## Problem Formulation and Preliminaries  
### Fsm  
In this section, the FSM and the related definitions will be introduced. Consider the FSM \(G=(\Xi,\Sigma,\zeta)\) described by  
\[\left\{ \begin{array}{l}q(t_{k+1})\in\varphi(q(t_{k}),\sigma(t_{k+1}))\\ \sigma(t_{k+1})\in\varphi(q(t_{k}),x(t_{k+1}))\\ \psi(t_{k+1})=\zeta(q(t_{k}),\sigma(t_{k+1}))\end{array} \right. \tag{1}\]  
where \(q(t_{k})\in\Xi\), \(\sigma(t_{k})\in\Sigma\), and \(\psi(t_{k})\in\Psi\) are, respectively, the discrete state, the input event and the output of FSM at time \(t_{k}\), and \(t_{k}\) is the \(k-th\) jumping instant. \(\Xi=\{q_{1},\ldots,q_{m}\}\), \(\Sigma=\{\sigma_{1},\ldots,\sigma_{m}\}\) and \(\Psi\subseteq\Xi\) are the finite sets of states, input events and outputs, respectively. \(\varphi:\Xi\times\Sigma\to 2^{\Xi}\) is the FSM transition function, \(\phi:\Xi\times X\to 2^{\Xi}\) is the function specifying the possible events, and \(\zeta:\Xi\times\Sigma\to\Psi\) is the output function. \(x(\cdot)\in X\) is the continuous system state, \(X\in\mathbb{R}^{n}\), and \(q(t)\) is the current node of FSM. **Remark 1**. : _Note that the state transitions of the hybrid systems depend on the current node evolution of FSM, and the current node \(q(t)\in\Xi\) specifies the corresponding subsystem of continuous dynamics being active. \(q(t)=q(t^{+})\) means there is no jumping, and \(q(t)\
eq q(t^{+})\) means the object will jump from the current node to another._  
**Remark 2**. : _From (1), it can be seen that the state transition may be not unique when a deterministic input event occurs. This type of FSM is the non-deterministic FSM, which can be described by a state transition graph. For example, Figure 1 shows an FSM with five nodes, where the set of node is \(\Xi=\{q_{1},q_{2},q_{3},q_{4},q_{5}\}\), and the set of the input event is \(\Sigma=\{\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4}\}\). The jumping rule of FSM is that the current node jumps from one node to another when a particular input event occurs. However, there may be more than one node that the current node could jump to, thus, except the jump which is needed, all other jumps should be limited via controlling. After the jump, the limitation should be released to maintain the liveliness of FSM. For the FSM shown in Figure 1, the current node is \(q_{5}\), and when the input event \(\sigma_{4}\) occurs, the current node jumps from node \(q_{5}\) to \(q_{4}\), then the current node is \(q_{4}\). At this time, if the input event \(\sigma_{1}\) occurs, the current node can jump from \(q_{4}\) to \(q_{3}\) or to \(q_{2}\). The jump from \(q_{4}\) to \(q_{3}\) can be limited, and then the current node jumps from \(q_{4}\) to \(q_{2}\). After that, the limitation will be released._  
**Definition 1**. : _Given FSM (1) and any \(q_{i},q_{j}\in\Xi\), if there is an input event sequence \(\{\sigma^{1},\cdots,\sigma^{\ell}\}\in\Sigma\) resulting in the current node jumping along the node sequence \(\{q^{0},q^{1},\cdots,q^{\ell}\}\), where \(\ell\in\mathbb{N}\), \(q^{0}=q_{i}\) and \(q^{\ell}=q_{j}\), then the node sequence \(\{q^{0},\cdots,q^{\ell}\}\) is called the path from \(q_{i}\) to \(q_{j}\). The jumping number \(\ell\) is the length of the path from \(q_{i}\) to \(q_{j}\). Let \(h(q_{i},q_{j})\) denote the set of all paths from node \(q_{i}\) to \(q_{j}\). If the path \(\{q^{0},q^{1},\cdots,q^{\ell}\}\in h(q_{i},q_{j})\) has the least number of nodes, then it is called the shortest path from \(q_{i}\) to \(q_{j}\). Furthermore, the length of the shortest path from \(q_{i}\) to \(q_{j}\) is called the distance from \(q_{i}\) to \(q_{j}\), and it can be denoted by \(d(q_{i},q_{j})=\ell\). In the case that there is no path from \(q_{i}\) to \(q_{j}\), the distance is denoted by \(d(q_{i},q_{j})=\infty\), \(i,j,\ell\in\mathbb{N}\)._  
**Remark 3**.