## 3 System Model and Problem Formulation
### The Specification of Petri Net  
Petri nets are usually applied to characterize asynchronous and concurrent behaviors, which depict the logical relationship between events and deduce the state activities of the system with an algebraic matrix based on network theory. The classical Petri net consists of place, transition, token, and incidence matrix. On this basis, we extended it to be suitable for describing wireless power transmission. For more details, please refer to our previous work [41; 42]. Furthermore, in order to character the state and behavior of multi-UAV aided wireless power and information transfer, we propose a hybrid colored cyber Petri net system, and the definition is as follows. **Definition 1**. : _A 11-tuple\(\sum=(S,T;F,C,W,M;Rd,Wr,Inhibit,Permit,K)\) is said to be a hybrid colored cyber Petri net system (HCCPNS), if it satisfied the following conditions:_  
1. \(N=(S,T;F)\) _is a directed net, which is named the basic net of_ \(\sum\)_;_
2. \(C\) _is the vector of colors with red and green,_ \(C=\{c_{1},c_{2}\}\)_;_
3. _The place, transition, and weight function of directed arc all contain two colors; the red one represents the continuous part and the green one represents the discrete part, i.e.,_ \(M(S)=[M_{C}(S),M_{D}(S)]\)_,_ \(T=[T_{C},T_{D}]\)_,_ \(W=[W_{C},W_{D}]\)_;_
4. \(F\subseteq S\times T\cup T\times S\)_,_ \(Rd\subseteq S\times T\)_,_ \(Wr\subseteq T\times S\) _and_ \(F\cap Rd\cap Wr=\Phi\)_;_ \(F\)_,_ \(Rd\)_,_ \(Wr\)_,_ \(Inhibit\) _and Permit is the flow, read, write, inhibitor, and permission relationship;_
5. \(K=\{k_{L},k_{H}\}\) _is the lower and upper capacity function of_ \(N\)_, and_ \(k_{L}:S\rightarrow\mathbb{R};k_{H}:S\rightarrow\mathbb{R}\)_, where_ \(\mathbb{R}\) _is a set of real numbers;_
6. \(W:F\rightarrow\mathbb{R}\cup Exp(S,\tau)\) _is the weight function, where_ \(Exp(S,\tau)\) _is the set of functional expressions for_ \(\tau\)_;_
7. \(M:F\rightarrow\mathbb{R}\) _is the marking, and_ \(M_{0}\) _is the initial marking._  
**Definition 2**. : _The condition that transition \(t\) will be fired:_  
\[ \begin{array}{l}\forall s\in{}^{\bullet}t:(M(s)-W_{M}(s,t))\in[k_{L}(s),k_{H} (s)]\wedge\\ \forall s\in{}t^{\bullet}:(M(s)+W_{M}(t,s))\in[k_{L}(s),k_{H}(s)]\\ \qquad\qquad\wedge\forall s\in{}t^{\bullet}_{W}:W(t,s)\in[k_{L}(s),k_{H}(s)] \end{array}  \tag{10}\]_If transition \(t\) is fired, it can be recorded as \(M[t\,>\)._  
**Definition 3**. : _Result of transition firing. After transition \(t\) is fired, the original marking \(M\) will be changed to \(M^{\prime}\), then_  
\[M(s)^{\prime}=\left\{ \begin{array}{cl}M(s)-W(s,t)&\quad s\in{{}^{\bullet}}t-{{} ^{\bullet}}\\ M(s)+W(t,s)&\quad s\in{{}^{\bullet}}t-{{}^{\bullet}}t\\ M(s)-W(s,t)+W(t,s)&\quad s\in{{}^{\bullet}}t\cap{{}^{\bullet}}\\ W(t,s)&\quad s\in{{}^{\bullet}}W{{}^{\bullet}}\\ M(s)&\quad s\
otin\left({{}^{\bullet}}t\cap{{}^{\bullet}}\cup{{}^{\bullet}}t{{ }^{\bullet}_{W}}\right)\end{array} \right. \tag{11}\]  
_The dynamic equation of HCCPNS is:_  
\[M(t_{n})=\left[ \begin{array}{c}M_{\mathrm{C}}(t_{n})\\ M_{\mathrm{D}}(t_{n})\end{array} \right]+\left[ \begin{array}{c}A_{\mathrm{C}} \cdot\left[\int_{t_{0}}^{t_{n}}\left(\,\right)\right]d\tau\\ A_{\mathrm{D}}\end{array} \right] \tag{12}\]  
_where \(A_{\mathrm{C}}\) and \(A_{\mathrm{D}}\) are the incidence matrix of the continuous and discrete part, respectively._  
A simple HCCPNS model is shown in Figure 2. Figure 2a is the folded specification, and Figure 2b is the unfolded specification as well as Figure 2c. They represent the continuous parts and the discrete part denoted in red and green, respectively, i.e., \(M(S_{1})=[M_{\mathrm{C}}(S_{1}),M_{\mathrm{D}}(S_{1})]\), \(M(S_{2})=[M_{\mathrm{C}}(S_{2}),M_{\mathrm{D}}(S_{2})]\), \(T=[T_{\mathrm{C}},T_{\mathrm{D}}]\), \(W(S_{1},T)=[W_{\mathrm{C}}(S_{1},T_{\mathrm{C}}),W_{\mathrm{D}}(S_{1},T_{ \mathrm{D}})]\), \(W(T,S_{2})=[W_{\mathrm{C}}(T_{\mathrm{C}},S_{2}),W_{\mathrm{D}}(T_{\mathrm{D}},S_{2})]\).