On equilibrium in control problems with applications to evolution systems

Radu 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 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

Abstract.

In this paper we examine a mutual control problem for systems of two abstract evolution equations subject to a proportionality final condition. Related observability and semi-observability problems are discussed. The analysis employs a vector fixed-point approach, using matrices rather than constants, and applies the technique of Bielecki equivalent norms.

Key words and phrases:

evolution equation; control; fixed point; semigroup

2010 Mathematics Subject Classification:

47J35, 34K35, 47H10

1. Introduction

In a recent paper [16], we introduced the concept of mutual control related to systems whose unknowns exert some control over each other. More exactly, we have considered four sets $D_{1}$ , $D_{2}$ , $C_{1}$ , $C_{2}$ with $C_{1}\subset D_{1}\times D_{2}$ , $C_{2}\subset D_{2}\times D_{1}$ , a linear space $Z,$ two mappings $E_{1}$ , $E_{2}:D_{1}\times D_{2}\rightarrow Z,$ and the problem

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

In this context, $y$ controls the state $x$ in the first equation, and $x$ controls the state $y$ in the second. The controllability conditions on $x$ and $y$ are expressed by the appartenence to the sets $C_{1}$ and $C_{2}$ , respectively. We call this problem the mutual control problem.

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.

Also, in [16], we showed that the concept of mutual controllability is related to the notion of equilibrium in general, and includes in particular the concept of a Nash equilibrium ([2, 8, 10, 11, 14, 15, 17, 18, 19]).

In this paper, we illustrate the notion of a mutual control by studying a system of abstract evolution equations

(1.1)

\begin{cases}x^{\prime}(t)=A\,x(t)+F\left(x(t),y(t)\right)\\ y^{\prime}(t)=A\,y(t)+G\left(x(t),y(t)\right)\end{cases}t\in[0,T],

together with the controllability condition

(1.2)

x\left(T\right)-ax(0)=k\left(y\left(T\right)-by\left(0\right)\right).

Here, $a,b,k>0$ are given numbers, $A$ is a linear operator generating a semigroup of operators, and $F,G:X^{2}\rightarrow X$ are continuous mappings.

We note that the controllability condition (1.2) under consideration is non-standard. Unlike the conventional requirement [3], [21], where the final values $x\left(T\right)$ and $y\left(T\right)$ of the two unknowns are specified, it instead demands a proportional relationship between their deviations from the initial states $x\left(0\right)$ and $y\left(0\right),$ respectively.

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 proliferations of 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 results target two aspects:

a):: The problem as an observability one, consisting in determination of the initial states $x\left(0\right)$ and $y\left(0\right)$ from the observable final relations;
b):: A semi-observability problem, that is, finding a solution of the mutual problem when only one of the initial states is given, consequently leading to the determination of the other initial state.

We solve the problem (1.1)-(1.2) using an equivalent formulation as a fixed-point equation. This approach allows for the application of vector techniques based on fixed-point theorems, Bielecki-type norms, and matrices instead of constants. The advantages of each fixed-point method are emphasized, considering specific assumptions about $F$ and $G$ to ensure uniqueness and localization of the solutions.

2. Preliminaries

Let $\left(X,|\cdot|_{X}\right)$ be a Banach space, and let $\operatorname{\mathcal{L}}(X)$ be the set of all bounded linear operators from $X$ to $X$ . Endowed with the norm

|U|_{\mathcal{L}(X)}=\sup_{x\in X\setminus\{0\}}\frac{|Ux|_{X}}{|x|_{X}},

$\operatorname{\mathcal{L}(X)}$ is a Banach space.

2.1. Abstract evolution equations

Let $T>0$ and $A\colon D(A)\subset X\rightarrow X$ be the generator of a $C_{0}$ -semigroup $\{S(t)\,:\,t\geq 0\}$ .

A function $u\in C\left([0,T];X\right)$ is said to be a mild solution of the equation

u^{\prime}(t)=Au(t)+F(t,u(t))\ \ \ \,(t\in\left[0,T\right]),

if it satisfies

u\left(t\right)=S\left(t\right)u(0)+\int_{0}^{t}S\left(t-s\right)F\left(s,u% \left(s\right)\right)ds,\,\text{ for all $t\in[0,T]$.}

Throughout this paper, the number $C_{A}$ stands for upper bound for the norm in $\mathcal{L}(X)$ of the operators $S(t)$ , uniform with respect to $t\in\left[0,2T\right]$ , that is

(2.1)

\left|S(t)\right|_{\mathcal{L}(X)}\leq C_{A},\text{ }

for all $t\in\left[0,2T\right].$ If $A$ generates a semigroup of contractions, then $C_{A}=1.$

For details about semigroups of linear operators and abstract evolution equations we refer to the books [7] and [20].

2.2. Bielecki-type norms

For each number $\theta\geq 0$ , on the space $C([0,T];X)$ , we define the Bielecki norm

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

Note that when $\theta=0,$ the Bieleki norm $\left|\cdot\right|_{0}$ is equivalent to the usual Chebyshev (uniform) norm $\left|\cdot\right|_{\infty}$ . We mention that with a suitable choice of $\theta$ , the Bielecki-type norms allow us to relax the strict conditions on constants such as Lipschitz or growth conditions required by fixed point theorems.

2.3. Matrices convergent to zero

Dealing with systems of equations it is convenient (see, e.g., [9, 13]) to use a vector approach based on matrices instead of constants.

A square matrix $M\in\mathcal{M}_{n\times n}\left(\mathbb{R}_{+}\right)$ 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., [9]).

Lemma 2.1.

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

(a):: The matrix $M$ is convergent to zero.
(b):: The spectral radius of $M$ is less than $1$ , i.e., $\rho(M)<1$ .
(c):: The matrix $I-M,$ where $I$ is the unit matrix of the same size, is invertible and its inverse has nonnegative entries, i.e., $\left(I-M\right)^{-1}\in\mathcal{M}_{n\times n}\left(\mathbb{R}_{+}\right).$

In case $n=2$ , we have the following characterization.

Lemma 2.2.

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

a_{11},\ \ a_{22}<1

and

\text{tr}(M)<1+\text{det}(M),\,\ \text{i.e.,}\,\ a_{11}+a_{22}<1+a_{11}a_{22}-% a_{12}a_{21}.

In the next Section 3, we 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 aim to find $\theta$ such that $M(\theta)$ is convergent to zero. Here, $a_{ij}\,(i,j=1,2)$ are nonnegative numbers with $\ a_{11}<1$ 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 $\theta\geq 0.$

The next lemma proved in [16] deals with the existence of a $\theta\geq 0$ for which matrix $M\left(\theta\right)$ is convergent to zero. It shows us that there can be values of $\theta>0$ for which the matrix $M\left(\theta\right)$ is convergent to zero, although the matrix $M(0)$ corresponding to the Chebyshev norm is not convergent to zero.

From Lemma 2.2, the matrix $M(\theta)$ is convergent to zero if and only if $h(\theta)<0$ , where

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

Lemma 2.3.

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.

2.4. Fixed point theorems

Next, we recall two fixed point theorems, which, along with the well-known Schauder fixed point theorem, will play a key role in our analysis. The first result is Perov’s fixed point theorem (see, e.g., [13]) for mappings on the Cartesian product of two metric spaces.

Theorem 2.4 (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)\\ 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)\\ 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).

The second result is Avramescu’s fixed point theorem [1].

Theorem 2.5 (Avramescu).

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

(a)

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

(b)

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

d(N_{2}(x,y),N_{2}(x,y^{\prime}))\leq L\,d(y,y^{\prime})

for all $x\in D_{1}$ and $y,y^{\prime}\in D_{2}$ .

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], [5] and [6].

3. Mutual control for abstract evolution equations

In this section, we aim to find a mild solution of the control problem (1.1)-(1.2). More exactly, we look for $x,y\in C\left(\left[0,T\right];X\right)$ such that, for all $t\in[0,T]$ , the following relations hold

(3.1)

\begin{cases}x\left(t\right)=S\left(t\right)x\left(0\right)+\int_{0}^{t}S\left% (t-s\right)F\left(x\left(s\right),y\left(s\right)\right)ds\\ y\left(t\right)=S\left(t\right)y\left(0\right)+\int_{0}^{t}S\left(t-s\right)G% \left(x\left(s\right),y\left(s\right)\right)ds,\end{cases}

(3.2)

x\left(T\right)-ax\left(0\right)=k\left(y\left(T\right)-by\left(0\right)\right).

3.1. Fixed point formulation.

Our first aim is to combine the controllability condition (3.2) with (3.1) into a single system that takes the form of a fixed-point equation. Assume $\left(x,y\right)$ is a solution of the mutual control problem. Then, we have

	$\displaystyle x\left(T\right)-ax\left(0\right)=S\left(T\right)x\left(0\right)-% ax\left(0\right)+\int_{0}^{T}S\left(T-s\right)F\left(x\left(s\right),y\left(s% \right)\right)ds,$
	$\displaystyle y\left(T\right)-by\left(0\right)=S\left(T\right)y\left(0\right)-% by\left(0\right)+\int_{0}^{T}S\left(T-s\right)G\left(x\left(s\right),y\left(s% \right)\right)ds,$

and using (3.2) yields

(3.3)		$\displaystyle 0$	$\displaystyle=S\left(T\right)x\left(0\right)-kS\left(T\right)y\left(0\right)-% ax\left(0\right)+kby\left(0\right)$
		$\displaystyle\quad+\int_{0}^{T}S\left(T-s\right)\left(F\left(x\left(s\right),y% \left(s\right)\right)-kG\left(x\left(s\right),y\left(s\right)\right)\right)ds.$

When we express $x\left(0\right)$ and $y\left(0\right)$ from $\left(\ref{eq1}\right)$ and replace them in (3.1), we obtain

(3.4)

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

where $N_{1},N_{2}\colon C\left(\left[0,T\right];X\right)^{2}\rightarrow C\left(\left% [0,T\right];X\right)^{2}$ are given by

	$\displaystyle N_{1}(x,y)(t)$	$\displaystyle=\frac{1}{a}S(t)\left[S(T)x(0)-kS(T)y(0)+kby(0)\right]$
		$\displaystyle\quad+\frac{1}{a}\int_{0}^{T}S(T-s)\left(F(x(s),y(s))-kG(x(s),y(s% ))\right)ds$
		$\displaystyle\quad+\int_{0}^{t}S(t-s)F(x(s),y(s))ds,$
	$\displaystyle N_{2}(x,y)(t)$	$\displaystyle=\frac{1}{kb}S(t)\left[-S(T)x(0)+kS(T)y(0)+ax(0)\right]$
		$\displaystyle\quad-\frac{1}{kb}\int_{0}^{T}S(T-s)\left(F(x(s),y(s))-kG(x(s),y(% s))\right)ds$
		$\displaystyle\quad+\int_{0}^{t}S(t-s)G(x(s),y(s))ds.$

Thus, the pair $\left(x,y\right)$ is a fixed point of the operator $\left(N_{1},N_{2}\right)$ . In the following lemma, we show that the converse also holds, i.e., any fixed point of the operator $\left(N_{1},N_{2}\right)$ satisfies both $\left(\ref{el}\right)$ and $\left(\ref{3.1'}\right)$ .

Lemma 3.1.

Any fixed point of the operator $(N_{1},N_{2})$ satisfy $\left(\ref{el}\right)$ and $\left(\ref{3.1'}\right)$ .

Proof.

Assume that $\left(x,y\right)$ is a solution of (3.4). Setting $t=0$ , we obtain

$\displaystyle ax\left(0\right)$	$\displaystyle=$	$\displaystyle S\left(T\right)x\left(0\right)-kS\left(T\right)y\left(0\right)+% kby\left(0\right)$
		$\displaystyle+\int_{0}^{T}S\left(T-s\right)\left(F\left(x\left(s\right),y\left% (s\right)\right)-kG\left(x\left(s\right),y\left(s\right)\right)\right)ds,$
$\displaystyle kby\left(0\right)$	$\displaystyle=$	$\displaystyle-S\left(T\right)x\left(0\right)+kS\left(T\right)y\left(0\right)+% ax\left(0\right)$
		$\displaystyle-\int_{0}^{T}S\left(T-s\right)\left(F\left(x\left(s\right),y\left% (s\right)\right)-kG\left(x\left(s\right),y\left(s\right)\right)\right)ds,$

whence

	$\displaystyle ax\left(0\right)-kby\left(0\right)$	$\displaystyle=2S\left(T\right)x\left(0\right)-2kS\left(T\right)y\left(0\right)% -\left(ax\left(0\right)-kby\left(0\right)\right)$
		$\displaystyle\quad+2\int_{0}^{T}S\left(T-s\right)\left(F\left(x\left(s\right),% y\left(s\right)\right)-kG\left(x\left(s\right),y\left(s\right)\right)\right)ds,$

that is

(3.5)		$\displaystyle ax\left(0\right)-kby\left(0\right)$	$\displaystyle=S\left(T\right)x\left(0\right)-kS\left(T\right)y\left(0\right)$
		$\displaystyle\quad+\int_{0}^{T}S\left(T-s\right)\left(F\left(x\left(s\right),y% \left(s\right)\right)-kG\left(x\left(s\right),y\left(s\right)\right)\right)ds.$

Letting $t=T$ in (3.4) gives

	$\displaystyle x\left(T\right)$	$\displaystyle=\frac{1}{a}\left[S\left(T\right)\left(S\left(T\right)x\left(0% \right)-kS\left(T\right)y\left(0\right)+kby\left(0\right)\right.\right]$
		$\displaystyle\quad+\frac{1}{a}S\left(T\right)\int_{0}^{T}S\left(T-s\right)% \left(F\left(x\left(s\right),y\left(s\right)\right)-kG\left(x\left(s\right),y% \left(s\right)\right)\right)ds$
		$\displaystyle\quad+\int_{0}^{T}S\left(T-s\right)F\left(x\left(s\right),y\left(% s\right)\right)ds,$

and

	$\displaystyle ky\left(T\right)$	$\displaystyle=\frac{1}{b}\left[S\left(T\right)\left(-S\left(T\right)x\left(0% \right)+kS\left(T\right)y\left(0\right)+ax\left(0\right)\right.\right]$
		$\displaystyle\quad-\frac{1}{b}S\left(T\right)\int_{0}^{T}S\left(T-s\right)% \left(F\left(x\left(s\right),y\left(s\right)\right)-kG\left(x\left(s\right),y% \left(s\right)\right)\right)ds$
		$\displaystyle\quad+k\int_{0}^{T}S\left(T-s\right)G\left(x\left(s\right),y\left% (s\right)\right)ds.$

Whence

	$\displaystyle x\left(T\right)-ky\left(T\right)$	$\displaystyle=\frac{1}{a}S\left(T\right)\big{[}S\left(T\right)x\left(0\right)-% kS\left(T\right)y\left(0\right)+kkby\left(0\right)$
		$\displaystyle\quad+\int_{0}^{T}S\left(T-s\right)\left(F\left(x\left(s\right),y% \left(s\right)\right)-kG\left(x\left(s\right),y\left(s\right)\right)\right)ds% \big{]}$
		$\displaystyle\quad+\int_{0}^{T}S\left(T-s\right)F\left(x\left(s\right),y\left(% s\right)\right)ds$
		$\displaystyle\quad+\frac{1}{b}S\left(T\right)\big{[}S\left(T\right)x\left(0% \right)-kS\left(T\right)y\left(0\right)-ax\left(0\right)$
		$\displaystyle\quad+\int_{0}^{T}S\left(T-s\right)\left(F\left(x\left(s\right),y% \left(s\right)\right)-kG\left(x\left(s\right),y\left(s\right)\right)\right)ds% \big{]}$
		$\displaystyle\quad-k\int_{0}^{T}S\left(T-s\right)G\left(x\left(s\right),y\left% (s\right)\right)ds.$

Since, by (3.3), the expressions within the square brackets equal $ax\left(0\right)$ and $-kby\left(0\right)$ , respectively, we deduce:

(3.6)		$\displaystyle x\left(T\right)-ky\left(T\right)$	$\displaystyle=S\left(T\right)x\left(0\right)-kS\left(T\right)y\left(0\right)$
		$\displaystyle+\int_{0}^{T}S\left(T-s\right)\left(F\left(x\left(s\right),y\left% (s\right)\right)-kG\left(x\left(s\right),y\left(s\right)\right)\right)ds.$

Now (3.5) and (3.6) imply

ax\left(0\right)-kby\left(0\right)=x\left(T\right)-ky\left(T\right),

and so the controllability condition is satisfied.

Finally, (3.5) and (3.4) imply that $\left(x,y\right)$ satisfies (3.1). ∎

3.2. Existence

Using Perov’s and Leray-Schauder’s fixed point theorems, we obtain the following existence result.

Theorem 3.2.

Assume that the following conditions are satisfied:

(i)

There are constants $a_{F},b_{F},a_{G},b_{G}\geq 0$ such that

(3.7)		$\displaystyle\|F(x,y)-F(\overline{x},\overline{y})\|_{X}\leq a_{F}\|x-\overline{x% }\|_{X}+b_{F}\|y-\overline{y}\|_{X},$
	$\displaystyle\|G(x,y)-G(\overline{x},\overline{y})\|_{X}\leq a_{G}\|x-\overline{x% }\|_{X}+b_{G}\|y-\overline{y}\|_{X},$

for all $x,\overline{x},y,\overline{y}\in X$ and $t\in\left[0,T\right];$

(ii)

The map $H:=F-kG$ is bounded on $X^{2};$

(iii)

The matrix

(3.8)

M:=TC_{A}\left[\begin{array}[]{ll}\frac{1}{a}C_{A}\left(a_{F}+ka_{G}\right)+a_% {F}&\frac{1}{a}C_{A}\left(b_{F}+kb_{G}\right)+b_{F}\vskip 3.0pt plus 1.0pt % minus 1.0pt\\ \frac{1}{kb}C_{A}\left(a_{F}+ka_{G}\right)+a_{G}&\frac{1}{kb}C_{A}\left(b_{F}+% kb_{G}\right)+b_{G}\end{array}\right]

is convergent to zero;

(iv)

Semigroup $\left\{S\left(t\right);\ t\geq 0\right\}$ is compact and $\left|S\left(T\right)\right|_{\mathcal{L}\left(X\right)}<\frac{2}{\frac{1}{a}+% \frac{1}{b}}.$

Then problem (3.1)-(3.2) has at least one solution $\left(x,y\right)$ in $C\left(\left[0,T\right];X\right)^{2}.$

Proof.

We prove the result in two steps: (a) First, using Perov’s fixed point theorem we prove that for any $\alpha,\beta\in X,$ there exists in $C\left(\left[0,T\right];X\right)^{2}$ a unique fixed point $\left(x_{\alpha,\beta},y_{\alpha,\beta}\right)$ of the operator $(N_{1},N_{2})$ defined as follows:

(3.9)	$\displaystyle N_{1}\left(x,y\right)\left(t\right)$	$\displaystyle=\frac{1}{a}S\left(t\right)\left[S\left(T\right)\alpha-kS\left(T% \right)\beta+kb\beta\right]$
	$\displaystyle\quad+\frac{1}{a}S(t)\int_{0}^{T}S\left(T-s\right)\left(F\left(x% \left(s\right),y\left(s\right)\right)-kG\left(x\left(s\right),y\left(s\right)% \right)\right)ds$
	$\displaystyle\quad+\int_{0}^{t}S\left(t-s\right)F\left(x\left(s\right),y\left(% s\right)\right)ds.$

	$\displaystyle N_{2}\left(x,y\right)\left(t\right)$	$\displaystyle=\frac{1}{kb}S\left(t\right)\left[-S\left(T\right)\alpha+kS\left(% T\right)\beta+a\alpha\right]$
		$\displaystyle\quad-\frac{1}{kb}S\left(t\right)\int_{0}^{T}S\left(T-s\right)% \left(F\left(x\left(s\right),y\left(s\right)\right)-kG\left(x\left(s\right),y% \left(s\right)\right)\right)ds$
		$\displaystyle\quad+\int_{0}^{t}S\left(t-s\right)G\left(x\left(s\right),y\left(% s\right)\right)ds.$

For any $x,\overline{x},y,\overline{y}\in C\left(\left[0,T\right];X\right),$ one has

	$\displaystyle\left\|N_{1}\left(x,y\right)-N_{1}\left(\overline{x},\overline{y}% \right)\right\|_{0}$	$\displaystyle\leq TC_{A}\left(\frac{1}{a}C_{A}\left(a_{F}+ka_{G}\right)+a_{F}% \right)\left\|x-\overline{x}\right\|_{0}$
		$\displaystyle\quad+TC_{A}\left(\frac{1}{a}C_{A}\left(b_{F}+kb_{G}\right)+b_{F}% \right)\left\|y-\overline{y}\right\|_{0},$

and

	$\displaystyle\left\|N_{2}\left(x,y\right)-N_{2}\left(\overline{x},\overline{y}% \right)\right\|_{0}$	$\displaystyle\leq TC_{A}\left(\frac{1}{kb}C_{A}\left(a_{F}+ka_{G}\right)+a_{G}% \right)\left\|x-\overline{x}\right\|_{0}$
		$\displaystyle\quad+TC_{A}\left(\frac{1}{kb}C_{A}\left(b_{F}+kb_{G}\right)+b_{G% }\right)\left\|y-\overline{y}\right\|_{0}.$

Writing the above relations in the vector form, we obtain

\left[\begin{array}[]{c}\left|N_{1}\left(x,y\right)-N_{1}\left(\overline{x},% \overline{y}\right)\right|_{0}\\[5.0pt] \left|N_{2}\left(x,y\right)-N_{2}\left(\overline{x},\overline{y}\right)\right|% _{0}\end{array}\right]\leq M\left[\begin{array}[]{c}\left|x-\overline{x}\right% |_{0}\\[5.0pt] \left|y-\overline{y}\right|_{0}\end{array}\right].

Consequently, $N=(N_{1},N_{2})$ is a Perov contraction on $C\left(\left[0,T\right];X\right)^{2}$ and thus it has a unique fixed point.

(b) Using Leray-Schauder’s fixed point theorem we prove that the map $H:X^{2}\rightarrow X^{2},$

\left(\alpha,\beta\right)\mapsto\left(x_{\alpha,\beta}\left(0\right),y_{\alpha% ,\beta}\left(0\right)\right),

has at least one fixed point $\left(\alpha^{\ast},\beta^{\ast}\right).$ Then the pair $\left(x_{\alpha^{\ast},\beta^{\ast}},y_{\alpha^{\ast},\beta^{\ast}}\right)$ is a solution of the mutual control problem. To start, we show that $H$ is continuous. Indeed, denoting

	$\displaystyle\gamma_{1}(\alpha,\beta)$	$\displaystyle=\frac{1}{a}S(t)\left(S(T)\alpha-kS(T)\beta+kb\beta\right),$
	$\displaystyle\gamma_{2}(\alpha,\beta)$	$\displaystyle=\frac{1}{kb}S(t)\left(-S(T)\alpha+kS(T)\beta+a\alpha\right),$

one has

	$\displaystyle\left\|\gamma_{1}(\alpha,\beta)-\gamma_{1}(\overline{\alpha},% \overline{\beta})\right\|_{X}$	$\displaystyle\leq C\left(\left\|\alpha-\overline{\alpha}\right\|_{X}+\left\|\beta% -\overline{\beta}\right\|_{X}\right),$
	$\displaystyle\left\|\gamma_{2}(\alpha,\beta)-\gamma_{2}(\overline{\alpha},% \overline{\beta})\right\|_{X}$	$\displaystyle\leq C\left(\left\|\alpha-\overline{\alpha}\right\|_{X}+\left\|\beta% -\overline{\beta}\right\|_{X}\right).$

for a suitable constant $C>0.$ Then

	$\displaystyle\left\|x_{\alpha,\beta}-x_{\overline{\alpha},\overline{\beta}}% \right\|_{0}$	$\displaystyle\leq C\left(\left\|\alpha-\overline{\alpha}\right\|_{X}+\left\|\beta% -\overline{\beta}\right\|_{X}\right)$
		$\displaystyle\quad+m_{11}\left\|x_{\alpha,\beta}-x_{\overline{\alpha},\overline% {\beta}}\right\|_{0}+m_{12}\left\|y_{\alpha,\beta}-y_{\overline{\alpha},% \overline{\beta}}\right\|_{0},$
	$\displaystyle\left\|y_{\alpha,\beta}-y_{\overline{\alpha},\overline{\beta}}% \right\|_{0}$	$\displaystyle\leq C\left(\left\|\alpha-\overline{\alpha}\right\|_{X}+\left\|\beta% -\overline{\beta}\right\|_{X}\right)$
		$\displaystyle\quad+m_{21}\left\|x_{\alpha,\beta}-x_{\overline{\alpha},\overline% {\beta}}\right\|_{0}+m_{22}\left\|y_{\alpha,\beta}-y_{\overline{\alpha},% \overline{\beta}}\right\|_{0}.$

where $m_{ij}$ are the entries of matrix $M.$ Hence

\begin{bmatrix}\left|x_{\alpha,\beta}-x_{\overline{\alpha},\overline{\beta}}% \right|_{0}\\[7.0pt] \left|y_{\alpha,\beta}-y_{\overline{\alpha},\overline{\beta}}\right|_{0}\end{% bmatrix}\leq M\begin{bmatrix}\left|x_{\alpha,\beta}-x_{\overline{\alpha},% \overline{\beta}}\right|_{0}\\[7.0pt] \left|y_{\alpha,\beta}-y_{\overline{\alpha},\overline{\beta}}\right|_{0}\end{% bmatrix}+\begin{bmatrix}C\left(\left|\alpha-\overline{\alpha}\right|_{X}+\left% |\beta-\overline{\beta}\right|_{X}\right)\\[7.0pt] C\left(\left|\alpha-\overline{\alpha}\right|_{X}+\left|\beta-\overline{\beta}% \right|_{X}\right)\end{bmatrix},

equivalently

\begin{bmatrix}\left|x_{\alpha,\beta}-x_{\overline{\alpha},\overline{\beta}}% \right|_{0}\\[7.0pt] \left|y_{\alpha,\beta}-y_{\overline{\alpha},\overline{\beta}}\right|_{0}\end{% bmatrix}\leq\left(I-M\right)^{-1}\begin{bmatrix}C\left(\left|\alpha-\overline{% \alpha}\right|_{X}+\left|\beta-\overline{\beta}\right|_{X}\right)\\[7.0pt] C\left(\left|\alpha-\overline{\alpha}\right|_{X}+\left|\beta-\overline{\beta}% \right|_{X}\right)\end{bmatrix},

which clearly shows that $x_{\alpha,\beta}$ and $y_{\alpha,\beta}$ depend continuously on $\alpha$ and $\beta.$ This implies the continuity of $H.$ Next we show that $H$ maps bounded sets into bounded sets. This follows from the estimate

\left[\begin{array}[]{c}\left|x_{\alpha,\beta}-x_{0,0}\right|_{0}\vskip 3.0pt % plus 1.0pt minus 1.0pt\\ \left|y_{\alpha,\beta}-y_{0,0}\right|_{0}\end{array}\right]\leq\left(I-M\right% )^{-1}\left[\begin{array}[]{c}C\left(\left|\alpha\right|_{X}+\left|\beta\right% |_{X}\right)\vskip 3.0pt plus 1.0pt minus 1.0pt\\ C\left(\left|\alpha\right|_{X}+\left|\beta\right|_{X}\right)\end{array}\right].

Third, since the semigroup is compact, $H$ maps any bounded set into a compact set. Therefore, the map $H$ is completely continuous. The final step is to establish the boundedness of the set of all solutions to the family of equations

H\left(\alpha,\beta\right)=\lambda\left(\alpha,\beta\right),\ \ \ \lambda>1.

If $\left(\alpha,\beta\right)$ is such a solution, then

	$\displaystyle\lambda\alpha$	$\displaystyle=\frac{1}{a}\big{[}S\left(T\right)\alpha-kS\left(T\right)\beta+kb\beta$
(3.10)			$\displaystyle\quad+\int_{0}^{T}S\left(T-s\right)\left(F\left(x_{\alpha,\beta}% \left(s\right),y_{\alpha,\beta}\left(s\right)\right)-kG\left(x_{\alpha,\beta}% \left(s\right),y_{\alpha,\beta}\left(s\right)\right)\right)ds\big{]},$

	$\displaystyle\lambda\beta$	$\displaystyle=\frac{1}{kb}\big{[}-S\left(T\right)\alpha+kS\left(T\right)\beta+a\alpha$
		$\displaystyle\quad-\int_{0}^{T}S\left(T-s\right)\left(F\left(x_{\alpha,\beta}% \left(s\right),y_{\alpha,\beta}\left(s\right)\right)-kG\left(x_{\alpha,\beta}% \left(s\right),y_{\alpha,\beta}\left(s\right)\right)\right)ds\big{]}.$

When we add the above two relations, we find

\lambda\left(a\alpha+kb\beta\right)=a\alpha+kb\beta,

and since $\lambda>1$ , this implies $a\alpha+kb\beta=0$ . Substituting $kb\beta$ by $-a\alpha$ in (3.10) and passing to the norm, in virtue of (ii), we obtain

2\left|\alpha\right|_{X}<\left(\lambda+1\right)\left|\alpha\right|_{X}\leq% \left(\frac{1}{a}+\frac{1}{b}\right)\left|S\left(T\right)\right|_{\mathcal{L}% \left(X\right)}\left|\alpha\right|_{X}+\widetilde{C},

for some constant $\widetilde{C}.$ Using the condition on $\left|S\left(T\right)\right|_{\mathcal{L}\left(X\right)}$ from (iv), we deduce the boundedness of $\alpha$ independently of $\lambda.$ Next from $\alpha\alpha+kb\beta=0,$ we also have that $\beta$ is bounded. Therefore Leray-Schauder’s theorem applies and gives the result. ∎

Remark 3.3.

The previous existence result can be seen as an observability result. Indeed, having observed the final relation (3.2), we may guess one of the possible initial states $\left(x\left(0\right),y\left(0\right)\right)$ of the process.

In the next section we deal with a semi-observability problem for the situation that one of the initial states, say $y\left(0\right),$ is known while the second one $x\left(0\right)$ is found after obtaining a solution of the mutual control problem.

4. A semi-observability problem

In this section, we aim to find a mild solution of the control problem (1.1)-(1.2) assuming that the initial state $y\left(0\right)=\beta$ is known. More exactly, we look for $x,y\in C\left(\left[0,T\right];X\right)$ such that, for all $t\in[0,T]$ , the following relations hold:

(4.1)

\begin{cases}x\left(t\right)=S\left(t\right)x\left(0\right)+\int_{0}^{t}S\left% (t-s\right)F\left(x\left(s\right),y\left(s\right)\right)ds\\ y\left(t\right)=S\left(t\right)y\left(0\right)+\int_{0}^{t}S\left(t-s\right)G% \left(x\left(s\right),y\left(s\right)\right)ds,\end{cases}

(4.2)

y\left(0\right)=\beta,

(4.3)

x\left(T\right)-ax\left(0\right)=k\left(y\left(T\right)-by\left(0\right)\right).

4.1. Fixed point formulation of the problem

Similar to the previous section, we try to incorporate all the equations in only one fixed point problem. Let $\left(x,y\right)$ be a solution of (4.1)-(4.3). Then, using (4.1) and (4.3) gives

S\left(T\right)\left(x\left(0\right)-k\beta\right)+\int_{0}^{T}S\left(T-s% \right)\left(F\left(x(s),y(s)\right)-kG\left(x(s),y(s)\right)\right)ds=ax\left% (0\right)-kb\beta,

whence

(4.4)		$\displaystyle x\left(0\right)$	$\displaystyle=\frac{1}{a}\left(S\left(T\right)\left(x\left(0\right)-k\beta% \right)+kb\beta\right)$
		$\displaystyle\quad+\frac{1}{a}\int_{0}^{T}S\left(T-s\right)\left(F\left(x(s),y% (s)\right)ds-kG\left(x(s),y(s)\right)\right)\,ds.$

When we substitute it in (4.1), one has

(4.5)

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

where

	$\displaystyle N_{1}(x,y)(t)$	$\displaystyle=\frac{1}{a}S(t+T)\left(x(0)-k\beta\right)+k\frac{b}{a}S(t)\beta$
		$\displaystyle\quad+\int_{0}^{t}S(t-s)F(x(s),y(s))\,ds$
		$\displaystyle\quad+\frac{1}{a}\int_{0}^{T}S(t+T-s)\left(F(x,y-kG(x,y))\right),$
	$\displaystyle N_{2}(x,y)(t)$	$\displaystyle=S(t)\beta+\int_{0}^{t}S(t-s)G(x(s),y(s))\,ds.$

The converse implication also holds.

Lemma 4.1.

A pair $\left(x,y\right)$ solves the system $(\ref{E0})$ if and only if it solves $(\ref{E1})$ - $(\ref{E3})$ .

Proof.

We only have to prove that (4.5) implies (4.1)-(4.3). Clearly, (4.2) follows directly from the second equation in (4.5). Setting $t=0$ in the first equation of (4.5) gives (4.4), while setting $t=T$ provides

	$\displaystyle x(T)$	$\displaystyle=S(T)\left[\frac{1}{a}S(T)\left(x(0)-k\beta\right)+k\frac{b}{a}% \beta\right]$
		$\displaystyle\quad+\frac{1}{a}S(T)\int_{0}^{T}S(T-s)\left(F(x(s),y(s))-kG(x(s)% ,y(s))\right)ds$
		$\displaystyle\quad+\int_{0}^{T}S(T-s)F(x(s),y(s))ds.$

From (4.4), one has

S\left(T\right)x\left(0\right)=x\left(T\right)-\int_{0}^{T}S\left(T-s\right)F% \left(x(s),y(s)\right)ds,

and using it in (4.4) yields

	$\displaystyle x(0)$	$\displaystyle=\frac{1}{a}x(T)-\frac{1}{a}\int_{0}^{T}S(T-s)F(x(s),y(s))ds-k% \frac{1}{a}S(T)\beta+k\frac{b}{a}\beta$
		$\displaystyle\quad+\frac{1}{a}\int_{0}^{T}S(T-s)\left(F(x(s),y(s))-kG(x(s),y(s% ))\right)ds$
		$\displaystyle=\frac{1}{a}x(T)-k\frac{1}{a}S(T)\beta+k\frac{b}{a}\beta-k\frac{1% }{a}\int_{0}^{T}S(T-s)G(x(s),y(s))ds$
		$\displaystyle=\frac{1}{a}x(T)-k\frac{1}{a}\left[S(T)\beta+\int_{0}^{T}S(T-s)G(% x(s),y(s))ds\right]+k\frac{b}{a}\beta$
		$\displaystyle=\frac{1}{a}x(T)-k\frac{1}{a}y(T)+k\frac{b}{a}\beta,$

that is (4.3). Finally, combining (4.4) and the first equation of (4.5) results in the first equation of (4.1). ∎

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

Using Perov’s fixed point theorem, we obtain the following existence and uniqueness result.

Theorem 4.2.

Assume that the following conditions are satisfied:

(i)

There are nonnegative constants $a_{11},a_{12},a_{21},a_{22}$ such that

	$\displaystyle\|F(x,y)-F(\overline{x},\overline{y})\|_{X}\leq a_{11}\|x-\overline{% x}\|_{X}+a_{12}\|y-\overline{y}\|_{X},$
	$\displaystyle\|G(x,y)-G(\overline{x},\overline{y})\|_{X}\leq a_{21}\|x-\overline{% x}\|_{X}+a_{22}\|y-\overline{y}\|_{X},$

for all $x,\overline{x},y,\overline{y}\in X$ and $t\in\left[0,T\right];$

(ii)

There exists $\theta\geq 0$ such that the matrix

M(\theta):=C_{A}\begin{bmatrix}\frac{1}{a}+\left(1+\frac{1}{a}\right)Ta_{11}+% \frac{k}{a}Ta_{21}&\left(\left(1+\frac{1}{a}\right)a_{12}+\frac{k}{a}a_{22}% \right)\frac{e^{\theta T}-1}{\theta}\\[3.0pt] Ta_{21}&a_{22}\,\frac{1-e^{-\theta T}}{\theta}\end{bmatrix}

is convergent to zero.

Then, for every $\beta\in X$ , problem (3.1)-(3.2) has a unique solution $\left(x,y\right)$ in $C\left(\left[0,T\right];X\right)^{2}$ with $y\left(0\right)=\beta.$

Proof.

We apply Perov’s fixed point theorem to the operator $(N_{1},N_{2})$ with the following choice of spaces: $X_{1}=$ $C([0,T];X)$ endowed with the uniform norm, and $X_{2}=\left\{y\in C([0,T];X):\ y\left(0\right)=\beta\right\}$ endowed with the metric induced by the Bielecki norm $\left|\cdot\right|_{\theta}$ on $C([0,T];X)$ . Clearly, $N_{1}\left(X_{1}\times X_{2}\right)\subset X_{1},$ while the inclusion $N_{2}\left(X_{1}\times X_{2}\right)\subset X_{2}$ is obvious from the definition of $N_{2}.$ Next, let $x,\overline{x}\in X_{1}$ and $y,\overline{y}\in X_{2}$ . One has

	$\displaystyle\left\|N_{1}(x,y)\left(t\right)-N_{1}(\overline{x},\overline{y})% \left(t\right)\right\|_{X}$
	$\displaystyle\leq\frac{1}{a}C_{A}\left\|x-\overline{x}\right\|_{0}+\left(\left(1% +\frac{1}{a}\right)C_{A}Ta_{11}+\frac{k}{a}C_{A}Ta_{21}\right)\left\|x-% \overline{x}\right\|_{0}$
	$\displaystyle\quad+\left(\left(1+\frac{1}{a}\right)C_{A}a_{12}+\frac{k}{a}C_{A% }a_{22}\right)\int_{0}^{T}\left\|y\left(s\right)-\overline{y}\left(s\right)% \right\|_{X}ds$
	$\displaystyle\leq m_{11}\left\|x-\overline{x}\right\|_{0}+m_{12}\frac{e^{\theta T% }-1}{\theta}\|y-\overline{y}\|_{\theta},$

where

	$\displaystyle m_{11}$	$\displaystyle=C_{A}\left(\frac{1}{a}+\left(1+\frac{1}{a}\right)Ta_{11}+\frac{k% }{a}Ta_{21}\right),$
	$\displaystyle m_{12}$	$\displaystyle=C_{A}\left(\left(1+\frac{1}{a}\right)a_{12}+\frac{k}{a}a_{22}% \right).$

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

(4.6)

\left|N_{1}(x,y)-N_{1}(\overline{x},\overline{y})\right|_{0}\leq m_{11}|x-% \overline{x}|_{0}+m_{12}\frac{e^{\theta T}-1}{\theta}|y-\overline{y}|_{\theta}.

For the second operator $N_{2}$ , we estimate

	$\displaystyle\left\|N_{2}(x,y)\left(t\right)-N_{2}(\overline{x},\overline{y})% \left(t\right)\right\|_{X}$	$\displaystyle\leq\,\,C_{A}Ta_{21}\|x-\overline{x}\|_{0}$
		$\displaystyle+a_{22}\,C_{A}\int_{0}^{t}\|y(s)-\overline{y}(s)\|_{X}ds$
		$\displaystyle\leq m_{21}\|x-\overline{x}\|_{0}+m_{22}\frac{e^{\theta t}-1}{% \theta}\|y-\overline{y}\|_{\theta},$

where

m_{21}=\,C_{A}Ta_{21},\,\ \ m_{22}=C_{A}a_{22}.

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

(4.7)

\left|N_{2}(x,y)-N_{2}(\overline{x},\overline{y})\right|_{\theta}\leq m_{21}|x% -\overline{x}|_{0}+m_{22}\frac{1-e^{-\theta T}}{\theta}|y-\overline{y}|_{% \theta}.

Now, writing inequalities (4.6) and (4.7) in a vector form, gives

\begin{bmatrix}\left|N_{1}(x,y)-N_{1}(\overline{x},\overline{y})\right|_{0}% \vskip 3.0pt plus 1.0pt minus 1.0pt\\ \left|N_{2}(x,y)-N_{2}(\overline{x},\overline{y})\right|_{\theta}\end{bmatrix}% \leq M(\theta)\begin{bmatrix}|x-\overline{x}|_{0}\vskip 3.0pt plus 1.0pt minus% 1.0pt\\ |y-\overline{y}|_{\theta}\end{bmatrix},

where from assumption (ii), the matrix $M(\theta)$ is convergent to zero. Therefore, Perov’s fixed point theorem applies and guarantees the existence of a unique fixed point $\left(x,y\right)\in X_{1}\times X_{2}$ of the operator $(N_{1},N_{2})$ . ∎

4.3. Existence via Schauder’s fixed point theorem

The next result does not require Lipschitz conditions for $F$ and $G,$ but only linear growth ones. Although this sacrifices uniqueness, it offers the advantage of localizing the solution.

Theorem 4.3.

Assume the there are nonnegative constants $a_{ij},$ $i=1,2;$ $j=1,2,3$ such that

(4.8)		$\displaystyle\|F(x,y)\|_{X}\leq a_{11}\|x\|_{X}+a_{12}\|y\|_{X}+a_{13},$
	$\displaystyle\|G(x,y)\|_{X}\leq a_{21}\|x\|_{X}+a_{22}\|y\|_{X}+a_{23},$

for all $x,y\in X$ and $t\in\left[0,T\right]$ . In addition, assume that $\left\{S\left(t\right);t\geq 0\right\}$ is a compact semigroup and condition (ii) in Theorem 3.2 holds.

Then, for each $\beta\in X,$ problem (3.1)-(3.2) has at least one solution $\left(x,y\right)$ in $C\left(\left[0,T\right];X\right)^{2}$ with $y\left(0\right)=\beta.$

Proof.

We shall apply the Schauder’s fixed point theorem to the operator $(N_{1},N_{2})$ on a closed convex bounded subset of $X_{1}\times X_{2}$ of the form

(4.9)

D_{\beta;R_{1},R_{2}}:=\left\{(x,y)\in C([0,T];X)^{2}\,:\,y\left(0\right)=% \beta,\ |x|_{0}\leq R_{1},\,|y|_{\theta}\leq R_{2}\right\},

where the positive numbers $R_{1},$ $R_{2}$ will be determined in a such way that the operator $(N_{1},N_{2})$ is invariant over $D_{\beta;R_{1},R_{2}}$ , i.e.,

\left|N_{1}(x,y)\right|_{0}\leq R_{1},\ \left|N_{2}(x,y)\right|_{\theta}\leq R% _{2}\text{ \ whenever\ }|x|_{0}\leq R_{1},\ \,|y|_{\theta}\leq R_{2}.

Let $(x,y)\in D_{\beta;R_{1},R_{2}}.$ Then, for all $t\in[0,T]$ , we have

	$\displaystyle\left\|N_{1}(x,y)\left(t\right)\right\|_{X}$	$\displaystyle\leq\frac{1}{a}C_{A}\|x\|_{0}+\frac{1}{a}C_{A}\left(1+b\right)k% \left\|\beta\right\|_{X}$
		$\displaystyle+C_{A}\,\frac{k}{a}\int_{0}^{T}\left(a_{21}\|x(s)\|_{X}+a_{22}\|y(s)% \|_{X}+a_{23}\right)ds$
		$\displaystyle+C_{A}\left(1+\frac{1}{a}\right)\int_{0}^{T}\left(a_{11}\|x(s)\|_{X% }+a_{12}\|y(s)\|_{X}+a_{13}\right)ds$
		$\displaystyle\leq m_{11}\|x\|_{0}+m_{12}\int_{0}^{T}e^{\theta s}e^{-\theta s}\|y(% s)\|_{X}+C_{1}$
		$\displaystyle\leq m_{11}R_{1}+m_{12}R_{2}\frac{e^{\theta T}-1}{\theta}+C_{1},$

where

m_{11}=C_{A}\left(\frac{1}{a}+\left(1+\frac{1}{a}\right)Ta_{11}+\frac{k}{a}Ta_% {21}\right),\ \ m_{12}=C_{A}\left(\left(1+\frac{1}{a}\right)a_{12}+\frac{k}{a}% a_{22}\right),\text{ }

and

C_{1}=\frac{1}{a}C_{A}\left(1+b\right)k\left|\beta\right|_{X}+C_{A}\frac{k}{a}% Ta_{23}+C_{A}\left(1+\frac{1}{a}\right)a_{13}.

Hence,

(4.10)

\left|N_{1}(x,y)\right|_{0}\leq m_{11}R_{1}+m_{12}R_{2}\frac{e^{\theta T}-1}{% \theta}+C_{1}.

For the second operator $N_{2}$ , we compute

	$\displaystyle\left\|N_{2}(x,y)\left(t\right)\right\|_{X}$	$\displaystyle\leq C_{A}\left\|\beta\right\|_{X}+C_{A}\int_{0}^{t}\left(a_{21}\|x(% s)\|_{X}+a_{22}\|y(s)\|_{X}+a_{23}\right)ds$
		$\displaystyle\leq\,C_{A}\,Ta_{21}\|x\|_{0}+a_{22}C_{A}\int_{0}^{t}e^{\theta s}e^% {-\theta s}\|y(s)\|_{X}ds+C_{2}$
		$\displaystyle\leq m_{21}\|x\|_{0}+m_{22}\|y\|_{\theta}\frac{e^{\theta t}-1}{\theta% }+C_{2},$

where $m_{21}=\,C_{A}\,Ta_{21}$ , $m_{22}=C_{A}a_{22}$ and $C_{2}=C_{A}\left|\beta\right|_{X}+C_{A}Ta_{23}.$ Dividing by $e^{\theta t}$ and taking the supremum over $t\in[0,T]$ , we obtain

(4.11)

\left|N_{2}(x,y)\right|_{\theta}\leq m_{21}R_{1}+m_{22}\frac{1-e^{-\theta T}}{% \theta}R_{2}+C_{2}.

In the matrix form, relations (4.10) and (4.11) read as

\begin{bmatrix}\left|N_{1}(x,y)\right|_{0}\\ \left|N_{2}(x,y)\right|_{\theta}\end{bmatrix}\leq M(\theta)\begin{bmatrix}R_{1% }\\ R_{2}\end{bmatrix}+\begin{bmatrix}C_{1}\\ C_{2}\end{bmatrix},

where $M(\theta)$ is given in (3.8). For the invariance condition it suffices to have

M(\theta)\begin{bmatrix}R_{1}\\ R_{2}\end{bmatrix}+\begin{bmatrix}C_{1}\\ C_{2}\end{bmatrix}\leq\begin{bmatrix}R_{1}\\ R_{2}\end{bmatrix},

or equivalently

\begin{bmatrix}C_{1}\\ C_{2}\end{bmatrix}\leq\left(I-M\left(\theta\right)\right)\begin{bmatrix}R_{1}% \\ R_{2}\end{bmatrix}.

From assumption (ii), since the matrix $M(\theta)$ is convergent to zero, $(I-M(\theta))^{-1}$ has nonnegative entries. Therefore, the last inequality is equivalent to

(4.12)

(I-M(\theta))^{-1}\begin{bmatrix}C_{1}\\ C_{2}\end{bmatrix}\leq\begin{bmatrix}R_{1}\\ R_{2}\end{bmatrix},

which clearly allows us to choose positive numbers $R_{1},R_{2}$ such that the set $D_{\beta;R_{1},R_{2}}$ is invariant under the operator $(N_{1},N_{2}).$

Note that, as the semigroup generated by $A$ is assumed to be compact, it follows from standard arguments that $N_{1}$ and $N_{2}$ are completely continuous operators (see, e.g., [13]). Consequently, Schauder’s fixed point theorem applies and guarantees the existence of at least one fixed point of $(N_{1},N_{2})$ in $D_{\beta;R_{1},R_{2}}$ . ∎

Remark 4.4.

From the above result we have not only the existence of a solution, but also its localization, namely

(4.13)

|x(t)|_{X}\leq R_{1}\ \ \ \text{ and\ \ \ }|y(t)|_{X}\leq e^{\theta t}R_{2}\ % \ \text{for all }t\in[0,T],

where $\theta$ is given in assumption (ii), and $R_{1},R_{2}$ satisfy (4.12). Note that, due to the use of the Bielecki norm on the second component, we only have an exponential localization for $y$ .

4.4. Existence via Avramescu’s fixed point theorem

We apply Avramescu’s fixed point theorem to obtain a fixed point for the operator $(N_{1},N_{2})$ . It is noteworthy that, given the fixed point obtained in Theorem 4.5 below, the uniqueness of the second component is guaranteed by Banach’s principle, while both components $x$ and $y$ satisfy (4.13) with $\theta$ given in assumption (ii) of Theorem 4.2, and $R_{1},R_{2}$ satisfy (4.12).

Theorem 4.5.

Assume that there are nonnegative constants $a_{11},a_{12},a_{13},a_{21},a_{22}$ such that

(4.14)		$\displaystyle\|F(x,y)\|_{X}\leq a_{11}\|x\|_{X}+a_{12}\|y\|_{X}+a_{13},$
	$\displaystyle\|G(x,y)-G(\overline{x},\overline{y})\|_{X}\leq a_{21}\|x-\overline{% x}\|_{X}+a_{22}\|y-\overline{y}\|_{X},$

for all $x,\overline{x},y,\overline{y}\in X$ and $t\in\left[0,T\right].$ In addition assume that $\left\{S\left(t\right);t\geq 0\right\}$ is a compact semigroup and condition (ii) in Theorem 4.2 holds.

Then, for each $\beta\in X,$ problem (3.1)-(3.2) has at least one solution $\left(x,y\right)$ in $C\left(\left[0,T\right];X\right)^{2}$ with $y\left(0\right)=\beta.$

Proof.

Let us consider the sets,

	$\displaystyle D_{1}:=\left\{x\in C([0,T];X)\,:\,\|x\|_{0}\leq R_{1}\right\},$
	$\displaystyle D_{2}:=\left\{y\in C([0,T];X)\,:\,y\left(0\right)=\beta,\ \|y\|_{% \theta}\leq R_{1}\right\},$

where $R_{1},R_{2}$ are positive numbers which are chosen below. We observe that, from (4.14), function $G$ satisfies the growth condition

|G(x,y)|_{X}\leq a_{21}|x|_{X}+a_{22}|y|_{X}+a_{23},

where $a_{23}=\left|G\left(0,0\right)\right|_{X}.$ Consequently, performing the same estimates as in the proof of Theorem 4.3 and using (ii), we conclude that

\left|N_{1}(x,y)\right|_{0}\leq R_{1},\ \ \ \left|N_{2}(x,y)\right|_{\theta}% \leq R_{2},

for all $(x,y)\in D_{1}\times D_{2}$ , where $R_{1},R_{2}$ are obtained as in the proof of the previous theorem. This guarantees that $N_{i}\left(D_{1}\times D_{2}\right)\subset D_{i}$ for $i=1,2.$

Since the matrix $M(\theta)$ converges to zero, the diagonal elements are strictly less than $1$ , hence

\ a_{22}C_{A}\frac{1-e^{-\theta T}}{\theta}<1.

This guarantees that $N_{2}(x,\cdot)$ is a contraction for all $x\in C([0,T];X)$ . Indeed, following a similar reasoning as in the proof of Theorem 4.2, we deduce

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

for all $x\in D_{1}$ and $y,\overline{y}\in D_{2}$ . In addition, as above, $N_{1}\left(D_{1}\times D_{2}\right)$ is relatively compact in $C\left(\left[0,T\right];X\right).$ Therefore, Avramescu’s theorem applies and guarantees the existence of a pair $(x,y)\in D_{1}\times D_{2}$ such that $N_{1}(x,y)=x$ and $N_{2}(x,y)=y$ . ∎

5. Application

The results established in Section 4 can be easily applied to the diffusion system,

(5.1)

\begin{cases}u_{t}=\Delta u+f(u,v)\\ v_{t}=\Delta v+g(u,v)\,\text{ in }\Omega,\\ u=v=0\,\text{ on }\partial\Omega\end{cases}(t\in[0,T]),

with the controllability condition

(5.2)

u(T)-au\left(0\right)=k\left(v(T)-bv\left(0\right)\right),

where $a,b,k>0.$

The functions $f,g\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ are assumed to be continuous, and there exist nonnegative constants $a_{11},a_{12},a_{21},a_{22},c_{f},c_{g}$ such that one of the following three conditions is satisfied:

	$\displaystyle(\text{c}_{1}):\begin{cases}\left\|f(p,q)-f(\overline{p},\overline% {q})\right\|\leq a_{11}\|p-\overline{p}\|+a_{12}\|q-\overline{q}\|\\ \left\|g(p,q)-g(\overline{p},\overline{q})\right\|\leq a_{21}\|p-\overline{p}\|+a_% {22}\|q-\overline{q}\|\end{cases}$
	$\displaystyle(\text{c}_{2}):\begin{cases}\left\|f(p,q)\right\|\leq a_{11}\|p\|+a_{% 12}\|q\|+c_{f}\\ \left\|g(p,q)\right\|\leq a_{21}\|p\|+a_{22}\|q\|+c_{g}\end{cases}$
	$\displaystyle(\text{c}_{3}):\begin{cases}\left\|f(p,q)\right\|\leq a_{11}\|p\|+a_{% 12}\|q\|+c_{f}\\ \left\|g(p,q)-g(\overline{p},\overline{q})\right\|\leq a_{21}\|p-\overline{p}\|+a_% {22}\|q-\overline{q}\|\end{cases}$

for all $p,q,\overline{p},\overline{q}\in\mathbb{R}$ .

Here, $\Omega\subset\mathbb{R}^{n}$ is a bounded open set, $X$ is the Banach space $L^{2}(\Omega)$ endowed with the usual $|\cdot|_{L^{2}}$ norm, $A=\Delta$ , with

D(A)=\left\{u\in H_{0}^{1}(\Omega)\,:\,\Delta u\in L^{2}(\Omega)\right\}.

Note that $A$ is the infinitesimal generator of a compact semigroup of contractions in $L^{2}(\Omega)$ (cf, [20, Theorem 7.2.5]). Hence, the constant $C_{A}$ given in (2.1) is $1.$ Also $F$ and $G$ are the superposition operators $\ F,G:L^{2}\left(\Omega\right)^{2}\rightarrow L^{2}\left(\Omega\right),$

F\left(u,v\right)\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right),% \ \ \ G\left(u,v\right)\left(x\right)=g\left(u\left(x\right),v\left(x\right)% \right)\ \ \ \left(x\in\Omega\right).

Simple computations show that if either $($ c ${}_{1})$ , $($ c ${}_{2})$ , or $($ c ${}_{3})$ holds, then assumption (i) of Theorem 4.2, Theorem 4.3, or Theorem 4.5 is satisfied with $a_{13}=c_{f}m(\Omega)^{1/2},\ a_{23}=c_{g}m(\Omega)^{1/2}$ and the constants $a_{11},a_{12},a_{21},a_{22}$ .

Additionally, if we assume there exists $\theta\geq 0$ such that the matrix

M(\theta):=\begin{bmatrix}\frac{1}{a}+\left(1+\frac{1}{a}\right)Ta_{11}+\frac{% k}{a}Ta_{21}&\left(\left(1+\frac{1}{a}\right)a_{12}+\frac{k}{a}a_{22}\right)% \frac{e^{\theta T}-1}{\theta}\\[3.0pt] Ta_{21}&a_{22}\,\frac{1-e^{-\theta T}}{\theta}\end{bmatrix}

converges to zero, then condition (ii) of the Theorem 4.2 is verified. Consequently, depending on the conditions imposed ((c₁), (c₂), or (c₃)), there exists a weakly solution for the mutual control problem (5.1)-(5.2), which is either unique; localized; or unique in one component and localized in both.

References

[1] C. Avramescu, Asupra unei teoreme de punct fix (in Romanian), St. Cerc. Mat. 22 (2) (1970), 215–221.
[2] M. Beldinski and M. Galewski, Nash type equilibria for systems of non-potential equations, Appl. Math. Comput. 385 (2020), pp. 125456.
[3] M. Coron, Control and Nonlinearity, AMS, Providence, 2007.
[4] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
[5] A. Granas, Fixed Point Theory, Springer, New York, 2003.
[6] M.A. Krasnoselskii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl. 10 (1958), 345–409.
[7] P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer, 2018.
[8] S. Park, Generalizations of the Nash equilibrium theorem in the KKM theory, Fixed Point Theor. Appl. 2010 (2010).
[9] R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Model. 49 (2009), 703–708.
[10] R. Precup, Nash-type equilibria and periodic solutions to nonvariational systems, Adv. Nonlinear Anal. 3 (2014), no. 4, 197–207.
[11] R. Precup, A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems, J. Math. Anal. Appl. 463 (2018), 412–431.
[12] R. Precup, On some applications of the controllability principle for fixed point equations, Results Appl. Math. 13 (2022) 100236, 1–7.
[13] R. Precup, Methods in Nonlinear Integral Equations, Dordrecht, Springer, 2002.
[14] R. Precup and A. Stan, Stationary Kirchhoff equations and systems with reaction terms, AIMS Mathematics, 7 (2022), Issue 8, 15258–15281.
[15] R. Precup and A. Stan, Linking methods for componentwise variational systems, Results Math. 78 (2023), 1-25.
[16] R. Precup and A. Stan, A mutual control problem for semilinear systems via fixed point approach, submitted.
[17] A. Stan, Nonlinear systems with a partial Nash type equilibrium, Studia Univ. Babeş-Bolyai Math. 66 (2021), 397–408.
[18] A. Stan, Nash equilibria for componentwise variational systems, J. Nonlinear Funct. Anal. 6 (2023), 1-10.
[19] A. Stan, Localization of Nash-type equilibria for systems with partial variational structure, J. Numer. Anal. Approx. Theory, 52 (2023) , 253–272.
[20] I.I. Vrabie, $C_{0}$ -Semigroups and Applications , Elsevier, Amsterdam, 2003.
[21] J. Zabczyk, Mathematical Control Theory, Springer, Cham, 2020.

	$\displaystyle\left\|\gamma_{1}(\alpha,\beta)-\gamma_{1}(\overline{\alpha},% \overline{\beta})\right\|_{X}$	$\displaystyle\leq C\left(\left\|\alpha-\overline{\alpha}\right\|_{X}+\left\|\beta% -\overline{\beta}\right\|_{X}\right),$
	$\displaystyle\left\|\gamma_{2}(\alpha,\beta)-\gamma_{2}(\overline{\alpha},% \overline{\beta})\right\|_{X}$	$\displaystyle\leq C\left(\left\|\alpha-\overline{\alpha}\right\|_{X}+\left\|\beta% -\overline{\beta}\right\|_{X}\right).$

	$\displaystyle\left\|x_{\alpha,\beta}-x_{\overline{\alpha},\overline{\beta}}% \right\|_{0}$	$\displaystyle\leq C\left(\left\|\alpha-\overline{\alpha}\right\|_{X}+\left\|\beta% -\overline{\beta}\right\|_{X}\right)$
		$\displaystyle\quad+m_{11}\left\|x_{\alpha,\beta}-x_{\overline{\alpha},\overline% {\beta}}\right\|_{0}+m_{12}\left\|y_{\alpha,\beta}-y_{\overline{\alpha},% \overline{\beta}}\right\|_{0},$
	$\displaystyle\left\|y_{\alpha,\beta}-y_{\overline{\alpha},\overline{\beta}}% \right\|_{0}$	$\displaystyle\leq C\left(\left\|\alpha-\overline{\alpha}\right\|_{X}+\left\|\beta% -\overline{\beta}\right\|_{X}\right)$
		$\displaystyle\quad+m_{21}\left\|x_{\alpha,\beta}-x_{\overline{\alpha},\overline% {\beta}}\right\|_{0}+m_{22}\left\|y_{\alpha,\beta}-y_{\overline{\alpha},% \overline{\beta}}\right\|_{0}.$

	$\displaystyle\left\|N_{1}(x,y)\left(t\right)-N_{1}(\overline{x},\overline{y})% \left(t\right)\right\|_{X}$
	$\displaystyle\leq\frac{1}{a}C_{A}\left\|x-\overline{x}\right\|_{0}+\left(\left(1% +\frac{1}{a}\right)C_{A}Ta_{11}+\frac{k}{a}C_{A}Ta_{21}\right)\left\|x-% \overline{x}\right\|_{0}$
	$\displaystyle\quad+\left(\left(1+\frac{1}{a}\right)C_{A}a_{12}+\frac{k}{a}C_{A% }a_{22}\right)\int_{0}^{T}\left\|y\left(s\right)-\overline{y}\left(s\right)% \right\|_{X}ds$
	$\displaystyle\leq m_{11}\left\|x-\overline{x}\right\|_{0}+m_{12}\frac{e^{\theta T% }-1}{\theta}\|y-\overline{y}\|_{\theta},$

	$\displaystyle\left\|N_{2}(x,y)\left(t\right)-N_{2}(\overline{x},\overline{y})% \left(t\right)\right\|_{X}$	$\displaystyle\leq\,\,C_{A}Ta_{21}\|x-\overline{x}\|_{0}$
		$\displaystyle+a_{22}\,C_{A}\int_{0}^{t}\|y(s)-\overline{y}(s)\|_{X}ds$
		$\displaystyle\leq m_{21}\|x-\overline{x}\|_{0}+m_{22}\frac{e^{\theta t}-1}{% \theta}\|y-\overline{y}\|_{\theta},$

	$\displaystyle\left\|N_{1}(x,y)\left(t\right)\right\|_{X}$	$\displaystyle\leq\frac{1}{a}C_{A}\|x\|_{0}+\frac{1}{a}C_{A}\left(1+b\right)k% \left\|\beta\right\|_{X}$
		$\displaystyle+C_{A}\,\frac{k}{a}\int_{0}^{T}\left(a_{21}\|x(s)\|_{X}+a_{22}\|y(s)% \|_{X}+a_{23}\right)ds$
		$\displaystyle+C_{A}\left(1+\frac{1}{a}\right)\int_{0}^{T}\left(a_{11}\|x(s)\|_{X% }+a_{12}\|y(s)\|_{X}+a_{13}\right)ds$
		$\displaystyle\leq m_{11}\|x\|_{0}+m_{12}\int_{0}^{T}e^{\theta s}e^{-\theta s}\|y(% s)\|_{X}+C_{1}$
		$\displaystyle\leq m_{11}R_{1}+m_{12}R_{2}\frac{e^{\theta T}-1}{\theta}+C_{1},$

On equilibrium in control problems with applications to evolution systems

Abstract

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On equilibrium in control problems with applications to evolution systems

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

2. Preliminaries

2.1. Abstract evolution equations

2.2. Bielecki-type norms

2.3. Matrices convergent to zero

Lemma 2.1.

Lemma 2.2.

Lemma 2.3.

2.4. Fixed point theorems

Theorem 2.4 (Perov).

Theorem 2.5 (Avramescu).

3. Mutual control for abstract evolution equations

3.1. Fixed point formulation.

Lemma 3.1.

Proof.

3.2. Existence

Theorem 3.2.

Proof.

Remark 3.3.

4. A semi-observability problem

4.1. Fixed point formulation of the problem

Lemma 4.1.

Proof.

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

Theorem 4.2.

Proof.

4.3. Existence via Schauder’s fixed point theorem

Theorem 4.3.

Proof.

Remark 4.4.

4.4. Existence via Avramescu’s fixed point theorem

Theorem 4.5.

Proof.

5. Application

References

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