\frac{e^{\theta T}-1}{\theta}\\[5.0pt] a_{21}&a_{22}\frac{1}{\theta}\end{bmatrix}.

Since $M\left(\theta\right)\leq\widetilde{M}(\theta)$ componentwise, if $\widetilde{M}(\theta)$ is convergent to zero, then $M\left(\theta\right)$ is so too. Following similar reasoning as above, the matrix $\widetilde{M}(\theta)$ converges to zero if and only if $\ \widetilde{h}(\theta)<0$ , where

\widetilde{h}(\theta)=a_{11}+\frac{a_{22}}{\theta}-1-\frac{a_{11}a_{22}}{\theta}+a_{12}a_{21}\frac{e^{\theta T}-1}{\theta}\ \ \left(\theta>0\right).

Note that $\widetilde{h}$ cannot be extended at zero by continuity and in general is not convex. However, under suitable conditions on $a_{ij},$ a similar result to the previous lemma holds true. In the following, $W:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ represents the Lambert function restricted to $\mathbb{R}_{+}$ , i.e., the inverse of the function $ze^{z}\ \left(z\in\mathbb{R}_{+}\right).$ For details we send to [7].

Lemma 2.2.

Assume that $a_{11}<1$ .

(i):

(2.1)

\tau:=a_{22}(1-a_{11})-a_{12}a_{21}>0

then the function $\tilde{h}$ is strictly convex on $(0,\infty)$ .

(ii):

a_{12}a_{21}Te^{\theta_{1}T}<1-a_{11},

where $\theta_{1}=\frac{1}{T}\left(W\left(\frac{\tau}{e\,a_{12}a_{21}}\right)+1\right)$ , then $\widetilde{h}(\theta_{1})<0$ and there is a vicinity $V=\left(\sigma_{1},\sigma_{2}\right)$ of $\theta_{1}$ with $0<\sigma_{1}<\sigma_{2}<+\infty$ such that matrix $\widetilde{M}(\theta)$ is convergent to zero for every $\theta\in V$ and is not so for $\theta\notin V.$

Proof.

(i) Simple computations yield

\widetilde{h}^{\prime}(\theta)=\frac{1}{\theta^{2}}\left(a_{12}a_{21}e^{\theta T}\left(\theta T-1\right)-\tau\right).

Differentiating again $h^{\prime}(\theta)$ with respect to $\theta\in(0,\infty)$ , and using (2.1), we deduce

	$\displaystyle\tilde{h}^{\prime\prime}(\theta)$	$\displaystyle=\frac{1}{\theta^{3}}\left(a_{12}a_{21}e^{\theta T}\theta^{2}T^{2}-2\left(-\tau+a_{12}a_{21}e^{\theta T}\theta T-a_{12}a_{21}e^{\theta T}\right)\right)$
		$\displaystyle=\frac{a_{12}a_{21}}{\theta^{3}}e^{\theta T}\left(\theta^{2}T^{2}-2\theta T+2\right)+\frac{2\tau}{\theta^{3}}\left(-a_{11}a_{22}-a_{12}a_{21}+a_{22}\right)$
		$\displaystyle=\frac{a_{12}a_{21}}{\theta^{3}}e^{\theta T}\left(1+(\theta T-1)^{2}\right)+\frac{2\tau}{\theta^{3}}>0.$

(ii) Note that from $a_{22}(1-a_{11})>0$ , we find $\lim_{\theta\rightarrow 0}\widetilde{h}(\theta)=+\infty$ , while from $a_{12}a_{21}>0$ , we have $\lim_{\theta\rightarrow\infty}\widetilde{h}(\theta)=+\infty$ . Therefore, $\widetilde{h}$ has a minimum $\theta_{1}\in(0,\infty)$ . Since $\widetilde{h}$ is convex, we have $\widetilde{h}^{\prime}(\theta_{1})=0$ , which leads to

a_{12}a_{21}e^{\theta_{1}T}(\theta_{1}T-1)=\tau.

Letting $z:=\theta_{1}T-1$ , we obtain

ze^{z}=\frac{\tau}{e\,a_{12}a_{21}},

thus $z=W\left(\frac{\tau}{e\,a_{12}a_{21}}\right)$ . Consequently,

\theta_{1}=\frac{1}{T}\left(W\left(\frac{\tau}{e\,a_{12}a_{21}}\right)+1\right).

Next, evaluating $\widetilde{h}$ at $\theta_{1}$ , we find

\widetilde{h}(\theta_{1})=a_{11}-1+a_{12}a_{21}Te^{\theta_{1}T}.

Therefore, under our assumption, $\widetilde{h}(\theta_{1})<0$ . The conclusion follows from the convexity of $\widetilde{h}$ and its limits as $\theta$ approaches 0 and infinity. ∎

Remark 2.

The advantage of using $\widetilde{M}(\theta)$ instead of $M(\theta)$ is that we have an analytical expression of the point where the minimum of $\tilde{h}$ is attained. However, in applications, due to numerical computer power, it may be more advantageous to find an approximate solution of the equation $h^{\prime}(\theta)=0$ , and evaluate $h$ at that point.

Since $\widetilde{M}(\theta)$ is an approximation of $M(\theta)$ , there are cases where $\widetilde{M}\left(\theta\right)$ does not converge to zero for any $\theta>0$ , however, there exists a $\theta_{0}>0$ such that $M(\theta_{0})$ does. The example below illustrates this situation.

Example 1.

Let

a_{11}=0.3,\,a_{12}=0.62,\,a_{21}=0.45,\,a_{22}=0.63,\text{ and }T=0.98.

Simple computations show that $\tau=0.162>0$ and $\theta_{1}\approx 1.1931$ , but conditions (ii) is not satisfied, since

0.88\approx a_{12}a_{21}Te^{\theta_{1}T}>1-a_{11}=0.7.

On the other hand, solving numerically $h^{\prime}(\theta)=0$ gives an approximate solution $\theta_{0}\approx 0.3505$ , and hence $h(\theta_{0})\approx-0.00798<0,$ i.e., $M\left(\theta_{0}\right)$ is convergent to zero. Additionally, note that $h(0)=0.0056>0$ , i.e., $M(0)$ is not convergent to zero.

2.2. Fixed point formulation of the mutual control problem

Throughout this section, let $\beta\in\mathbb{R}^{n}\$ be an arbitrarily given value, and define

C_{\beta}([0,T];\mathbb{R}^{n}):=\left\{y\in C([0,T];\mathbb{R}^{n})\;:\;y(0)=\beta\right\}.

Additionally, we denote

X_{\beta}=C([0,T];\mathbb{R}^{n})\times C_{\beta}([0,T];\mathbb{R}^{n}),

which is a complete metric space with respect to the metric inherited from $C([0,T];\mathbb{R}^{n})\times C([0,T];\mathbb{R}^{n})$ . Under the initial condition $y\left(0\right)=\beta,$ the system (1.1) is equivalent to

(2.2)

\begin{cases}x\left(t\right)=S_{A}\left(t\right)x(0)+\int_{0}^{t}S_{A}\left(t-s\right)f\left(x(s),y(s)\right)ds\\[5.0pt] y\left(t\right)=S_{B}\left(t\right)\beta+\int_{0}^{t}S_{B}\left(t-s\right)g\left(x(s),y(s)\right)ds,\end{cases}

while the controllability condition (1.2) becomes

(2.3)		$\displaystyle x(0)$	$\displaystyle=kS_{A}(-T)S_{B}\left(T\right)\beta+k\int_{0}^{T}S_{A}(-T)S_{B}\left(T-s\right)g\left(x(s),y(s)\right)ds$
		$\displaystyle\quad-\int_{0}^{T}S_{A}\left(-s\right)f\left(x\left(s\right),y\left(s\right)\right)ds.$

Thus, substituting relation (2.3) in (2.2), for every $t\in[0,T]$ , we have

(2.4)

\begin{cases}x\left(t\right)=N_{1}(x,y)(t)\\ y\left(t\right)=N_{2}(x,y)(t),\end{cases}

where $\left(N_{1},N_{2}\right)\colon X_{\beta}\rightarrow X_{\beta}$ ,

	$\displaystyle N_{1}(x,y)(t)$	$\displaystyle=kS_{A}(t-T)S_{B}\left(T\right)\beta+k\int_{0}^{T}S_{A}(t-T)S_{B}\left(T-s\right)g\left(x(s),y(s)\right)ds$
		$\displaystyle\quad-\int_{t}^{T}S_{A}\left(t-s\right)f\left(x(s),y(s)\right)ds,$

and

N_{2}(x,y)(t)=S_{B}\left(t\right)\beta+\int_{0}^{t}S_{B}\left(t-s\right)g\left(x(s),y(s)\right)ds.

Clearly, the operator $(N_{1},N_{2})$ is well defined from $X_{\beta}~$ to $X_{\beta}$ . Consequently, the system (2.4) can be viewed as a fixed-point equation in $X_{\beta}$ for the operator $(N_{1},N_{2})$ .

The next result establishes the equivalence between the fixed points of the operator $(N_{1},N_{2})$ and the solutions of our mutual control problem.

Lemma 2.3.

A pair $\left(x,y\right)\in X_{\beta}$ is a solution of the mutual control problem (1.1)-(1.2), i.e., it satisfies both $\left(\ref{semi-observability}\right)$ and $\left(\ref{cc1}\right),$ if and only if it is a fixed point for the operator $\left(N_{1},N_{2}\right).$

Proof.

The necessity has already been explained. For the sufficiency, assume $(x,y)\in X_{\beta}$ is a fixed point of $(N_{1},N_{2})$ , i.e., it satisfies (2.4). Letting $t=0$ in the first relation, we derive the controllability condition (2.3). Next, applying the operator $S_{A}(t)$ to (2.3), we deduce

	$\displaystyle S_{A}(t)x(0)+\int_{0}^{T}S_{A}(t-s)f(x(s),y(s))ds\,ds$
	$\displaystyle=kS_{A}(t-T)S_{B}(T)\beta+k\int_{0}^{T}S_{A}(t-T)S_{B}(T-s)g(x(s),y(s))\,ds,$

which, when used in the definition of $N_{1}$ leads to the first relation of (2.2), i.e.,

x(t)=S_{A}(t)x(0)+\int_{0}^{t}S_{A}(t-s)f(x(s),y(s))\,ds,

The second relation in (2.2) follows directly from the definition of $N_{2}$ . ∎

2.3. Existence and uniqueness via Perov’s fixed point theorem

For our first existence result, we apply Perov’s fixed point theorem, which requires that $f$ and $g$ satisfy global Lipschitz conditions.

Theorem 2.4.

Assume that there are constants $a,b,c,d\geq 0$ such that

	$\displaystyle\left\|f\left(x,y\right)-f\left(\overline{x},\overline{y}\right)\right\|$	$\displaystyle\leq$	$\displaystyle a\left\|x-\overline{x}\right\|+b\left\|y-\overline{y}\right\|,$
	$\displaystyle\left\|g\left(x,y\right)-g\left(\overline{x},\overline{y}\right)\right\|$	$\displaystyle\leq$	$\displaystyle c\left\|x-\overline{x}\right\|+d\left\|y-\overline{y}\right\|,$

for all $x,\overline{x},y,\overline{y}\in\mathbb{R}^{n}.$ If there exists $\theta\geq 0$ such that the matrix

(2.5)

M(\theta)=\left[\begin{array}[]{ll}TC_{A}\left(a+kcC_{B}\right)&C_{A}\left(b+kdC_{B}\right)\frac{e^{\theta T}-1}{\theta}\\[5.0pt] cTC_{B}&dC_{B}\frac{1-e^{-\theta T}}{\theta}\end{array}\right]

is convergent to zero, then the mutual control problem (1.1)-(1.2) has a unique solution in $X_{\beta}$ .

Proof.

We apply Perov’s fixed point theorem to the operators $N_{1},N_{2}$ on the space $X_{1}=C([0,T];\mathbb{R}^{n})$ and $X_{2}=C_{\beta}([0,T];\mathbb{R}^{n})$ . The first space, $X_{1}$ , is equipped with the uniform norm, while the second space, $X_{2}$ , is equipped with a Bielecki norm $\|\cdot\|_{\theta}$ , where $\theta$ is given in the hypothesis. For any $x,\overline{x}\in C\left(\left[0,T\right];\mathbb{R}^{n}\right)$ and $y,\overline{y}\in C_{\beta}\left([0,T];\mathbb{R}^{n}\right)$ , one has

	$\displaystyle\left\|N_{1}\left(x,y\right)\left(t\right)-N_{1}\left(\overline{x},\overline{y}\right)\left(t\right)\right\|$	$\displaystyle\leq kC_{A}C_{B}\int_{0}^{T}\left\|g\left(x\left(s\right),y\left(s\right)\right)-g\left(\overline{x}\left(s\right),\overline{y}\left(s\right)\right)ds\right\|$
		$\displaystyle+C_{A}\int_{t}^{T}\left\|f\left(x\left(s\right),y\left(s\right)\right)-f\left(\overline{x}\left(s\right),\overline{y}\left(s\right)\right)\right\|ds$
		$\displaystyle\leq TC_{A}\left(a+kcC_{B}\right)\left\\|x-\overline{x}\right\\|$
		$\displaystyle\quad+C_{A}\left(b+kdC_{B}\right)\int_{0}^{T}e^{\theta s}e^{-\theta s}\left\|y(s)-\overline{y}(s)\right\|ds,$
		$\displaystyle\leq a_{11}\left\\|x-\overline{x}\right\\|+a_{12}\frac{e^{\theta T}-1}{\theta}\left\\|y-\overline{y}\right\\|,$

where

a_{11}=TC_{A}\left(a+kcC_{B}\right)\text{ and }a_{12}=C_{A}\left(b+kdC_{B}\right).

Therefore, taking the supremmum over $t\in[0,T]$ yields

\left\|N_{1}(x,y)-N_{1}(\overline{x},\overline{y})\right\|\leq a_{11}\|x-\overline{x}\|+a_{12}\frac{e^{\theta T}-1}{\theta}\|y-\overline{y}\|_{\theta}.

For the second operator $N_{2}$ , for every $t\in[0,T]$ , we estimate

	$\displaystyle\left\|N_{2}\left(x,y\right)\left(t\right)-N_{2}\left(\overline{x},\overline{y}\right)\left(t\right)\right\|$	$\displaystyle\leq C_{B}\int_{0}^{t}\left\|g\left(x\left(s\right),y\left(s\right)\right)-g\left(\overline{x}\left(s\right),\overline{y}\left(s\right)\right)\right\|ds$
		$\displaystyle\leq cTC_{B}\\|x-\overline{x}\\|+dC_{B}\frac{e^{t\theta}-1}{\theta}\\|y-\overline{y}\\|_{\theta}.$

Multiplying both sides by $e^{-\theta t}$ and taking the supremum over $t\in[0,T]$ , we obtain

\left\|N_{2}\left(x,y\right)-N_{2}\left(\overline{x},\overline{y}\right)\right\|_{\theta}\leq a_{21}\|x-\overline{x}\|+a_{22}\frac{1-e^{-\theta T}}{\theta}\|y-\overline{y}\|_{\theta},

where

a_{21}=cTC_{B}\text{ and }a_{22}=dC_{B}.

Writing the above relations in the vector form, one has

\begin{bmatrix}\left\|N_{1}\left(x,y\right)-N_{1}\left(\overline{x},\overline{y}\right)\right\|\\[5.0pt] \ \left\|N_{2}\left(x,y\right)-N_{2}\left(\overline{x},\overline{y}\right)\right\|_{\theta}\end{bmatrix}\leq M(\theta)\begin{bmatrix}\left\|x-\overline{x}\right\|\\[5.0pt] \ \left\|y-\overline{y}\right\|_{\theta}\end{bmatrix}.

Consequently, since the matrix $M(\theta)$ is convergent to zero, the operator $(N_{1},N_{2})$ is a Perov contraction on $X_{\beta}$ . Thus, in view of Lemma 2.3, there exists a unique solution $(x,y)\in X_{\beta}$ for the problem $\left(\ref{se}\right)$ - $\left(\ref{cc0}\right).$ ∎

2.4. Existence and localization via Schauder’s fixed point theorem

Our second result allows linear growths on the functions $f$ and $g$ instead of the Lipschitz coditions, but at the cost of losing the uniqueness of the fixed point for the operator $(N_{1},N_{2})$ .

Theorem 2.5.

Assume that $f$ and $g$ satisfy the following linear growth conditions

(2.6)		$\displaystyle\left\|f(x,y)\right\|$	$\displaystyle\leq a\left\|x\right\|+b\left\|y\right\|+\gamma,$
	$\displaystyle\left\|g(x,y)\right\|$	$\displaystyle\leq c\left\|x\right\|+d\left\|y\right\|+\delta,$

for all $x,y\in\mathbb{R}^{n}$ and some nonnegative constants $a,b,c,d,\gamma,\delta$ . If there exists $\theta\geq 0$ such that the matrix $M\left(\theta\right)$ given in $\left(\ref{Matrice Mc}\right)$ is convergent to zero, then the mutual control problem (1.1)-(1.2) has at least one solution $(x,y)\in X_{\beta}$ , satisfying some bounds of the form

\left|x\left(t\right)\right|\leq R_{1}\quad\text{and}\quad\left|y\left(t\right)\right|\leq e^{t\theta}R_{2},\ \left(t\in[0,T]\right),

where $R_{1}$ and $R_{2}$ are chosen appropriately below.

Proof.

We apply Schauder’s fixed-point theorem to the operator $(N_{1},N_{2})$ on the bounded, closed, and convex set $B_{R_{1}}\times B_{R_{2}}$ . The sets $B_{R_{1}}$ and $B_{R_{2}}$ are balls centered at the origin with radii $R_{1}$ and $R_{2}$ in the Banach space $C([0,T];\mathbb{R}^{n})$ with the uniform norm, and in the metric space $C_{\beta}([0,T];\mathbb{R}^{n})$ with the Bielecki norm $\left||\cdot|\right|_{\theta}$ , respectively. Here, $R_{1}$ and $R_{2}$ are positive real numbers that will be determined later.

To this aim, we need to show that the operator $(N_{1},N_{2})$ is completely continuous on $X_{\beta}$ , and that it maps the set $B_{R_{1}}\times B_{R_{2}}$ into itself, i.e.,

(2.7)

\left\|N_{1}(x,y)\right\|\leq R_{1},\ \left\|N_{2}(x,y)\right\|_{\theta}\leq R_{2}\text{ \ whenever\ }\|x\|\leq R_{1},\ \,\|y\|_{\theta}\leq R_{2}.

Using standard arguments, the complete continuity is immediate (see, e.g., [13]).

Let $\left(x,y\right)\in X_{\beta}$ . Then,

	$\displaystyle\left\\|N_{1}\left(x,y\right)\right\\|$	$\displaystyle\leq kC_{A}C_{B}\left\|\beta\right\|+kC_{A}C_{B}\int_{0}^{T}\left\|g\left(x(s\right),y\left(s\right)\right\|ds$
		$\displaystyle\quad+C_{A}\int_{t}^{T}\left\|f\left(x(s\right),y\left(s\right)\right\|ds$
		$\displaystyle\leq TC_{A}\left(a+kcC_{B}\right)\left\\|x\right\\|+C_{A}\left(b+kdC_{B}\right)\frac{e^{\theta T}-1}{\theta}\left\\|y\right\\|_{\theta}+\eta_{1},$

where

\eta_{1}=C_{A}\left(kC_{B}\left|\beta\right|+kTC_{B}\delta+T\gamma\right).

Similarly,

(2.8)

\left|N_{2}\left(y,y\right)\left(t\right)\right|\leq C_{B}\left|\beta\right|+C_{B}\int_{0}^{t}\left|g\left(x\left(s\right),y\left(s\right)\right)\right|ds.

Multiplying (2.8) by $e^{-t\theta}$ and taking the supremum over $[0,T]$ yields

\left|N_{2}\left(y,y\right)\left(t\right)\right|\leq cTC_{B}\left\|x\right\|+dC_{B}\frac{1-e^{-\theta T}}{\theta}\left\|y\right\|_{\theta}+\eta_{2},

where

\eta_{2}=C_{B}\left|\beta\right|+TC_{B}\delta.

Thus

\left[\begin{array}[]{c}\left\|N_{1}\left(x,y\right)\right\|\\[5.0pt] \left\|N_{2}\left(y,y\right)\right\|_{\theta}\end{array}\right]\leq M\left(\theta\right)\left[\begin{array}[]{c}\left\|x\right\|\\[5.0pt] \left\|y\right\|_{\theta}\end{array}\right]+\left[\begin{array}[]{c}\eta_{1}\\[5.0pt] \eta_{2}\end{array}\right].

Note that the invariance condition $\left(\ref{invarianta}\right)\$ holds if $R_{1},R_{2}$ satisfy

M\left(\theta\right)\left[\begin{array}[]{c}R_{1}\\[5.0pt] R_{2}\end{array}\right]+\left[\begin{array}[]{c}\eta_{1}\\[5.0pt] \eta_{2}\end{array}\right]\leq\left[\begin{array}[]{c}R_{1}\\[5.0pt] R_{2}\end{array}\right],

or equivalently

(2.9)

\left[\begin{array}[]{c}\eta_{1}\\[5.0pt] \eta_{2}\end{array}\right]\leq\left(I-M\left(\theta\right)\right)\left[\begin{array}[]{c}R_{1}\\[5.0pt] R_{2}\end{array}\right].

Since the matrix $M\left(\theta\right)$ is convergent to zero, Lemma 1.1 guarantees that the matrix $\left(I-M\left(\theta\right)\right)^{-1}$ has nonnegative entries. Thus, we can multiply $\left(\ref{eqmu}\right)\$ with $\left(I-M\left(\theta\right)\right)^{-1}$ without changing the sign of inequality. If we choose $R_{1},R_{2}$ large enough such that

(I-M\left(\theta\right))^{-1}\begin{bmatrix}\eta_{1}\\[5.0pt] \eta_{2}\end{bmatrix}\leq\begin{bmatrix}R_{1}\\ R_{2}\end{bmatrix},

the invariance condition is satisfied. Consequently, Schauder’s fixed point theorem in $B_{R_{1}}\times B_{R_{2}}$ provides the result. ∎

2.5. Existence via Avramescu’s fixed point theorem

Assuming that both functions $f,g$ have linear growth, and $g$ is Lipschitz continuous with respect to the second variable $y,$ we obtain the following existence result.

Theorem 2.6.

Assume there exists constants $a,c,b,d,\gamma,\delta\geq 0$ such that

	$\displaystyle\left\|f(x,y)\right\|\leq a\left\|x\right\|+b\left\|y\right\|+\gamma,$
	$\displaystyle\left\|g(x,y)-g\left(x,\overline{y}\right)\right\|\leq d\left\|y-\overline{y}\right\|,$

and

\left|g\left(x,0\right)\right|\leq c\left|x\right|+\delta,

for all $x,y,\overline{y}\in\mathbb{R}^{n}$ . If there exists $\theta\geq 0$ such that the matrix $M\left(\theta\right),$ given in (2.5), is convergent to zero, then the mutual control problem $\left(\emph{\ref{se}}\right)$ - $\left(\emph{\ref{cc0}}\right)$ has at least one solution in $X_{\beta}.$

Proof.

Clearly,

	$\displaystyle\left\|f(x,y)\right\|\leq a\left\|x\right\|+b\left\|y\right\|+\gamma,$
	$\displaystyle\left\|g(x,y)\right\|\leq c\left\|x\right\|+d\left\|y\right\|+\delta,$

for all $x,y\in\mathbb{R}^{n}$ . Then, as in the proof of Theorem 2.5, we find $R_{1},R_{2}>0$ such that

\left\|N_{1}(x,y)\right\|\leq R_{1}\ \ \ \text{and\ \ \ }\left\|N_{2}(x,y)\right\|_{\theta}\leq R_{2}

for all $\left(x,y\right)\in X_{\beta}$ with $\left\|x\right\|\leq R_{1}$ and $\left\|y\right\|_{\theta}\leq R_{2}$ . Moreover, $N_{1}\left(B_{R_{1}}\times B_{R_{2}}\right)$ is relatively compact in $C\left([0,T];\mathbb{R}^{n}\right)$ . It remains to show that $N_{2}(x,\cdot)$ is a contraction on $B_{R_{2}}$ with a contraction coefficient independent of $x\in B_{R_{1}}$ . For any $x\in C\left([0,T];\mathbb{R}^{n}\right)$ and any $y,\overline{y}\in C_{\beta}\left([0,T];\mathbb{R}^{n}\right)$ , we have

\left|N_{2}(x,y)(t)-N_{2}(x,\overline{y})(t)\right|\leq dC_{B}\int_{0}^{t}\left|y(s)-\overline{y}(s)\right|ds,

whence

\left\|N_{2}(x,y)(t)-N_{2}(x,\overline{y})\right\|_{\theta}\leq dC_{B}\frac{1-e^{-\theta T}}{\theta}\left\|y-\overline{y}\right\|_{\theta},

where

dC_{B}\frac{1-e^{-\theta T}}{\theta}<1,

since the matrix $M(\theta)$ is convergent to zero. Thus, Avramescu’s theorem applies and gives the final conclusion. ∎

Remark 3.

Under the conditions of the previous theorem, if $\left(x,y\right),\ \left(x,\overline{y}\right)$ are two solutions of the mutual control problem $\left(\ref{se}\right)$ - $\left(\ref{cc0}\right)$ , then necessarily $y=\overline{y}.$

3. Conclusions

The concept of mutual control introduced in this paper and illustrated on the case of semi-linear systems of the first order with a final condition of proportionality of the states is likely to be used for various other classes of systems and other controllability conditions. In this context, problems of observability or partial observability can be formulated. The working technique, as suggested by this paper, is mainly based on the fixed point theory and could be completed by considering other abstract existence principles.

References

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[8] S. Park, Generalizations of the Nash equilibrium theorem in the KKM theory, Fixed Point Theor. Appl. 2010 (2010), 234706.
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	$\displaystyle\left\|f\left(x,y\right)-f\left(\overline{x},\overline{y}\right)\right\|$	$\displaystyle\leq$	$\displaystyle a\left\|x-\overline{x}\right\|+b\left\|y-\overline{y}\right\|,$
	$\displaystyle\left\|g\left(x,y\right)-g\left(\overline{x},\overline{y}\right)\right\|$	$\displaystyle\leq$	$\displaystyle c\left\|x-\overline{x}\right\|+d\left\|y-\overline{y}\right\|,$

	$\displaystyle\left\|N_{1}\left(x,y\right)\left(t\right)-N_{1}\left(\overline{x},\overline{y}\right)\left(t\right)\right\|$	$\displaystyle\leq kC_{A}C_{B}\int_{0}^{T}\left\|g\left(x\left(s\right),y\left(s\right)\right)-g\left(\overline{x}\left(s\right),\overline{y}\left(s\right)\right)ds\right\|$
		$\displaystyle+C_{A}\int_{t}^{T}\left\|f\left(x\left(s\right),y\left(s\right)\right)-f\left(\overline{x}\left(s\right),\overline{y}\left(s\right)\right)\right\|ds$
		$\displaystyle\leq TC_{A}\left(a+kcC_{B}\right)\left\\|x-\overline{x}\right\\|$
		$\displaystyle\quad+C_{A}\left(b+kdC_{B}\right)\int_{0}^{T}e^{\theta s}e^{-\theta s}\left\|y(s)-\overline{y}(s)\right\|ds,$
		$\displaystyle\leq a_{11}\left\\|x-\overline{x}\right\\|+a_{12}\frac{e^{\theta T}-1}{\theta}\left\\|y-\overline{y}\right\\|,$

(2.6)		$\displaystyle\left\|f(x,y)\right\|$	$\displaystyle\leq a\left\|x\right\|+b\left\|y\right\|+\gamma,$
	$\displaystyle\left\|g(x,y)\right\|$	$\displaystyle\leq c\left\|x\right\|+d\left\|y\right\|+\delta,$

	$\displaystyle\left\\|N_{1}\left(x,y\right)\right\\|$	$\displaystyle\leq kC_{A}C_{B}\left\|\beta\right\|+kC_{A}C_{B}\int_{0}^{T}\left\|g\left(x(s\right),y\left(s\right)\right\|ds$
		$\displaystyle\quad+C_{A}\int_{t}^{T}\left\|f\left(x(s\right),y\left(s\right)\right\|ds$
		$\displaystyle\leq TC_{A}\left(a+kcC_{B}\right)\left\\|x\right\\|+C_{A}\left(b+kdC_{B}\right)\frac{e^{\theta T}-1}{\theta}\left\\|y\right\\|_{\theta}+\eta_{1},$

A Mutual Control Problem for Semilinear Systems via Fixed Point Approach

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A mutual control problem for semilinear systems via fixed point approach

Abstract.

1. Introduction and preliminaries

1.1. Bielecki-type norms

1.2. Matrices convergent to zero

Lemma 1.1.

Theorem 1.2 (Perov).

Theorem 1.3 (Avramescu).

2. Controllability of the mutual control problem

2.1. On a type of matrices convergent to zero

Lemma 2.1.

Proof.

Remark 1.

Lemma 2.2.

Proof.

Remark 2.

Example 1.

2.2. Fixed point formulation of the mutual control problem

Lemma 2.3.

Proof.

2.3. Existence and uniqueness via Perov’s fixed point theorem

Theorem 2.4.

Proof.

2.4. Existence and localization via Schauder’s fixed point theorem

Theorem 2.5.

Proof.

2.5. Existence via Avramescu’s fixed point theorem

Theorem 2.6.

Proof.

Remark 3.

3. Conclusions

References

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