This section introduces some relevant notations, assumptions, and definitions that are necessary for developing the results of this paper. Let \(\mathbb {R}\) denote the set of real numbers, let \(\mathbb {R}_{+}\) be the set of all nonnegative real numbers, let \(\mathbb {R}^{n}\) denote the ndimensional phase space, \(n\geq 1\). We write \(\\cdot \\) for the Euclidean vector norm, that is, for a vector \(x=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R}^{n}\), \(\x\=\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}\). \(\mathcal{ \mathcal{B}}_{\varepsilon }(\alpha )\), \(\alpha \in \mathbb {R}^{n}\), \(\varepsilon >0\), denote the open ball centered at α with radius ε. \(\langle \cdot ,\cdot \rangle \) is the standard inner product in \(\mathbb {R}^{n}\). For \(x\in \mathbb {R}^{n}\) and \(\mathcal{K}\subset \mathbb {R}^{n}\), let \(d_{\mathcal{K}}(x)\) denote the distance of the point x to the set \(\mathcal{K}\) defined by
$$ \mathrm{d}_{\mathcal{K}}(x)=\inf_{y\in \mathcal{K}} \Vert xy \Vert . $$
Firstly, we consider the nonlinear dynamical system
$$ \dot{x}(t)=f\bigl(x(t)\bigr), \qquad x(t_{0})=x_{0}, \quad t\in \mathbb {R}, $$
(2.1)
\(f: \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is Lipschitz continuous on \(\mathbb {R}^{n}\), and I is the maximal interval of existence for the solution \(x(t)\) of (2.1). For all \(t\in \mathbb {R}\), let \(\pi :I\times \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}\) be the flow generated by (2.1), where π is a continuous function, \(\pi (t_{0},x)=x\), and \(\pi (s, \pi (t,x))=\pi (t+s,x)\) for all \(x\in \mathbb {R}^{n}\), and \(t, s\in I\). We define the continuous function \(\pi _{x_{0}}:\mathbb {R}\rightarrow \mathbb {R}^{n}\) by \(\pi _{x_{0}}(t)\triangleq \pi (t,x_{0})\), which is called the nonlinear dynamical system (2.1) with initial condition \(x(t_{0})=x_{0}\). Note that we use the notation \(\pi _{x_{0}}\), \(t\in \mathbb {R}\), and \(x(t)\), \(t\in I\), interchangeably to denote the solution of (1.1) with initial condition \(x(t_{0}^{+})=x_{0}\).
The positive orbit of (2.1) through the point \(x_{0}\) is given by
$$ \varPi ^{+}(x_{0},t)\triangleq \bigl\{ \pi (t,x_{0})\pi (t_{0}, x_{0})=x_{0}, t \geq t_{0}\bigr\} . $$
We define
$$ \mathcal{M}^{+}(x_{0})= \biggl(\bigcup _{t>t_{0}}\varPi ^{+}(x_{0},t) \biggr) \cap \mathcal{M} $$
and the exit function (the resetting time) \(\tau (x):\mathcal{K} \rightarrow (t_{0},+\infty ]\), where \(\tau (x)\) is defined as follows. For a point \((\bar{t},\bar{x})\) on the trajectory of (2.1), \(\tau (\bar{x})=\hat{\tau }>\hat{t}\) means that \(\pi _{\bar{x}}( \hat{\tau })\notin \mathcal{M}\) for \(\bar{t}< t<\bar{t}+\hat{\tau }\). This means that \(\tau (x)\) is the time of the trajectory of (2.1) from the initialization to the first intersection with the impulsive set (the resetting set) \(\mathcal{M}\).
Remark 2.1
According to the definition of \(\tau (\cdot )\), we may know that \(\tau (x)>0\) for \(x\notin \mathcal{M}\) and \(\tau (x)=0\) for \(x\in \mathcal{M}\). Furthermore, if \(\mathcal{M}^{+}(x)=\emptyset \), then \(\tau (x)=\infty \).
The impulsive dynamical system (1.1) is called discontinuous dynamical system (DDS) when \(\mathcal{M}^{+}(x_{0}) \neq \emptyset \) [24, 31, 32].
Definition 2.1
(Viable solution)
A solution \(\pi _{x_{0}}(t)\) of (2.1) on \(\mathbb {R}_{+}\) with initial condition \(x(0)=x_{0} \in \mathcal{K}\) is said to be viable in the viability constraints \(\mathcal{K}\subset \mathbb {R}^{n}\) on \(\mathbb {R}_{+}\) if, for every time \(t\geq 0\), \(\pi _{x_{0}}(t)\in \mathcal{K}\).
It is apparent that DDS (1.1) is equal to the continuoustime dynamical system (2.1) if \(\mathcal{M}^{+}(x _{0})=\emptyset \). In this case, for every point \(x\in \mathcal{M}\), we consider the Bouligand tangent cone
$$ T_{\mathcal{K}}(x):= \biggl\{ v\in \mathbb {R}^{n}\Big\lim _{h\rightarrow 0^{+}} \frac{1}{h}d_{\mathcal{K}}(x+hv)=0 \biggr\} . $$
(2.2)
Condition (2.2) means that the vector field f is tangent to \(\mathcal{M}\). If \(f(x)\in T_{\mathcal{K}}(x)\) holds for all \(x\in \mathcal{M}\), then all solutions \(\pi _{x}(t)\) of (2.1) are viable in \(\mathcal{K}\) (see [28, 29]). If \(f(x)\notin T_{\mathcal{K}}(x)\), then the trajectory \(\varPi ^{+}(x_{0},t)\) of (2.1) through \(x\in \mathcal{M}\) is, in some sense, transversal to \(\mathcal{M}\), hence there exists at least one solution of (2.1) leaving the viability constraints \(\mathcal{K}\).
For statedependent impulsive system (1.1), we make the following hypotheses.

\(\mathit{(H1)}\)
:

\(\mathcal{M}\neq \emptyset \) and there exists a continuously differentiable function \(H:\partial {\mathcal{K}}\rightarrow \mathbb {R}\) such that the hypersurface \(\mathcal{M}\) is defined by
$$ \mathcal{M}\triangleq \bigl\{ x\in \partial {\mathcal{K}}H(x)=0 \text{ and } \nabla H(x)\neq 0\bigr\} . $$
(2.3)

\(\mathit{(H2)}\)
:

\(J: \mathcal{M}\rightarrow \mathcal{N}\) is a continuous differentiable function, and \(\det [\frac{\partial J(x)}{ \partial x} ]\neq 0\) for \(x\in \mathcal{M}\).
Assume that \(\mathit{(H1)}\) holds, then it follows from the implicit function theorem [33] that, for every \(x\in \mathcal{M}\), there exist a number j and a function \(h_{x}(x_{1}, \ldots ,x_{j1},x_{j+1},\ldots ,x_{n})\) such that \(\mathcal{M}\) is the graph of the function \(x_{j}=h_{x}(x_{1},\ldots ,x_{j1},x_{j+1}, \ldots ,x_{n})\) in the neighborhood of x. Assume that \(\mathit{(H2)}\) holds, then it follows from the inverse function theorem [33] that there exists a unique function \(J^{1}:\mathcal{N}\rightarrow \mathcal{M}\) such that \((J^{1}\circ J)(x)=x\) and \(J^{1}(x)\neq x\) for all \(x\in \mathcal{N}\). Furthermore, if \(\widetilde{H}(x)=H(J^{1}(x))\) for any \(x\in \mathcal{N}\), then \(\mathcal{N}=\{x\in \mathcal{K} \widetilde{H}(x)=0\}\). It follows from (2.3) that we can easily prove that \(\nabla \widetilde{H}(x)\neq 0\) [24, 31]. Furthermore, we make the following assumptions:

\(\mathit{(H3)}\)
:

\(\overline{\mathcal{N}}\cap \mathcal{M}=\emptyset \), where \(\overline{\mathcal{N}}\) is the closure of \(\mathcal{N}\).

\(\mathit{(H4)}\)
:

The vector field \(f(x)\) satisfies the following transversality condition: \(\langle \nabla H (x),f(x)\rangle \neq 0 \) for all \(x\in \mathcal{M}\).

\(\mathit{(H5)}\)
:

For \(x\in \mathcal{N}\), \(\langle \nabla \widetilde{H}(x),f(x)\rangle \neq 0\).
Hypothesis \(\mathit{(H3)}\) ensures that when the trajectory of (1.1) meets the surface of \(\mathcal{M}\), it instantaneously leaves \(\mathcal{M}\). This means that the points of \(\mathcal{M}\) are isolated on every trajectory of (1.1). If hypothesis \(\mathit{(H4)}\) holds, then \(\mathcal{M}\) is said to be transversal to the vector field f, it is also called a cross section. In other words, the vector field \(f(x)\) is not tangent to the surface \(\mathcal{M}\) (see Fig. 1). Hypothesis \(\mathit{(H5)}\) implies that the hyperplane \(\mathcal{N}\) is not tangent to the solution of (1.1).
In order to define a solution of (1.1), we need the following definition.
Definition 2.2
A function \(\widetilde{\pi }_{x_{0}}:[t_{0},t_{f})\rightarrow \mathcal{K}\), \(t_{f}\in \mathbb {R}\cup {\infty }\), \(t_{f}>t_{0}\), is a solution of (1.1) with initial condition \(x(t_{0}^{+})=x_{0} \in \mathcal{K}\) if the following conditions are satisfied:

(i)
\(\widetilde{\pi }_{x_{0}}(t)\) is right continuous on \([t_{0},t_{f})\);

(ii)
For every \(t\in [t_{0},t_{f})\), left and right limits of \(\widetilde{\pi }_{x_{0}}(t)\) exist, denoted by \(\widetilde{\pi }_{x _{0}}^{}(t)\triangleq \lim_{s\rightarrow t^{}}\widetilde{\pi }_{x _{0}}(s)\) and \(\widetilde{\pi }_{x_{0}}^{+}(t)\triangleq \lim_{s\rightarrow t^{+}}\widetilde{\pi }_{x_{0}}(s)\);

(iii)
There exists a closed discrete subset \(\mathcal{I}_{x _{0}}\subset [t_{0},t_{f})\) called impulsive times such that (a) for \(t\notin \mathcal{I}_{x_{0}}\), \(\widetilde{\pi } _{x_{0}}(t)\) is differentiable, \(\frac{\mathrm {d}\widetilde{\pi }_{x_{0}}(t)}{\mathrm {d}t}=f(\widetilde{\pi } _{x_{0}}(t))\), and \(\widetilde{\pi }_{x_{0}}(t)\notin \mathcal{M}\); (b) for \(t\in \mathcal{I}_{x_{0}}\), \(\widetilde{\pi }_{x _{0}}^{}(t)\in \mathcal{M}\) and \(\widetilde{\pi }_{x_{0}}^{+}(t)=J( \widetilde{\pi }_{x_{0}}^{}(t))\).
If \(\mathcal{M}^{+}(x_{0})=\emptyset \), then \(\widetilde{\pi }_{x_{0}}(t)= \pi _{x_{0}}(t)\), that is, the trajectory \(\widetilde{\varPi }^{+}(x_{0},t)\) does not intersect with impulse surface \(\mathcal{M}\), there is no impulsive effect. Thus, the trajectory \(\widetilde{\varPi }^{+}({x_{0}},t)\) starting at the initial point \({x_{0}}\in \mathcal{K}\) will remain in the viability constraints \(\mathcal{K}\) forever. Therefore, by the existence and uniqueness theorem for ordinary differential equation, \(\widetilde{\pi }_{x_{0}}(t)\) exists and is unique on an interval \([0,t_{f})\) as a viable solution of system (2.1).
However, if \(\mathcal{M}^{+}({x_{0}})\neq \emptyset \), then \(\tau (x_{0})<+\infty \). Thus, there exists a smallest positive time \(\tau _{1}\triangleq \tau (x_{0})\) such that \(x_{1}\triangleq \pi _{x _{0}}(\tau _{1})\in \mathcal{M}\) and \(\pi _{x_{0}}(t)\notin \mathcal{M}\) for \(t_{0}< t<\tau _{1}\). Furthermore, \(x_{1}\) is instantaneously transferred to \(x_{1}^{+}\triangleq J(x_{1})\). Then we define \(\widetilde{\pi }_{x}\) on \([t_{0},t_{1}]\) by
$$\begin{aligned}& \widetilde{\pi }_{x_{0}}(t)= \textstyle\begin{cases} \pi _{x_{0}}(t), & t_{0}\leq t< t_{1}, \\ x_{1}^{+}, &t=t_{1}, \end{cases}\displaystyle \end{aligned}$$
where \(\widetilde{\pi }_{x_{0}}(0^{+})=x_{0}\) and \(t_{1}\triangleq \tau _{1}\). Further, if \(\mathcal{M}^{+}({x_{1}^{+}})=\emptyset \), then we define \(\widetilde{\pi }_{x_{0}}(t)=\pi _{x_{1}^{+}}(t\tau _{1})\) for \(\tau _{1}\leq t<+\infty \) and \(\tau (x_{1}^{+})=+\infty \). That is to say, the trajectory \(\widetilde{\varPi }^{+}({x_{0}},t)\) starting at the initial point \({x_{0}}\in \mathcal{K}\) meets the surface \(\mathcal{M}\) only once and does not hit the surface \(\mathcal{M}\) beyond the time \(t=\tau _{1}\). On the other hand, if \(\mathcal{M}^{+}({x_{1}^{+}}) \neq \emptyset \), then there exists a smallest positive time \(\tau _{2}\triangleq \tau (x_{1}^{+})\) such that \(x_{2}\triangleq \pi _{x_{1}^{+}}(\tau _{2})\in \mathcal{M}\) and \(\pi _{x_{1}^{+}}(t\tau _{1})\notin \mathcal{M}\), for \(\tau _{1}< t<\tau _{1}+\tau _{2}\). Moreover, \(x_{2}\) jumps to point \(x_{2}^{+}\triangleq J(x_{2})\). Therefore, we define \(\widetilde{\pi }_{x}\) on \([t_{1}, t_{2}]\) by
$$\begin{aligned}& \widetilde{\pi }_{x_{0}}(t)= \textstyle\begin{cases} \pi _{x_{1}^{+}}(tt_{1}), & t_{1}\leq t< t_{2}, \\ x_{2}^{+}, &t=t_{2}, \end{cases}\displaystyle \end{aligned}$$
where \(t_{2}=\tau _{1}+\tau _{2}\). Repeating this process for \(x_{k}^{+}\), \(k=2,3,\ldots \) , we can define \(\widetilde{\pi }_{x_{0}}\) on each \([t_{k},t_{k+1}]\) by the following:
$$\begin{aligned}& \widetilde{\pi }_{x_{0}}(t)= \textstyle\begin{cases} \pi _{x_{k}^{+}}(tt_{k}), & t_{k}\leq t< t_{k+1}, \\ x_{k+1}^{+}, &t=t_{k+1}, \end{cases}\displaystyle \end{aligned}$$
where \(t_{k}=\sum_{i=1}^{k}\tau _{i}\), \(\tau _{i}\triangleq \tau (x_{i1} ^{+})\), \(t_{0}=0\), and \(x_{k}^{+}\triangleq J(x_{k})\). Therefore, the solution \(\widetilde{\pi }_{x_{0}}(t)\) of (1.1) is defined on the interval \([t_{0},t_{k+1}]\) (see Fig. 2). If \(\mathcal{M}^{+}(x_{k}^{+})=\emptyset \) for some k, then the trajectory \(\widetilde{\varPi }^{+}(x_{0},t)\) of (1.1) with initial condition \(x(t_{0}^{+})=x_{0}\in \mathcal{K}\) will intersect the impulsive set \(\mathcal{M}\) finitely many times (k times) and will remain in the viability constraints \(\mathcal{K}\) forever. Then there exists a solution of (1.1), and \(\widetilde{\pi }:[\tau _{k},t_{f})\rightarrow \mathcal{K}\) is a maximal solution of (2.1). If \(\mathcal{M}^{+}(x_{k}^{+})\neq \emptyset \) for all \(k=1,2,\ldots \) , then \(\widetilde{\pi }_{x_{0}}(t)\) is defined on the interval \([t_{0},t_{f})\). Furthermore, a maximal interval of the existence of a solution does not exist since \([t_{0},t_{f})\) involves a sequence \(\{t_{k}\}_{k=1}^{\infty }\) of impulsive times, where \(t_{k}=\sum_{i=1}^{\infty }\tau _{i}\) and \(\lim_{k\rightarrow \infty }= \infty \).
Remark 2.2
Note that \(\widetilde{\pi }_{x_{0}}(t_{k})\in \mathcal{M}\), \(\widetilde{\pi }_{x_{0}}(t_{k}^{+})\in \mathcal{N}\). Moreover, \(\widetilde{\pi }_{x_{0}}(t_{k})\in \mathcal{M}\) for \(t_{k}>t_{0}\) and \(\widetilde{\pi }_{x_{0}}(t_{k})\in \mathcal{N}\) for \(t_{k}>t_{0}\), where \(t_{k}\in \mathcal{I}_{x_{0}}\).
For given \(x_{0}\in \mathcal{K}\), the positive orbit of (1.1) with initial condition \(x(0^{+})=x_{0}\in \mathcal{K}\) is defined by
$$ \widetilde{\varPi }^{+}(x_{0},t)=\bigl\{ \widetilde{\pi }_{x_{0}}(t)t\in [t _{0},t_{f})\bigr\} . $$
We let \(t_{k}\) denote the kth instant of time at which \(\widetilde{\varPi }^{+}(x_{0},t)\) intersects \(\mathcal{M}\), \(\mathcal{I} _{x_{0}}\) is denoted by \(\{t_{1},t_{2},\ldots ,t_{k},\ldots \}\), where \(t_{0}< t_{1}< t_{2}<\cdots <t_{k}<\cdots \) and \(\lim_{k\rightarrow \infty }t_{k}=\infty \).
Remark 2.3
If \(t_{f}<\infty \), then \(\widetilde{\pi }_{x_{0}}:[t_{0},t_{f}) \rightarrow \mathbb {R}^{n}\) is a maximal solution of (1.1), where \(\mathcal{I}_{x_{0}}\neq \emptyset \), \(\widetilde{\pi }_{x_{0}}:[ \max (\mathcal{I}_{x_{0}}),t_{f})\rightarrow \mathbb {R}^{n}\) is a maximal solution of (1.1), and when \(\mathcal{I}=\emptyset \), \(\widetilde{\pi }_{x_{0}}:[t_{0},t_{f})\rightarrow \mathbb {R}^{n}\) is a maximal solution of (2.1). If \(t_{f}=\infty \), then the solution is obviously maximal.
We shall use \(\mathcal{PC}([t_{0},t_{f}),\mathbb {R}^{n})\) to denote the class of piecewise continuous functions from \([t_{0},t_{f})\) to \(\mathbb {R}^{n}\), with discontinuities of the first kind only at \(t=t_{k}\), \(k=1,2, \ldots \) . Thus, \(\widetilde{\pi }_{x_{0}}(t)\in \mathcal{PC}^{1}([t _{0},t_{f}),\mathbb {R}^{n})\).
Now we give the Schauder fixed point theorem, the definitions of the impulsive viable solution and continuation of the solution of (1.1).
Theorem 2.1
(Schauder fixed point theorem [33])
Let
\(\mathcal{C}\subseteq \mathbb {R}^{n}\)
be a nonempty, convex, and closed set, let
\(f:\mathcal{C}\rightarrow \mathcal{C}\)
be continuous, and assume that
\(f(\mathcal{C})\)
is bounded. Show that there exists
\(x\in \mathcal{C}\)
such that
\(f(x)=x\).
Definition 2.3
(Impulsive viable solution)
A solution \(\widetilde{\pi }_{x_{0}}(t) \in \mathcal{PC}^{1}([t_{0},t_{f}),\mathbb {R}^{n})\) of (1.1) on the interval \([t_{0},t_{f})\) with initial condition \(x(0^{+})=x_{0}\) is said to be viable in the viability constraints \(\mathcal{K}\subset \mathbb {R}^{n}\) on \([t_{0},t_{f})\) if, for every time \(t\in [t_{0},t_{f})\backslash \mathcal{I}_{x_{0}}\), \(\widetilde{\pi }_{x_{0}}(t)\in \mathcal{K}\).
Definition 2.4
([24])
A solution \(\widetilde{\pi }_{x_{0}}(t)\) of (1.1) is said to be continuable to a set \(U\in \mathbb {R}^{n}\) as time decreases (increases) if there exists a time \(s\in \mathbb {R}\) such that \(s\leq 0\) (\(s\geq 0\)) and \(\widetilde{\pi }_{x_{0}}(s)\in U\).
In order to obtain the sufficient conditions of continuation of the solutions of (1.1), we make the following hypotheses:

\(\mathit{(H6)}\)
:

\(\sup \f(x)\<+\infty \) for all \(x\in \mathcal{K}\).

\(\mathit{(H7)}\)
:


\(\mathit{(a)}\)
:

Every solution \(\pi _{x_{0}}(t)\), \(x_{0}\in \mathcal{K}\), of (2.1) is continuable to either ∞ or \(\mathcal{M}\) as time increases.

\(\mathit{(b)}\)
:

Every solution \(\pi _{x_{0}}(t)\), \(x_{0}\in \mathcal{K}\), of (2.1) is continuable to either −∞ or \(\mathcal{N}\) as time decreases.