Controllability of nonlinear ordinary differential equations with non-instantaneous impulses

1.   Introduction

The dynamics of many evolving processes are subject to abrupt changes, such as shocks, harvesting and natural disaster. These phenomena involve short term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. Sometimes time abrupt changes may stay for time intervals such impulses are called non-instantaneous impulses. The theory of instantaneous impulsive differential equations is an important branch of differential equation theory, which has extensive physical, chemical, biological, engineering background and realistic mathematical model, and hence has been emerging as an important area of investigation in the last decades. Hernández and O'Regan ^[20] first studied the initial value problem for a new class of abstract evolution equations with non-instantaneous impulses in Banach spaces.

Control theory is an area of application-oriented mathematics which deals with basic principles underlying the analysis and design of control systems. It is well known that controllability plays a significant role in modern control theory and engineering since they are closely related to pole assignment, structural decomposition and quadratic optimal control. See functional analysis ^[1].

The notion of controllability means that it is possible to steer a dynamic system from an arbitrary initial state to an arbitrary final state using a set of admissible controls. There exist many criteria and definitions of controllability. They depend both on the constraints on the control signal and the state equation. It should be noticed that in infinite-dimensional spaces there exist linear subspaces which are not closed. We can distinguish two concepts of controllabilityin the case of infinite-dimensionalsystems. Exact controllability means that the system can be steered to an arbitrary final state. Approximate controllability enables us to steer the system to an arbitrary small neighborhood of the final state. It is self-evident that approximate controllability is essentially a weaker notion than exact controllability. As a result, the latter always implies the former, but the converse statement is not true in general. See the works of Bashirov and Kerimov (1997)^[2], Bashirov and Mahmudov (1999)^[3], Benchohra and Ouahab (2005)^[4]. Controllability of semilinear integrodifferential systems in Banach spaces was investigatedby Lasiecka and Triggiani (1991)^[5].

For nonlinear systems, controllability results can be shown via contraction mapping principle, Schauder's fixed point theorem and Schaefer's fixed point theorem by constructing a suitable control function (see ^[6,7,8,9]). In particular, it is summarized that the sufficient conditions for approximate controllability of various types of dynamic systems using Schauder's fixed-point theorem in Hilbert spaces by Babiarz ^[10]. In 2013, Liu et al. ^[11] used the Krasnoselskii's fixed point theorem to studied controllabily of nonlinear fractional impulsive evolution systems. Of course, using Nussbaum's fixed point theorem can also be used to investigate the controllability of nonlinear systems (see ^{[11,12,13,17,18,19]}). In this paper, on the basis of previous research^[14], ^[15], the controllability of the initial value problem is studied, and the fixed point theorem is applied to nonlinear ordinary differential equations with non-instantaneous impulses. Our work can be considered as a contribution to this nascent fields. The results obtained in this paper are a supplement to the existing literature and essentially extend some existing results in this area.

Next, we study the following controllability of nonlinear ordinary differential equation with non-instantaneous impulses by fixed point theorem in a Banach space E

where $0 < t_{1} < t_{2} < \cdots < t_{m} < t_{m+1}: = a, \; a > 0$ is a constant, $J = [0, a]$, $J^{\prime} = J\backslash \{t_{1}, t_{2}, \cdots t_{m}\}$ and $s_{0} = t_{0}: = 0, \; s_{i}\in(t_{i}, t_{i+1})$ for each $i = 1, 2, \cdots, m.$ $f: [0, a]\times E\rightarrow E$ is a continuous nonlinear function, $\; g_{i}:(t_{i}, s_{i}]\times E\rightarrow E, \; i = 1, 2, \cdots, m$ is non-instantaneous impulsive function for all $i = 1, 2, \cdots, m,$ and $u_{0}\in E$. The control function $h(\cdot)$ is given in $L^{2}(J^{\prime}, U)$ with $U$ as a Banach space and $B$ is bounded linear operator from $U$ into $E$.

2.   Preliminaries

Let $E$ be a Banach space with the norm $\|\cdot\|$, whose positive cone $P = \{x \in E \mid x \geq \theta\}$ is normal with normal constant $N$, where $\theta$ is the zero element in E. We denote by $C(J, E)$ the Banach space of all continuous functions from $J$ into $E$ endowed with the sup-norm

Then $C(J, E)$ is an ordered Banach space induced by the convex cone

and $P_{ C}$ is also a normal cone with normal constant $N$. Let

be a piecewise continuous function space. It is easy to see that $PC(J, E)$ is a Banach space endowed with the $PC$-norm

For any finite number $r > 0$, let

be a bounded convex closed set.

Let $L^{p}(J, R)(1\leq p\leq \infty)$ denote the Banach space of all Lebesgue measurable functions from J into E with $\|\varphi\|_{L^{p}(J, R)}: = (\int_{_{J}}|\varphi(t)|^{p}dt)^{\frac{1}{p}} < \infty$. And let $L^{p}(J, E)$ be the Banach space of functions $\varphi:J\rightarrow E$ which are Bochner integrable normed by $\|\varphi\|_{L^{p}(J, E)}$.

Definition 2.1. A function $u\in PC(J, E)$ is a solution of the system (1.1), it is equivalent to $u$ satisfies that

Lemma 2.2^[10] (Schauder's theorem)(Kulmin, 2004). Every continuous operator that maps a closed convex subset of a Banach space into a compact subset of itself has at least one fixed point.

Lemma 2.3^[11] (Krasnoselskii's fixed point theorem). Let $X$ be a Banach space, $\Omega$ a bounded closed and convex subset of $X$ and $F_{1}, \; F_{2}$ maps $\Omega$ into $X$ such that $F_{1}x+F_{2}y \in\Omega$ for every pair $x, y \in\Omega$. If $F_{1}$ is a contraction and $F_{2}$ is completely continuous, then the equation $F_{1}x+F_{2}x = x$ has a solution on $\Omega$.

Definition 2.4. The system (1.1) is said to be completely controllable on $[0, t_{f}](t_{f}\in (0, T])$ if for every $u_{0}, u_{t_{f}} \in E$, there exist a control $h \in L^{2}(J^{\prime}, U)$ such that the solution $u(t)$ of the system (1.1) satisfies $u({t_{f}}) = u_{t_{f}}$.

3.   Main results

we need the following additional assumption conditions:

(H1) There are arbitray constant $\; p_{0} > 0, \; p_{1}\geq0, \;$and $t\in{J}, u\in{E}$ such that

(H2) There exist positive constant $L_{i}\; (i = 1, 2, \cdots, m), \; \forall u, v \in E$ such that

(H3) The linear operator $B: L^{2}(J^{\prime}, U)\rightarrow L(J^{\prime}, U)$ is bounded, $W: L^{2}(J^{\prime}, U)\rightarrow E$ defined by

has an inverse operator $W^{-1}$ which takes value in $L^{2}(J^{\prime}, U)\setminus\ker W$ and there exist two positive constants $D_{1}, D_{2} > 0$ such that

Theorem 3.1. Assume that [H1]-[H3] and

hold, then system (1.1) has a solution $u\in PC(J, E)$ on $(0, T]$.

Proof. Define an operator $F:PC(J, E)\rightarrow PC(J, E)$ such that

and we define the control function $h_{u}(t)$ by

It is easy to see that the solution of the system (1.1) is equivalent to the fixed point of operator $\digamma$ defined by (3.1).

Now, we will prove that there exists a constant $R > 0$, $\Omega_{R} = \{u \in PC(J, E)\; |\; \|u(t)\|\leq R, t\in J\}$ such that $\digamma(\Omega_{R})\subset \Omega_{R}$. If this is not ture, then for each $r > 0$, there would exist $u_{r}\in \Omega_{r}$, $t_{r}\in J$ such that

If $\forall t_{r}\in[0, t_{1}]$ then by (3.1), (H1) and (H3), we know that

If $\forall t_{r}\in[t_{1}, s_{i}], \; i = 1, 2, \cdots, m,$ then by (3.1) and (H2), we know that

where $N = \max\limits_{i = 1, 2, \cdots, m}\sup\limits_{t\in J}\|g_{i}(t, \theta)\|$. If $\forall t_{r}\in(s_{i}, t_{i+1}], \; i = 1, 2, \cdots, m,$ then by (3.1) and (H1)-(H3), we know that

Combining (3.1)-(3.5) with the fact $\|(\digamma u_{_{r}})(t_{r})\| > r$, we obtain

Dividing both side of (3.6) by r and taking the lower limit as $r\rightarrow \infty$, we have

Next, we prove that $F$ is continuous in $\Omega_{R}$.

Let $u_{n}\in\Omega_{R}$ be a sequence, such that $\lim\limits_{n\rightarrow \infty}u_{n} = u$ in $\Omega_{R}$. By the continuity of nonlinear term f with respect to the second variable, for each $s\in J$ we have

If $\forall s\in(t_{i}, t_{i+1}], \; i = 1, 2, \cdots, m,$ by (3.4), (3.8) and Lebesgue dominated convergence theorem, we obtain

Then we concluded that $\|\digamma u_{n}-\digamma u\|_{pc}\rightarrow 0, \; (n\rightarrow \infty)$. Which meas that $\digamma$ is continuous in $\Omega_{R}$.

Now, we demonstrate that the operator $F:\Omega_{R}\rightarrow \Omega_{R}$ is equicontinuous. For any $u\in\Omega_{R}$ and $s_{i}\leq t^{\prime} < t^{\prime\prime}\leq t_{i+1}$, $i = 0, 1, 2, \cdots, m$, we can conclude that

Therefore, when $t^{\prime\prime}-t^{\prime}\rightarrow 0$, then $\|(\digamma u)(t^{\prime\prime})-(\digamma u)(t^{\prime})\|\rightarrow 0$. As a result, the operator $F :\Omega_{R}\rightarrow \Omega_{R}$ is equicontinuous. It follows the Arzela-Ascoli theorem that $F\Omega_{R}$ is sequentially compact in $PC(J, E)$. By Schauder's fixed point theorem (Lemma 2.2) $F$ has a fixed point in $F\Omega_{R}$. Therefore, $u$ are a solutions of the system (1.1). This completes the proof of Theorem 3.1.

Theorem 3.2. Assume that [H2], [H3] and

hold, then system (1.1) is completely controllable on $[0, t_{f}]$ for some $t_{f}\in[0, T].$

Proof. Let

by

By (H3), for any $u\in PC(J, E)$, From the references ^[14], we know that operators $F_{1}:\Omega_{R}\rightarrow \Omega_{R}$ is Lipschitz continuous and by the proof of Theorem 3.1, we can get that $F\Omega_{R}\subset \Omega_{R}$. According to the definition of completely continuous and the proof of Theorem 3.1, we get that $F_{2}$ is completely continuous Now, we prove that $F_{1}$ is contraction.

Let any $u, u'\in PC(J, E)$ and $t\in (s_{i}, t_{i+1}], \; i = 0, 1, 2, \cdots, m,$ we have

We derive that $\sum\limits_{i = 1}^{m}L_{i} < 1$, which implies $F_{1}$ is contraction. According to the Lemma 3, F has a fixed point $u\in \Omega_{R}$. Thus, If $t_{f}\in[0, t_{1}]$, it is easy to check that $u(t_{f}) = u_{t_{f}}$. Similarly if $t_{f}\in(t_{i}, s_{i}], \; t_{f}\in(s_{i}, t_{i+1}], \; i = 1, 2, \cdots, m, \; \; u(t_{f}) = u_{t_{f}}$. Consequently, the system (1.1) is completely controllable on $[0, t_{f}]$.

4.   An example

As an application of the abstract result, we consider the following controllability of nonlinear differential equations with non-instantaneous impulses

Where $E = C(I)$, $I = [0, 1]$, $J = [0, 1]$, $t_{0} = s_{0} = 0$, $t_{1} = \dfrac{1}{3}$, $s_{1} = \dfrac{2}{3}$, $t_{2} = 1$. $u\in C(I\times J, \mathbb{R})$. Let $f(t, u(t)) = \int_{0}^{1}\dfrac{e^{(s-t)}}{10}u(s)ds +\dfrac{1}{50}u(t)+1, \; g_{1}(t, u(t)) = \dfrac{t+|u(t)|}{1+t+|u(t)|}$, $h(t) = \frac{1}{2}e^{-(t-\frac{1}{2})}$. And then we're going to prove that we satisfy our assumption conditions (H1)-(H3).

Let $t\in[0, \dfrac{1}{3}]\bigcup(\dfrac{2}{3}, 1]$, we get

For any $u, v\in C(I\times J, \mathbb{R})$, $t\in (\dfrac{1}{3}, \dfrac{2}{3}]$, we have

Next, we define a continuous mapping $B:L^{2}(J^{\prime}, R) \rightarrow L^{2}(J^{\prime}, R)$, with domain

A direct calculation gives

To achieve $\|B\|\leq \frac{11}{20}$. Let's calculate

To achieve $\|W^{-1}\|\leq \frac{40}{11}$. Thus, where $p_{1} = \dfrac{1}{10}e-\dfrac{2}{25}$, $p_{0} = 1, \; K = \dfrac{3}{4}, \; N = \dfrac{2}{5}, \; D_{1} = \frac{11}{20}, \; D_{2} = \frac{40}{11}$, the initial value problem (4.1) satisfies the conditions (H1)-(H3). According to Formula (4.2)-(4.5) is established. By Theorem 3.2, the system (4.1) is controllable on $[0, 1]$.

5.   Conclusions

In this paper, the existence of solutions of ordinary differential equations with non-instantaneous impulses in Banach space is studied by using Schauder fixed point theory, and sufficient conditions for controllability results in infinite dimensional space are obtained by using Krasnoselskii fixed point theorem and the definition of complete controllability. However, the necessity of controllability cannot be obtained under this condition. Subsequent studies can obtain the approximate controllability of the system (1.1) under the condition that the controllable function satisfies the presolution formula.

Acknowledgment

Research supported by School-level project of Yili Normal University (2021YSYB069).

Conflict of interest

The authors declare that there is no interest in this paper.