TY - JOUR
AU - Nieto, Juan, J
AB - Abstract This paper is concerned with a kind of first-order singular differential system with impulses. Based on the Schaefer fixed-point theorem, some new verifiable algebraic criteria are given to ensure the controllability of bounded solutions for the considered system. The results obtained in this paper not only achieve the controllability of the singular differential system with impulses for the first time, but also complement the previous researches on singular differential system with impulses. Consequently, the results established are essentially new. Finally, the effectiveness of the obtained results are illustrated via a numerical example. 1. Introduction During the past years, impulsive differential equations have been studied by many authors. Some classical tools have been widely used to get the solutions of impulsive differential equations, such as fixed point theorems in cones, topological degree theory (including continuation method and coincidence degree theory), the method of lower and upper solutions and the critical point theory. For the theory and more related classical results, we assemble here some relevant results, such as Alzabut (2008), Georgieva et al. (2012), Liu (2007), Luo & Jing (2008), Saker & Alzabut (2007), Stamov & Alzabut (2011), Stamov et al. (2012) and Yang & Shen (2007) and monographs Benchohra et al. (2006), Lakshmikantham et al. (1989), Samoilenko & Perestyuk (1995) and Zavalishchin & Sesekin (1997). For example, Yang & Shen (2007) studied nonlinear boundary value problems for a class of first-order functional differential equations as follows $$\begin{align*} \begin{cases} x^{\prime}(t)=f(t,x(t),x(\theta(t))),& t\in J=[0,T], t\neq t_k,\\ \varDelta x(t_k)=I_k(x(t_k)),&k=1,2,...,m,\\ x(0)=px(T), x(t)=(0),&t\in[-r,0], \end{cases} \end{align*}$$ where |$f\in C(J\times{\mathbb{R}}^2,{\mathbb{R}})$|⁠, |$\theta (t)\in C(J,J^+)$|⁠, |$J^+=[-r,T]$|⁠, |$r>0$|⁠, |$0 < t_1 < t_2 < \cdot \cdot \cdot < t_m < T$|⁠, |$I_k\in C({\mathbb{R}},{\mathbb{R}})$|⁠, |$\varDelta x(t_k)=x(t^+_k)-x(t^-_k)$|⁠, |$k=1,2,...,m$|⁠. By using the lower and upper method and monotone iterative technique, the authors established new existence results of extreme solutions. Singular equations appear in a great deal of physical models and differential equations with singularities arise naturally in the study of the motion of particles under the influence of gravitational or electrostatic forces. Singular equation means that the equation becomes infinite at some value of the state variable. For example, in Mawhin (1993), the singularity models in which the restoring force caused by a compressed perfect gas, and in Pishkenari et al. (2008), Rützel et al. (2003) and Yang et al. (2005), the singular term can be regarded as a generalized Lennard-Jones potential or Van der Waals force and it is widely found in molecular dynamics to model the interaction between atomic particles. Compared with the classical first-order singular differential equations or impulsive differential equations, the study on first-order singular differential equations with impulses has been less considered in the literature, see Agarwal et al. (2005), Chu & Nieto (2007), Kong (2017), Kong & Luo (2018), Nieto & Uzal (2018a)and Nieto & Uzal (2018b). For example, in Agarwal et al. (2005), based on Schauder’s fixed point theorem, Agarwal et al. established the existence of at least one positive solution for the following first singular boundary value problems with impulses of the form: $$\begin{align*} \begin{cases} y^{\prime}=f(t,y), & \textrm{a.e. on}\ t\in J^{\prime} = J \setminus \{t_1,t_2,...,t_p\},\\ \varDelta y(t_k) = I_k(y(t_k)), &k = 1, 2,...,p,\\ y(0) = 0, \end{cases} \end{align*}$$ where |$J = [0,T]$|⁠, |$I_k: {\mathbb{R}} \rightarrow{\mathbb{R}}$| is continuous for each |$k =1, 2,...,p$| and the nonlinearity |$f$| may be singular in the independent variable and may also be singular at |$y = 0$|⁠. On the other hand, 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 the study of controllability plays an important role in the control theory and engineering (Aeyels, 1984; Rugh, 1993; Wang, 1999). The concept of controllability leads to some important conclusions regarding the behaviour of linear and nonlinear dynamical systems. Most of the practical systems are nonlinear in nature, and hence the study of nonlinear systems is important. For the basic theory on evolution system, the reader is referred to Tanabe’s book (Tanabe, 1979). As one of the most fundamental concepts in modern control theory, controllability has been studied by many scholars in the past several decades. For linear control systems, their controllability was completely solved in 1960s, which became a classical result. But for nonlinear control systems, although their controllability has been investigated extensively since 1970, and considerable significant progress has been obtained by introducing some powerful methods, including the well-known differential geometric method (Isidori, 1995), and the fixed point methods (Balachandran & Dauer, 1987). In the recent years, the study of impulsive control systems has received an increasing interest. There have been some researches undertaken dealing with the fundamental issues such as controllability for impulsive systems. See, to name a few, Arthi & Balachandran (2012), Arthi & Balachandran (2014), Fu (2003), Liu (2005), Radhakrishnan & Balachandran (2011) and Sakthivel et al. (2009). However, to the best of our knowledge, little attention has been devoted to the study of controllability of the singular differential systems with impulses. This may be due to the fact that the singular forces make the controllability of the impulsive differential systems more difficult and complex, and the mechanism on which how the controllability is influenced by the singularities and impulses associated with the first-order differential systems is far away from clarity. Therefore, at this stage, it is crucial and necessary to further research the dynamical relationship between the two models and fill this gap partially. Motivated by the above works and discussions, in this paper, we study the controllability for the following first-order singular differential equation with impulses of the form: $$\begin{equation} \begin{cases} x^{\prime}(t)= -a(t)x(t)+ f(t,x_t)+B(t)u(t), t\in J=[0,b], t\neq t_k, \\ \varDelta x(t_k)= x(t^+_k) -x(t^-_k)= I_k(x(t^-_k)), k = 1,2,...,m, \\ x(t)=\phi(t), t\in [-\tau,0], \end{cases} \end{equation}$$ (1.1) where the state variable |$x(\cdot )$| takes values in Banach space |$X$| with the norm |$|\cdot |$|⁠. The control function |$u(\cdot )$| is given in |$L^2(J,U)$|⁠, a Banach space of admissible control functions with |$U$| as a Banach space, |$a$| is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators |$T(t)$| in |$X$|⁠, |$B$| is a bounded linear functions from |$U$| into |$X$|⁠, |$f: J\times D \rightarrow X$| can be singular at |$x=0$|⁠, |$D = \big \{\varphi : [-\tau ,0] \rightarrow X, \varphi (t)\textrm{is continuous everywhere except a finite number} \textrm{of points} \widetilde{t} \textrm{at which} \varphi (\widetilde{t}^-), \varphi (\widetilde{t}^+)\ \textrm{exist and}\ \varphi (\widetilde{t}^-)=\varphi (\widetilde{t})\big \}$|⁠, |$I_k: X \rightarrow X $| is continuous, and |$\varDelta x(t_k) =x(t^+_k)-x(t^-_k)$|⁠, for all |$k = 1,2,...,m$|⁠, |$0 = t_0 < t_1 < t_2 <\cdot \cdot \cdot < t_m <t_{m+1} = b$|⁠, |$\phi : [-\tau ,0] \rightarrow X$|⁠. For every |$t\in J$|⁠, |$x_t\in D$| is defined by |$x_t(s)=x(t+s)$|⁠, |$-\tau \leq s \leq 0$|⁠. By means of the Schaefer fixed point theorem, the results on the controllability of (1.1) are established. We would like to remark that up to our knowledge the results on the controllability of first-order singular differential equations with impulses presented in this paper are the first ones available in the literature. The following theorem will be used to prove the main results in the next section. Theorem 1.1 (Schaefer fixed-point theorem, see Schaefer, 1955) Let S be a convex subset of a normed linear space |$E$| and |$0 \in S$|⁠. Let |$F: S\rightarrow S$| be a completely continuous operator and let $$\begin{align*} \zeta(F)=\big\{x\in S:x=\lambda\, Fx\ \textrm{for some}\ 0<\lambda<1 \big\}. \end{align*}$$ Then either |$\zeta (F)$| is unbounded or |$F$| has a fixed point. The remainder part of this paper is organized as follows. Section 2 is devoted to state some necessary definitions and lemmas. In Section 3, the proof of the main result is divided into the two lemmas. An example is given to illustrate the effectiveness of our results in Section 4. Finally, conclusions are drawn in Section 5. 2. Preliminaries Notations: $$\begin{gather*} a^+=\sup\limits_{t\in J}|a(t)|,\quad B^+=\sup\limits_{t\in J}|B(t)|;\\ a^-=\inf\limits_{t\in J}|a(t)|,\quad B^-=\inf\limits_{t\in J}|B(t)|. \end{gather*}$$ Denote |$J_0 = [0,t_1]$|⁠, |$J_k = (t_k,t_{k+1}]$|⁠, |$k = 1,2,...,m$|⁠. Let |$I\in{\mathbb{R}}$| be an interval. We define the following classes of functions: $$\begin{align*} PC(I,X) = \Big\{&x: I \rightarrow X: x(t) \ \textrm{is continuous every where except for some}\ t_k\ \textrm{at} \\ &\quad\textrm{which}\ x(t^+_k) \ \textrm{and}\ x(t^-_k) \ \textrm{exist and}\ x(t_k)=x(t^-_k),\ k=1,2,...,m \Big\}. \end{align*}$$ For |$x\in PC(I,X)$|⁠, take |$\|x\|_{PC}= \sup\nolimits _{-\tau \leq t\leq b} |x(t)|$|⁠, then it is clear that |$PC(I,X)$| is a Banach space. Definition 2.1 A solution |$x(\cdot )\in PC([-\tau ,b],X)$| is said to be a mild solution of (1.1) if |$x(t)\neq \phi (t)$| on |$[-\tau ,0]$|⁠; |$\varDelta x(t_k)=I_k(x(t^-_k))$|⁠, |$k=1,2,...,m$|⁠; the restriction of |$x(\cdot )$| to the interval |$J_k(k =1,...,m)$| is continuous and the following integral equation is verified: $$\begin{align*} x(t)=& T(t)\phi(0)+\int_{0}^{t}T(t-s)\big[ f(s,x_s)+ B(s)u(s)\big]\textrm{d}s +\sum_{0<t_k<t}T(t-t_k)I_k(x(t^-_k)), t\in J. \end{align*}$$ Definition 2.2 (1.1) is said to be controllable on the interval |$J$| if for every initial function |$\phi \in PC([-\tau ,0],X)$| and |$\hat{x} \in X$|⁠, there exist a control |$u\in L^2(J,U)$| such that the mild solution |$x(t)$| of (1.1) satisfies |$x(b) = \hat{x}$|⁠. Throughout this paper, we always assume that: (H1) |$a$| is the infinitesimal generator of a compact semigroup of bounded linear operators |$T(t)$| in |$X$| such that |$|T(t)| \leq M_1$|⁠, for some |$M_1\geq 1$|⁠. (H2) |$I_k \in C(X,X)$| and there exist constants |$\eta _k$| such that |$|I_k(x)|\leq \eta _k$|⁠, |$k = 1,2,...,m$|⁠, for each |$x \in X$|⁠. (H3) The linear operator |$W: L^2(J,U)\rightarrow X$|⁠, defined by $$\begin{align*} Wu=\int_{0}^{b}T(b-s)B(s)u(s)\textrm{d}s \end{align*}$$ has an invertible operator |$W^{-1}$|⁠, which takes values in |$L^2(J,U)\backslash \ker W$| and there exist positive constants |$M_2$| such that |$|W^{-1}|\leq M_2$|⁠. (H4) For almost all |$t\in J$|⁠, the function |$f(t,\cdot )$| is continuous and for each |$x\in D$|⁠, the function |$f(\cdot ,x): J \rightarrow X$| is strongly measurable. (H5) |$\lim\nolimits _{x\rightarrow 0^+} f(t,x)=-\infty $| and |$\lim\nolimits _{x\rightarrow +\infty } f(t,x)=+\infty $|⁠, which imply that, there exists a positive integer |$k$|⁠, and two positive functions |$\alpha _{k}$|⁠, |$\beta _{k} \in L^1(0,b)$| such that $$\begin{gather*} f(t,x)\leq -\alpha_{k}(t)<\alpha_{k}(t),\ \textrm{for}\ \|x\|_{PC}< k, t\in J\ a.e,\\ f(t,x)\geq \beta_{k}(t),\ \textrm{for}\ \|x\|_{PC}> k, t\in J\ a.e. \end{gather*}$$ (H6) There exist integrable functions |$h: J \rightarrow [0,\infty )$| such that |$ |f(t,\phi )|\leq h(t)\psi (\|\phi \|_{PC}), $| for almost all |$t \in J$|⁠, |$\phi \in PC([-\tau ,0],X)$|⁠, where |$\psi : (0,\infty ) \rightarrow (0,\infty )$| is a continuous nondecreasing function with $$\begin{align*} M_1\int_{0}^{b}h(s)\textrm{d}s< \int_{c}^{\infty} \frac{\textrm{d}s}{ \psi(s)}, \end{align*}$$ where |$c= M_1\|\phi \|_{PC}+ M_1Nb +\sum ^m_{k=1}M_1 \eta _k$|⁠, and $$\begin{align*} N:= B^+M_2 \left[ |\hat{x} |+M_1\|\phi \|_{PC} +M_1\int_{0}^{b}h(s)\psi(\|x_s\|_{PC})\textrm{d}s +\sum^m_{k=1}M_1\eta_k \right]. \end{align*}$$ Define $$\begin{equation} \begin{aligned} &\eta_1= \frac{\beta^-_{\eta_1} -B^+ M_2\Big( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\Big)}{a^+}, \\ &\eta_2=\frac{\alpha^+_{\eta_2} +B^+ M_2\Big( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\Big)}{a^-}, \end{aligned} \end{equation}$$ (2.1) where |$\alpha ^+_{\eta _2}=\sup _{t\in J}|\alpha _{\eta _2}(t)|$| and |$\beta ^-_{\eta _1}=\inf _{t\in J}|\beta _{\eta _1}(t)|$|⁠. Lemma 2.3 Assume that |$0<\eta _1< \eta _2$|⁠. Then the solution |$x(t)$| of (1.1) satisfies $$\begin{align*} 0<\eta_1 \leq x(t) \leq \eta_2,\ \textrm{for all}\ t\in [0, b]. \end{align*}$$ Proof. Let |$[0,b_1) \subseteq [0,b]$| be an interval such that |$x(t)>0$| for all |$t\in [0,b_1)$|⁠. Firstly, we prove $$\begin{equation} 0<x(t)\leq \eta_2,\ t\in [0,b_1). \end{equation}$$ (2.2) By way of contradiction, if (2.2) does not hold, then there exists |$\overline{t}\in [0,b_1)$| such that |$x(\overline{t})=\eta _2$| and |$0<x(t)< \eta _2$| for all |$t\in [0, \overline{t})$|⁠. Using hypothesis (H3) for an arbitrary function |$x(\cdot ) \in PC([0,b],X)$| define the control $$\begin{align*} u(t)=& W^{-1}\left[ \hat{x}-T(b)\phi(0) -\int_{0}^{b}T(b-s)f(s,x_s)\textrm{d}s -\sum^m_{k=1}T(b-t_k)I_k(x(t^-_k)) \right](t). \end{align*}$$ Calculating the derivative of |$x(t)$|⁠, then by (H5) and (2.1), we have that $$\begin{align*} 0\leq&\ x^{\prime}(\overline{t}) = -a(\overline{t})x(\overline{t}) +f(\overline{t},x_{\overline{t}}) +B(\overline{t})u(\overline{t})\\ =&\ -a(\overline{t})x(\overline{t}) +f(\overline{t},x_{\overline{t}}) +B(\overline{t})\\ &\cdot W^{-1}\left[ \hat{x}-T(b)\phi(0)-\int_{0}^{b}T(b-s)f(s,x_s)\textrm{d}s -\sum^m_{k=1}T(b-t_k)I_k(x(t^-_k)) \right](\overline{t}) \\ <& -a^- \eta_2+ \alpha_{\eta_2}(\overline{t}) +B^+ M_2\left[ |\hat{x}|+M_1\|\phi\|_{PC} +M_1 \int_{0}^{b} \alpha_{\eta_2}(s)\textrm{d}s +\sum^m_{k=1}M_1 \eta_k \right] \\ < & -a^- \eta_2+ \alpha^+_{\eta_2} + B^+ M_2\left( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\right)\\ =& -a^-\cdot \frac{\alpha^+_{\eta_2} +B^+ M_2\Big( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\Big)}{a^-}\\ &+\alpha^+_{\eta_2}+ B^+ M_2\left( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2}+\sum^m_{k=1}M_1 \eta_k\right)=0, \end{align*}$$ which leads to a contradiction. Then, (2.2) is satisfied. Secondly, we claim that the following inequality holds $$\begin{equation} x(t)\geq \eta_1>0, t\in [0, b_1). \end{equation}$$ (2.3) Otherwise, there exists |$\underline{t}\in [0,b_1)$| such that |$x(\underline{t})=\eta _1$| and |$x(t)\geq \eta _1$| for all |$t\in [0, \underline{t})$|⁠. Then from (2.2), we can see that |$\eta _1< x(t)< \eta _2$| for all |$t\in [0, \underline{t})$|⁠. Calculating the derivative of |$x(t)$|⁠, by (H5) and (2.1), we have that $$\begin{align*} 0\geq&\ x^{\prime}(\underline{t}) = -a(\underline{t})x(\underline{t}) +f(\underline{t},x_{\underline{t}}) +B(\underline{t})u(\underline{t})\\ =&\ -a(\underline{t})x(\underline{t}) +f(\underline{t},x_{\underline{t}})\\ &+B(\underline{t}) \cdot W^{-1}\left[ \hat{x}-T(b)\phi(0)-\int_{0}^{b}T(b-s)f(s,x_s)\textrm{d}s-\sum^m_{k=1}T(b-t_k)I_k(x(t^-_k)) \right](\underline{t}) \\> & -a^+ \eta_1 + \beta_{\eta_1}(\underline{t}) -B^+ M_2\left( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\right)\\ > & -a^+ \eta_1 + \beta^-_{\eta_1} -B^+ M_2\left( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\right)\\ =& -a^+ \cdot \frac{\beta^-_{\eta_1} -B^+ M_2\left( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\right)}{a^+}\\ &+ \beta^-_{\eta_1} -B^+ M_2\left( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\right) =0, \end{align*}$$ which also leads to a contradiction. Then, (2.3) is satisfied. Therefore, the proof is completed. Now, we are ready to present our main result. Theorem 2.4 If the conditions (H1)–(H6) are satisfied, then (1.1) is controllable on |$J$|⁠. 3. Proof of Theorem 2.4 In this section, we divide the proof into the following two lemmas. Lemma 3.1 If the conditions (H1)–(H6) are satisfied, then the mild solution of (1.1) is bounded, i.e., there exist positive constants |$0<K_2<K_1$| such that $$\begin{align*} K_2 \leq \|x \|_{PC}= \sup\limits_{-\tau\leq t\leq b} |x (t)|\leq K_1. \end{align*}$$ Proof. Using hypothesis (H3) for an arbitrary function |$x(\cdot ) \in PC([-\tau ,b],X)$| define the control $$\begin{equation} \begin{aligned} u(t)=& W^{-1}\left[ \hat{x}-T(b)\phi(0) -\int_{0}^{b}T(b-s)f(s,x_s)\textrm{d}s -\sum^m_{k=1}T(b-t_k)I_k(x(t^-_k)) \right](t). \end{aligned} \end{equation}$$ (3.1) We shall now show that when using this control the operator defined by $$\begin{equation} \begin{aligned} Fx(t)=&T(t)\phi (0)+\int_{0}^{t}T(t-s)\big[f(s,x_s)+ B(s)u(s) \big]\textrm{d}s +\sum_{0<t_k<t}T(t-t_k)I_k(x (t^-_k)), t\in J \end{aligned} \end{equation}$$ (3.2) has a fixed point. This fixed point is a solution of (1.1). Clearly, |$x (b) = (Fx )(b) =\hat{x} $|⁠, which implies that (1.1) is controllable. Moreover, in order to apply Theorem 1.1 to prove that the operator |$F$| defined in (3.2) has a fixed point, firstly we have to obtain a priori bounds for the following equation: $$\begin{equation} \begin{aligned} x (t)=&\ \lambda T(t)\phi (0) + \lambda \int_{0}^{t}T(t-\theta) B(t)W^{-1}\Bigg[ \hat{x} -T(b)\phi (0)- \int_{0}^{b}T(b-s)f(s,x_s)\textrm{d}s\\ &{\hskip152pt}- \sum^m_{k=1}T(b-t_k)I_k(x (t^-_k)) \Bigg](\theta)d\theta\\ &+ \lambda \int_{0}^{t} T(t-s) f(s,x_s)\textrm{d}s + \lambda \sum_{0<t_k<t}T(t-t_k)I_k(x(t^-_k)), t\in J, \end{aligned} \end{equation}$$ (3.3) which together with (H1), (H2), (H3) and (H6) yields $$\begin{align} |x (t)| \leq&\ M_1\|\phi \|_{PC} + M_1\int_{0}^{t} B^+M_2 \left[ |\hat{x} |+M_1\|\phi \|_{PC} +M_1\int_{0}^{b}h(s)\psi(\|x_s\|_{PC})\textrm{d}s+\sum^m_{k=1}M_1\eta_k \right](\theta)d\theta\nonumber \\ & + M_1\int_{0}^{t}h(s)\psi(\|x_s\|_{PC})\textrm{d}s+\sum_{0<t_k<t}M_1 \eta_k. \end{align}$$ (3.4) Consider the following function |$\mu $| given by $$\begin{equation} \begin{aligned} \mu(t)=\sup\big\{|x (s)|:-\tau \leq s \leq t, 0\leq t\leq b, i=1,2,...,n \big\}. \end{aligned} \end{equation}$$ (3.5) Let |$t^* \in [-\tau ,t]$| be such that |$\mu (t) = |x (t^*)|$|⁠. If |$t^* \in J$|⁠, from (3.4), (3.5) and (H6), it follows that $$\begin{align} \mu(t)\leq&\ M_1\|\phi \|_{PC} + M_1\int_{0}^{t} B^+M_2 \left[ |\hat{x} |+M_1\|\phi \|_{PC} +M_1\int_{0}^{b}h(s)\psi(\|x_s\|_{PC})\textrm{d}s+\sum^m_{k=1}M_1\eta_k \right](\theta)\textrm{d}\theta\nonumber\\ & + M_1\int_{0}^{t}h(s)\psi(\mu(s))\textrm{d}s+\sum_{0<t_k<t}M_1 \eta_k:=v(t), t\in J. \end{align}$$ (3.6) If |$t^*\in [-\tau ,0]$|⁠, then |$\mu (t) = \|\phi \|_{PC}$| and the previous inequality holds since |$M_1 \geq 1$|⁠. From (3.6), it follows that $$\begin{equation} \begin{aligned} M_1\|\phi \|_{PC}+ M_1Nb +\sum^m_{k=1}M_1 \eta_k:=v(0)=c, \end{aligned} \end{equation}$$ (3.7) $$\begin{equation} \mu(t) \leq v(t), t\in J, \end{equation}$$ (3.8) and $$\begin{equation} \begin{aligned} v^{\prime}(t)=&\ M_1 h(t)\psi(\mu(t)). \end{aligned} \end{equation}$$ (3.9) Since |$\psi $| is nondecreasing, in view of (3.8) and (3.9), we get $$\begin{equation} \begin{aligned} v^{\prime}(t)\leq & M_1h(t)\psi(v(t)), \end{aligned} \end{equation}$$ (3.10) thus, $$\begin{align*} \frac{v^{\prime}(t)}{\psi(v(t))}\leq M_1h(t), t\in J, \end{align*}$$ which leads to $$\begin{align*} \int_{v(0)}^{v(t)} \frac{\textrm{d}s}{\psi(s)}\leq M_1\int_{0}^{b}h(s)\textrm{d}s, t\in J, \end{align*}$$ by (H6), we obtain $$\begin{align*} \int_{v(0)}^{v(t)} \frac{\textrm{d}s}{\psi(s)}\leq M_1\int_{0}^{b}h(s)\textrm{d}s< \int_{v(0)}^{\infty} \frac{\textrm{d}s}{\psi(s)}, t\in J, \end{align*}$$ which implies that there exists a constant |$K^{\prime}$| such that |$v(t) \leq K^{\prime}$|⁠, |$t \in J=[0,b]$|⁠, and hence |$\mu (t)\leq K^{\prime}$|⁠, |$t\in J$|⁠. Since for every |$t\in J$|⁠, |$\|x \|_{PC}\leq \mu (t)$|⁠, we have $$\begin{equation} \|x \|_{PC}= \sup\limits_{-\tau\leq t\leq b} |x (t)|\leq K^{\prime}, \end{equation}$$ (3.11) where |$K^{\prime}$| depends on |$b$|⁠, the functions |$h$| and |$\psi $|⁠. Furthermore, let |$K_1=\max \{K^{\prime}, \eta _2\}$| and |$0<K_2<\eta _1<\eta _2$|⁠, then from Lemma 2.3 and (3.11), it follows that $$\begin{align*} K_2 \leq \|x \|_{PC}= \sup\limits_{-\tau\leq t\leq b} |x (t)|\leq K_1. \end{align*}$$ Therefore, the proof is completed. Lemma 3.2 If the conditions (H1)–(H6) are satisfied, then the operator |$F$| defined in (3.2) possesses a fixed point. Proof. For |$\phi \in PC([-\tau ,0],X)$| and define |$\widehat{\phi } \in PC([-\tau ,b],X)$| as follow: $$\begin{equation} \widehat{\phi} (t)= \begin{cases} T(t)\phi (0),&0\leq t\leq b,\\ \phi (t),&-\tau \leq t\leq0. \end{cases} \end{equation}$$ (3.12) If |$x (t)=y (t)+\widehat{\phi } (t)$|⁠, |$t\in [-\tau , b]$|⁠, then one can see that |$y $| satisfies |$y_0 = 0$| and $$\begin{equation} \begin{aligned} y(t)=&\ \int_{0}^{t}T(t-\theta) B(t)W^{-1}\Bigg[ \hat{x} -T(b)\phi (0)-\int_{0}^{b}T(b-s)f(s,y_s+\widehat{\phi}_s)\textrm{d}s\\ &{\hskip95pt}-\sum^m_{k=1}T(b-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \Bigg](\theta)\textrm{d}\theta\\ & +\int_{0}^{t} T(t-s)f(s,y_s+\widehat{\phi}_s)\textrm{d}s+\sum_{0<t_k<t}T(t-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big), t\in J, \end{aligned} \end{equation}$$ (3.13) if and only if |$x (t)$| satisfies $$\begin{align*} x (t)=& T(t)\phi (0)+ \int_{0}^{t}T(t-\theta) B(t)W^{-1}\Bigg[ \hat{x} -T(b)\phi (0)-\int_{0}^{b}T(b-s)f(s,x_s)\textrm{d}s\\ &{\hskip135pt}-\sum^m_{k=1}T(b-t_k)I_k\big(x(t^-_k)\big) \Bigg](\theta)\textrm{d}\theta \\ &+\int_{0}^{t} T(t-s)f(s,x_s)\textrm{d}s+\sum_{0<t_k<t}T(t-t_k)I_k\big(x(t^-_k)\big), t\in J, \end{align*}$$ and |$x (t)=\phi (t)$| for |$t\in [-\tau , 0]$|⁠. Define $$\begin{align*} PC_0=\Big\{y \in PC([-\tau,b],X):y_0=0 \Big\}, \end{align*}$$ and |$F:PC_0\rightarrow PC_0$| by |$(Fy )(t)=0$|⁠, |$-\tau \leq t\leq 0$|⁠, and $$\begin{equation} \begin{aligned} (Fy )(t)=& \int_{0}^{t}T(t-\theta)B(t)W^{-1}\Bigg[ \hat{x} -T(b)\phi (0)-\int_{0}^{b}T(b-s)f(s,y_s+\widehat{\phi}_s)\textrm{d}s \\ &{\hskip90pt}-\sum^m_{k=1}T(b-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \Bigg](\theta)\textrm{d}\theta\\ &+\int_{0}^{t} T(t-s) f(s,y_s+\widehat{\phi}_s)\textrm{d}s+\sum_{0<t_k<t}T(t-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big), t\in J. \end{aligned} \end{equation}$$ (3.14) Let $$\begin{align*} B_k=\Big\{y \in PC_0: \|y \|_{PC}\leq k, k\geq 1\Big\}. \end{align*}$$ Then, it is easy to see that the family |$B_k$| is uniformly bounded. In the following, we shall prove that |$F$| is a completely continuous operator. We divide the process of proof into the following three steps. Step 1. We show that |$F$| maps |$B_k$| into a equicontinuous family. Let |$y \in B_k$| and |$ \sigma _1$|⁠, |$\sigma _2\in J$|⁠. Then if |$0 < \sigma _1 < \sigma _2 \leq b$|⁠, $$\begin{align*} &\big|(Fy )(\sigma_1)-(Fy )(\sigma_2)\big|\\ &\leq \Bigg|\int_{0}^{\sigma_1} \big[T(\sigma_1-\theta)-T(\sigma_2-\theta)\big]B(\theta)W^{-1} \bigg[ \hat{x} -T(b)\phi (0)-\int_{0}^{b}T(b-s)f(s,y_s+\widehat{\phi}_s)\textrm{d}s\\ &\quad-\sum^m_{k=1}T(b-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \bigg](\theta)\textrm{d}\theta \Bigg| +\Bigg|\int_{\sigma_1}^{\sigma_2} T(\sigma_2-\theta) B(\theta)W^{-1}\bigg[ \hat{x} -T(b)\phi (0)\\ &\quad-\int_{0}^{b}T(b-s)f(s,y_s+\widehat{\phi}_s)\textrm{d}s -\sum^m_{k=1}T(b-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \bigg](\theta)\textrm{d}\theta \Bigg|\\ &\quad+\Bigg|\int_{0}^{\sigma_1} \big[T(\sigma_1-s)-T(\sigma_2-s)\big]\,f(s,y_s+\widehat{\phi}_s)\textrm{d}s\Bigg| +\Bigg|\int_{\sigma_1}^{\sigma_2} T(\sigma_2-s) f(s,y_s+\widehat{\phi}_s)\textrm{d}s\Bigg|\\ &\quad+\left|\sum_{0<t_k<\sigma_1}\big[ T(\sigma_1-t_k)-T(\sigma_2-t_k) \big]I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big)\right|+\left|\sum_{\sigma_1\leq t_k<\sigma_2}T(\sigma_2-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big)\right|, \end{align*}$$ which together with (H2), (H3) and (H5) give $$\begin{equation} \begin{aligned} &\big|(Fy )(\sigma_1)-(Fy )(\sigma_2)\big|\\ &\leq \int_{0}^{\sigma_1} \big|T(\sigma_1-\theta)-T(\sigma_2-\theta)\big|B^+M_2\bigg[ \hat{x} -T(b)\phi (0) +M_1\int_{0}^{b}\alpha_{k^{\prime}}(s)\textrm{d}s+\sum^m_{k=1}M_1\eta_k \bigg]\textrm{d}\theta \\ &\quad +\int_{\sigma_1}^{\sigma_2}|T(\sigma_2-\theta)| B^+M_2\left[ |\hat{x}|+M_1\|\phi\|_{PC} +M_1\int_{0}^{b}\alpha_{k^{\prime}}(s)\textrm{d}s+\sum^m_{k=1}M_1 \eta_k\right]\textrm{d}\theta\\ &\quad +\int_{0}^{\sigma_1} \big|T(\sigma_1-s)-T(\sigma_2-s)|\alpha_{k^{\prime}}(s)\textrm{d}s +\int_{\sigma_1}^{\sigma_2} |T(\sigma_2-s)| \alpha_{k^{\prime}}(s)\textrm{d}s \\ &\quad +\sum_{0<t_k<\sigma_1}\big|T(\sigma_1-t_k)-T(\sigma_2-t_k)\big|\eta_k +\sum_{\sigma_1\leq t_k<\sigma_2}|T(\sigma_2-t_k)|\eta_k, \end{aligned} \end{equation}$$ (3.15) where |$k^{\prime}=k+\|\widehat{\phi } \|_{PC}$|⁠. Since the compactness of |$T(t)$| for |$t> 0$|⁠, the compactness implies that the continuity in the uniform operator topology. Let |$\sigma _2-\sigma _1\rightarrow 0$|⁠, then we can see that $$\begin{align*} \big|(Fy )(\sigma_1)-(Fy )(\sigma_2)\big|\rightarrow 0. \end{align*}$$ Thus, |$F$| maps |$B_k$| into an equicontinuous family of functions. For the case |$\sigma _1 < \sigma _2 < 0$| or |$\sigma _1 < 0 < \sigma _2$|⁠, we can hand them similarly. Step 2. We show |$\overline{FB_k}$| is compact. Since we have shown |$FB_k$| is equicontinuous collection, it suffices by the Arzela–Ascoli theorem to show that |$F$| maps |$B_k$| into a precompact set in |$X$|⁠. Let |$0 < t \leq b$| be fixed and |$\epsilon $| a real number satisfying |$0 <\epsilon < t$|⁠. For |$y \in B_k$|⁠, we define $$\begin{align*} (F_\epsilon y )(t)=& \int_{0}^{t-\epsilon}T(t-\theta) B(t)W^{-1}\Bigg[\hat{x} -T(b)\phi (0) -\int_{0}^{b}T(b-s)f(s,y_s+\widehat{\phi}_s)\textrm{d}s\\ &{\hskip102pt}-\sum^m_{k=1}T(b-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \Bigg](\theta)\textrm{d}\theta\\ & +\int_{0}^{t-\epsilon} T(t-s) f(s,y_s+\widehat{\phi}_s)\textrm{d}s+\sum_{0<t_k<t}T(t-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big), \end{align*}$$ since |$T$| is bounded and linear, we have $$\begin{align*} (F_\epsilon y )(t)=&\ T(\epsilon)\int_{0}^{t-\epsilon}T(t-\theta-\epsilon) B(t)W^{-1}\Bigg[\hat{x} -T(b)\phi (0) -\int_{0}^{b}T(b-s)f(s,y_s+\widehat{\phi}_s)\textrm{d}s\\ &{\hskip140pt}-\sum^m_{k=1}T(b-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \Bigg](\theta)\textrm{d}\theta\\ &+T(\epsilon)\int_{0}^{t-\epsilon} T(t-s-\epsilon) f(s,y_s+\widehat{\phi}_s)\textrm{d}s+\sum_{0<t_k<t}T(t-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big). \end{align*}$$ Since |$T(t)$| is a compact operator, the set |$\varUpsilon _\epsilon (t) = \{(F_\epsilon y )(t): y \in B_k, i=1,2,...,n\}$| is precompact in |$X$|⁠, for every |$\epsilon $|⁠, |$0 < \epsilon < t$|⁠. Moreover, for every |$y \in B_k$|⁠, we have $$\begin{align*} \big|(Fy )(t)-(F_\epsilon y )(t)\big| \leq& \int_{t-\epsilon}^{t}|T(t-\theta)|B(t)W^{-1}\Bigg[ \hat{x} -T(b)\phi (0) -\int_{0}^{b}T(b-s)f(s,y_s+\widehat{\phi}_s)\textrm{d}s \\ &-\sum^m_{k=1}T(b-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \Bigg](\theta)\textrm{d}\theta +\int_{t-\epsilon}^{t} \Big| T(t-s) f(s,y_s+\widehat{\phi}_s) \Big|\textrm{d}s. \end{align*}$$ Therefore, there exist precompact sets arbitrarily close to the set |$\varUpsilon _\epsilon (t)$|⁠. Hence, the set |$\varUpsilon _\epsilon (t)$| is precompact in |$X$|⁠. Step 3. We show that |$F: PC_0 \rightarrow PC_0$| is continuous. Let |$\{y^{(l)} \}^\infty _0\sqsubseteq PC_0$| with |$y^{(l)} \rightarrow y $| in |$PC_0$|⁠. Then there is an integer |$r$| such that |$|y^{(l)} (t)| \leq r$| for all |$l$|⁠, |$t \in J$|⁠, so |$y^{(l)} \in B_r$| and |$y \in B_r$|⁠. By (H4), for each |$t \in J$|⁠, we have $$\begin{equation} f \big(t,y^{(l)}(t)+\widehat{\phi}(t)\big)\rightarrow f\big(t,y(t)+\widehat{\phi}(t)\big). \end{equation}$$ (3.16) From (H5), it follows that $$\begin{equation} \Big|f \big(t,y^{(l)}(t)+\widehat{\phi}(t)\big)-f \big(t,y(t)+\widehat{\phi}(t)\big)\Big| \leq 2 \alpha_{r^{\prime}}(t), \end{equation}$$ (3.17) where |$r^{\prime}=r+\|\widehat{\phi } \|_{PC}$|⁠. By the dominated convergence theorem, we have $$\begin{align*} &\|Fy^{(l)} -Fy \|_{PC}\\ &= \sup\limits_{t\in J} \Bigg| \int_{0}^{t}T(t-\theta)B(t)W^{-1} \cdot \Bigg\{ \int_{0}^{b}T(b-s) \bigg[ f\big(s,y^{(l)}(s)+\widehat{\phi}(s)\big)-f\big(s,y(s)+\widehat{\phi}(s)\bigg]\textrm{d}s \\ &\quad\qquad +\sum^m_{k=1}T(b-t_k)I_k\big(y^{(l)} (t^-_k)+\widehat{\phi} (t^-_k)\big)-\sum^m_{k=1}T(b-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \Bigg\}(\theta)\textrm{d}\theta\\ &\quad \qquad+\int_{0}^{t} T(t-s) \bigg[ f \big(s,y^{(l)}(s)+\widehat{\phi}(s)\big)-f \big(s,y(s)+\widehat{\phi}(s)\big) \bigg]\textrm{d}s\\ &\quad \qquad+\sum_{0<t_k<t}T(t-t_k)I_k\big(y^{(l)} (t^-_k)+\widehat{\phi} (t^-_k)\big)- \sum_{0<t_k<t}T(t-t_k)I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big) \Bigg|, \end{align*}$$ combining with (H1) and (H3), we can get $$\begin{align*} \|Fy^{(l)} -Fy \|_{PC} \leq& \int_{0}^{b}|T(t-\theta)|B^+M_2 \Bigg[ \int_{0}^{b}M_1 \Big| f\big(s,y^{(l)}(s)+\widehat{\phi}(s)\big)-f\big(s,y(s)+\widehat{\phi}(s)\big)\Big|\textrm{d}s \\ &+\sum^m_{k=1}|T(b-t_k)|\Big|I_k\big(y^{(l)} (t^-_k)+\widehat{\phi} (t^-_k)\big)-I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big)\Big| \Bigg](\theta)\textrm{d}\theta\\ &+\int_{0}^{b} |T(t-s)| \Big|\, f \big(s,y^{(l)}(s)+\widehat{\phi}(s)\big)-f \big(s,y(s)+\widehat{\phi}(s)\big) \Big| \textrm{d}s\\ &+\sum_{0<t_k<t}|T(t-t_k)|\Big| I_k\big(y^{(l)} (t^-_k)+\widehat{\phi} (t^-_k)\big)-I_k\big(y (t^-_k)+\widehat{\phi} (t^-_k)\big)\Big|, \end{align*}$$ which together with (3.16), (3.17), and |$y^{(l)} \rightarrow y $| in |$PC_0$|⁠, yields $$\begin{align*} \|Fy^{(l)} -Fy \|_{PC} \rightarrow 0. \end{align*}$$ Thus, |$F$| is continuous. This completes the proof that |$F$| is completely continuous. Finally, the set $$\begin{align*} \zeta(F)=\big\{x \in S:x =\lambda\ Fx\ \textrm{for some}\ 0<\lambda<1,i=1,2,...,n\big \} \end{align*}$$ is bounded, since for every solution |$y $| in |$\zeta (F)$| the function |$x =y +\widehat{\phi } $| is a mild solution of (1.1). From Lemma 3.1, we have proved that |$K_2 \leq \|x \|_{PC} \leq K_1$| and hence |$\|y\|_{PC}\leq K_1+\|\widehat{\phi } \|_{PC}$|⁠. Consequently, by Schaefer fixed-point theorem 1.1, the operator |$F$| has a fixed point in |$PC_0$|⁠. This implies that any fixed point of |$F$| is a solution of (1.1) on |$J$|⁠. Hence, according to Definition 2.2, we can conclude that (1.1) is controllable on |$J$|⁠. Up to now, the proof of Theorem 2.4 is completed. 4. An Illustrative Example In this section, we provide an example to illustrate our main results. Consider the following partial differential equation with singularities and impulses: $$\begin{equation} \begin{cases} \frac{\partial}{\partial t}z(y,t)=\frac{\partial^2}{\partial y^2}z(y,t)+B(t)u(t)+f(q(t,z(y,t-r))),\\ 0\leq y\leq \pi, t\in J=[0,b], t\neq t_k, \\ z(y,t^+_k) -z(y,t^-_k)= I_k(z(y,t^-_k)), k = 1,2,...,m, \\ z(0,t)=z(\pi,t)=0, t \geq 0,\\ z(y,t)=\phi(y,t), t\in [-\tau,0]. \end{cases} \end{equation}$$ (4.1) Take |$X=L^2[0,\pi ]$| and define |$\omega (t)=z(\cdot , t)$|⁠. Let $$\begin{align*} f(t,\omega_t)(y)=q(t,z(y,t-r))-\frac{1}{q(t,z(y,t-r))}, 0\leq y \leq \pi, \end{align*}$$ then we can see that |$q=0$| is the singularity. Define |$A:X\rightarrow X$| by |$A\omega =\omega ^{\prime\prime}$| with the domain $$\begin{align*} D(A)=\big\{\omega \in X: \omega, \omega^{\prime}\ \textrm{absolutely continuous},\ \omega^{\prime\prime} \in X, \omega(0) = \omega(\pi) = 0\big\}, \end{align*}$$ then, |$A\omega =\sum ^\infty _{n=1}n^2\langle \omega ,\omega _n\rangle \omega _n$|⁠, |$\omega \in D(A)$|⁠, where |$\omega _n=\sqrt{2/\pi } \sin ns$|⁠, |$n=1,2,...$| is the orthogonal set of eigenvectors of |$a$|⁠. It is easy to prove that |$a$| is the infinitesimal generator of an analytic semigroup |$T(t)$|⁠, |$t\geq 0$| in |$X$| as is given by $$\begin{align*} T(t)\omega =\sum^\infty_{n=1} \exp(-n^2t)\langle\omega,\omega_n\rangle\omega_n, \omega \in X. \end{align*}$$ Since the analytic semigroup |$T(t)$| is compact, according to Pazy (1983), there exists a constant |$M_1\geq 1 $| such that |$|T(t)| \leq M_1$|⁠. Assume that there exists an invertible operator |$W^{-1}$|⁠, which takes values in |$L^2(J,U)/\ker W$| such that $$\begin{align*} Wu=\int_{0}^{b}T(b-s)B(s)u(s)\textrm{d}s. \end{align*}$$ Further, there exists an integrable function |$h:J \rightarrow [0,\infty )$| satisfying $$\begin{align*} |f(t,\omega_t)|\leq h(t)\psi(\|\omega_t\|_{PC}), \end{align*}$$ where |$ \psi : [0,\infty ) \rightarrow (0,\infty )$| is continuous and nondecreasing. There exist constants |$\eta _k$| such that $$\begin{align*} |I_k(\omega)|\leq \eta_k, k = 1,2,...,m, \textrm{for each} \omega \in X, \end{align*}$$ where |$I_k(z(y,t^-_k))=I_k(\omega (t^-_k))(y)$|⁠. Also, we have $$\begin{align*} M_1\int_{0}^{b}h(s)\textrm{d}s< \int_{c}^{\infty} \frac{\textrm{d}s}{ \psi(s)}, \end{align*}$$ where |$c= M_1\|\phi \|_{PC}+ M_1Nb +\sum ^m_{k=1}M_1 \eta _k$|⁠. Specifically, let $$\begin{gather*} b=8, a(t)=4|\cos t|+4, B(t)=\sin t, m=9,\\ I_k(x)=6\sin x, M_1=1, \phi(t)= 9 \sin t, \end{gather*}$$ then, we have $$\begin{gather*} a^+=8, a^-=4,\ B^+=1,\ \eta_k=6,\ \|\phi\|_{PC}=9. \end{gather*}$$ Moreover, if we choose some |$\alpha _{\eta _2}(t)$|⁠, |$\beta _{\eta _1}(t)$| such that |$\alpha ^+_{\eta _2}=9$|⁠, |$\beta ^-_{\eta _1}=8$|⁠, choose some |$u$| such that |$|W^{-1}|\leq M_2=0.02$|⁠, and choose some |$\hat{x}$| such that |$|\hat{x}|=9$|⁠, thus, we obtain $$\begin{align*} &\eta_1= \frac{\beta^-_{\eta_1} -B^+ M_2\Big( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\Big)}{a^+}\approx 0.5039624, \\ &\eta_2=\frac{\alpha^+_{\eta_2} +B^+ M_2\Big( |\hat{x}|+M_1\|\phi\|_{PC} +M_1 b \alpha^+_{\eta_2} +\sum^m_{k=1}M_1 \eta_k\Big)}{a^-}\approx 3.242075. \end{align*}$$ Till now, all the assumptions in Lemma 2.3 and Theorem 2.4 are all satisfied. Therefore, we can see that (4.1) is controllable on |$J$|⁠. 5. Conclusion In this paper, we are concerned with the first-order singular differential systems with impulses. The result on the controllability of the considered system has been completely established by means of the Schaefer fixed point theorem. The new verifiable criterion have been given to guarantee the controllability. Finally, an example has been presented at the end of this paper to illustrate the effectiveness and feasibility of the proposed criterion. It must be mentioned that it is the first time to discuss the controllability of first-order singular differential systems with impulses like the form of (1.1). Consequently, the results established in the present paper are essentially new. In addition, the method provided in this paper can also be employed to analyse the controllability for other kinds of practical models. For example, the haematopoiesis model with impulses (Saker & Alzabut, 2009; Yao et al., 2018) of the form: $$\begin{align*} \begin{cases} x^{\prime}(t)= -a(t)x(t)+ \frac{b(t)}{1+x^n(t-\tau)}, t\in{\mathbb{R}}, t\neq t_k, \\ \varDelta x(t_k):=x(t^+_k) -x(t^-_k)=\gamma_kx(t_k)+\delta_k, k\in{\mathbb{N}}, \end{cases} \end{align*}$$ or, generalized hematopoiesis model with linear harvesting terms and impulses of the form: $$\begin{align*} \begin{cases} x^{\prime}(t)= -a(t)x(t)+\displaystyle\sum^P_{i=1}\frac{b_i(t)x^m(t-\tau_i(t))}{1+x^n(t-\tau_i(t))}-H( x(t-\delta(t))), t\in{\mathbb{R}}, t\neq t_k, \\ \varDelta x(t_k)= x(t^+_k) -x(t^-_k)= \gamma_kx(t_k)+I_k(x(t_k)), k\in{\mathbb{N}}, \end{cases} \end{align*}$$ where |$0 \leq m < n$|⁠, |$a\in C({\mathbb{R}},{\mathbb{R}}^+)$| is almost periodic function, |$b_i\in C({\mathbb{R}},{\mathbb{R}}^+)$| and |$\tau _i\in C({\mathbb{R}},{\mathbb{R}}^+)$| are continuous and pseudo almost periodic functions for |$i = 1, 2,..., P$|⁠; |$\delta \in C({\mathbb{R}},{\mathbb{R}}^+)$| is pseudo almost periodic function, |$H \in C({\mathbb{R}}^+,{\mathbb{R}}^+)$| is continuous and pseudo almost periodic, |${\mathbb{R}}^+=(0,+\infty )$|⁠. |$\gamma _k$|⁠, |$k\in{\mathbb{N}}$| is pseudo almost periodic sequence, |$\varDelta (x(t_k)) = x(t^+_k) -x(t^-_k)$| are impulses at moments |$t_k$| and |$t_1 < t_2 < \cdot \cdot \cdot $| is a strictly increasing sequence such that |$\lim \limits _{k\rightarrow \pm \infty } t_k =\pm \infty $|⁠. The sequence of functions |$\{I_k(x)\}_{k\in{\mathbb{Z}}}$| is pseudo almost periodic uniform for |$ x\in \varOmega $|⁠, where |$\varOmega $| is a subset of |${\mathbb{R}}$|⁠. The works mentioned above will be the topic of our future research. Authors’ contributions All authors read and approved the manuscript. Acknowledgements The authors thank the anonymous reviewers for their insightful suggestions, which improved this work significantly. References Aeyels , D. ( 1984 ) Local and global controllability for nonlinear systems . Systems Control Lett. , 5 , 19 – 26 . Google Scholar Crossref Search ADS WorldCat Agarwal , R. P. , Franco , D. & O’Regan , D. ( 2005 ) Singular boundary value problems for first and second order impulsive differential equations . Aequationes Mathematicae , 69 , 83 – 96 . Google Scholar Crossref Search ADS WorldCat Alzabut , J. O. ( 2008 ) Existence of periodic solutions of a type of nonlinear impulsive delay differential equations with a small parameter . J. 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TI - Control of bounded solutions for first-order singular differential equations with impulses
JO - IMA Journal of Mathematical Control and Information
DO - 10.1093/imamci/dnz033
DA - 2004-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/control-of-bounded-solutions-for-first-order-singular-differential-1WIR0LlN05
SP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -