In the last few decades, great attention has been paid to automatic control systems and their applications to computational mathematics and modeling. Many problems in control theory correspond to the maximal deviation of the regulated quantity. A classical example is that of an electric generator. In this case, the mechanism becomes active when the maximum voltage variation that is permitted is reached in an interval of time $I_{t}=[t-h,t]$ with $h$ a positive constant. The equation which describes the action of this regulator has the form

where $\delta$ and $p$ are constants that are determined by characteristic of system, $V(t)$ is the voltage and $F(t)$ is the effect of the perturbation that appears associated to the change of voltage [2]. Recently, there is an increasing interest towards equations which contain maxima. We mention the work in [1], [2], [5], [7]-[9], [14].

For some results concerning the asymptotic behavior and asymptotic equivalence of systems of differential equations, we refer in particular to Diamandescu [4], Gonzales and Pinto [6], Olaru [12], Piccinini et.al. [13], Talpalaru [15] and the references cited therein.

Consider the following differential system

and the perturbed differential system with maxima

In [9], the author proved existence and uniqueness, inequalities of Čaplygin type and data dependence (monotony, continuity) results for the solution of the Cauchy problem associated to the system (1.2), using weakly Picard operator technique. In this paper the hypotheses which will be imposed upon our equations are extensions of those suggested in [13]. Our results extend and improve corresponding theorems in the existing literature (see, e.g. [7]-[12]). Also, in the end we present an example to illustrate our results.

Let $\mathbb{R}^{n}$ be the Euclidian $n$-space. For $u=(u_{1},\ldots,u_{n})^{T}\in\mathbb{R}^{n}$, let $\left\|u\right\|:=\max\{\left|u_{1}\right|,\ldots,\left|u_{n}\right|\}$ be the norm of $u$. For a matrix $A\in M_{n\times n}(\mathbb{R}),\ A=(a_{ij}),$ we define the norm $\left|A\right|$ of $A$ by $\left|A\right|:=\underset{\left\|u\right\|\leq 1}{\sup}\left\|Au\right\|$. So,

In the above equations we consider that $y_{0}\in\mathbb{R}^{n}$, $f\in C([a,\infty[\times\mathbb{R}^{n}\times\mathbb{R}^{n},\mathbb{R}^{n})$ and let $BC([a,\infty[,\mathbb{R}^{n}):=\{y\in C([a,\infty[,\mathbb{R}^{n})|\ y$ bounded with $\left\|y\right\|:=\underset{t\in[a,\infty]}{\max}\left\{\left|y_{1}\right|,\ldots,\left|y_{n}\right|\right\}$.

Let $X_{1}$ denote the subspace of $\mathbb{R}^{n}$ consisting of all vectors which are values of bounded solutions of (1.1) for $t=0$ and let $X_{2\text{ }}$ an arbitrary closed fixed subspace of $\mathbb{R}^{n}$, supplementary to $X_{1}.$ We denote by $P_{1},$ the projection of $\mathbb{R}^{n}$ on $X_{1}$, (that is $P_{1}$ is a bounded linear operator, $P_{1}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n},\ P_{1}^{2}=P_{1},\$Ker$P_{1}=X_{2}$) and $P_{2}=I-P_{1}$ the projection onto $X_{2}$.

A solution of the equation (1.1) is a function $x\in C^{1}([a,\infty[,\mathbb{R}^{n})$ which satisfy the equation. Now, we give the definition of asymptotic equivalence that we need in the sequel.

We consider the following differential equation with maxima

with the condition

We suppose that the following condition holds:

• (C)

  there exists $L_{f}:[a,\infty[\rightarrow\mathbb{R}^{n}$ with $\int_{a}^{\infty}L_{f}(s)ds<\infty$ such that

  Report issue for preceding element

  $$\left\|f(t,u_{1},v_{1})-f(t,u_{2},v_{2})\right\|\leq L_{f}(t)\max\{\left|u_{1}-u_{2}\right|,\left|v_{1}-v_{2}\right|\},$$

  $\forall t\in[a,\infty[$ and $u_{i},v_{i}\in\mathbb{R}^{n},i=1,2$.

  Report issue for preceding element

Throughout this paper we note by $X(t)$ the fundamental matrix of the system (1.1). We remark that if $y\in C^{1}([a,\infty[,\mathbb{R}^{n})$ is a solution of the problem (2.2)–(2.3), then $y$ is a solution of

and if $y\in C([a,\infty[,\mathbb{R}^{n})$ is a solution of (2.4), then $y\in C^{1}([a,\infty[,\mathbb{R}^{n})$ and is a solution of (2.2)–(2.3).

Also, if $y\in C^{1}([a,\infty[,\mathbb{R}^{n})$ is a solution of the problem (2.2), then $y$ is a solution of

and if $y\in C([a,\infty[,\mathbb{R}^{n})$ is a solution of (2.5) then $y\in C^{1}([a,\infty[,\mathbb{R}^{n})$ and is a solution of (2.2).

Relative to problem (2.2)–(2.3) we have the following existence and uniqueness theorem (see [9], Theorem 3.1).

In this section we show that, under some conditions, for a given solution $x$ of (1.1) there exists a solution $y$ of (1.2) such that (2.1) holds. We consider the case when $P_{1}+P_{2}=1_{\mathbb{R}^{n}}$.

If $y\in BC([a,\infty[,\mathbb{R}^{n})$, then [7]

Let $x$ be a bounded solution of (1.1) and $y\in BC([a,\infty[,\mathbb{R}^{n})$. Then

So

Now we prove that $T$ is a contraction on $BC([a,\infty[,\mathbb{R}^{n})$.

So,

From Banach’s fixed point theorem, there exists a unique solution of equation (1.2). Let $y$ be a solution of (1.2) correspondent to $x$. Then

We fix $\varepsilon>0$ and determine $t_{1}\geq a$ such that the second integral in the right-hand side of the above inequality is less than $\varepsilon/2$. With $t_{1}$ fixed, from hypothesis (iii), we have that the first term on the right-hand side of the above inequality tends to zero as $x$ tends to infinity. We may therefore conclude that $\underset{t\rightarrow\infty}{\lim}\left\|x(t)-y(t)\right\|=0$.

Similarly, we take a bounded solution $y$ of (1.2) and we observe that

satisfies equation (1.1) and (2.1) holds.   

For the case when $P_{1}=P_{2}=1_{\mathbb{R}^{n}}$, we have the following result.

We fix $\varepsilon>0$ and determine $t_{1}\geq a$ such that the second integral in the right-hand side of the above inequality is less than $\varepsilon/2$. With $t_{1}$ fixed, from hypothesis (iii), we have that the first term on the right-hand side of the above inequality tends to zero as $x$ tends to infinity. We may therefore conclude that $\underset{t\rightarrow\infty}{\lim}\left\|x(t)-y(t)\right\|=0$, and the theorem is proved.

Similarly, we take a bounded solution $y$ of (1.2) and we observe that

satisfies equation (1.1) and (2.1) holds.   

In what follows we consider a differential system with maxima

where the unknown function $I(t)$ is the electric current, $L\neq 0,\ M\in M_{2\times 2}(\mathbb{R}),N$ and $k$ are constants. It is derived treating the original automatic regulation phenomenon ([3]) without linearization. Then we can formulate an initial-value problem for the above system as follows:

where $M(t)$ is the fundamental matrix of the nonperturbed system

The function$\ f(t,u,v)=L^{-1}\left(Nv+k\frac{u^{3}}{1+u^{2}}\right)$ satisfies condition $(C)$, i.e.,

with $c$=const. The condition $(iii)$ of Theorems 3.1 and 3.2 is satisfied with $P_{1}=\left(\begin{array}[c]{cc}1&0\\ 0&0\end{array}\right)$ and $P_{2}=\left(\begin{array}[c]{cc}1&0\\ 0&0\end{array}\right)$. By Theorems 3.1 and 3.2, for every bounded solution $J$ of equation (4.2), there exists a unique bounded solution $I$ of equation (4.1) such that $\underset{t\rightarrow\infty}{\lim}\left\|J(t)-I(t)\right\|=0$ holds and conversely, for every bounded solution $I$ of equation (4.1), there exists a unique bounded solution $J$ of equation (4.2) such that $\underset{t\rightarrow\infty}{\lim}\left\|J(t)-I(t)\right\|=0$ holds.