# Resource Allocation in Unmanned Aerial Vehicle (UAV)-Assisted Wireless-Powered Internet of Things
## 4 Optimal Solutions for Energy Transfer  
In this section, we discuss the solutions to the resource allocation problem given in (5) and (6). The UAV is considered to be the leader in the proposed dynamic game. The UAV should achieve optimal power control for energy transfer based on (5) and (6), and the optimal unit energy price can also be achieved. In order to solve the above problems, a feedback Nash equilibrium can be characterized as follows. **Definition 1. **_For each nodes category, there exists optimal feedback solutions, which are denoted by \(\left\{\pi_{k}^{*}(t),p_{k}^{*}(t)\right\}\)\((k=\{H,L\})\), if continuously differentiable functions \(V^{k}(t,x)\) exist for each nodes category \(k=\{H,L\}\), and \(V^{k}(t,x)\) should satisfy the following differential equations [29]:_  
\[-V_{t}^{H}(t,x)=\max_{\pi_{H}(t),p_{H}(t)}\left\{ \begin{array}{l}\left[\sum \limits_{i=1}^{N_{ll}}\left[\gamma_{L,i}(p_{H,i}(t))-\gamma_{th}\right]^{2}- \pi_{L}(t)p_{L,i}(t)\right]-c_{L}p_{L}^{2}(t)\\ -\varepsilon_{H}^{L}\sum\limits_{i=1}^{N_{ll}}p_{L,i}(t)-\varepsilon_{L}^{H} \sum\limits_{i=1}^{N_{ll}}p_{L,i}(t)-\mu_{H}x(t)\\ +V_{x}^{H}(t,x)\left[\sum\limits_{k=\{H,L\}}\eta_{k}p_{k}(t)+\sum\limits_{i=1 }^{N_{ll}}p_{H,i}(t)+\sum\limits_{i=1}^{N_{ll}}p_{L,i}(t)+\delta x(t)\right] \end{array} \right\} \tag{7}\]  
\[V^{H}(T,x)=s_{H}x(T)e^{-rT} \tag{8}\]  
\[-V_{t}^{L}(t,x)=\max_{\pi_{L}(t),p_{L}^{L}(t)}\left\{ \begin{array}{l}\left[ \sum\limits_{i=1}^{N_{ll}}\left[\left[\gamma_{L,i}(p_{L,i}(t))-\gamma_{th} \right]^{2}-\pi_{L}(t)p_{L,i}(t)\right]-c_{L}p_{L}^{2}(t)\\ -\varepsilon_{L}^{L}\sum\limits_{i=1}^{N_{ll}}p_{L,i}(t)-\varepsilon_{L}^{H} \sum\limits_{i=1}^{N_{ll}}p_{H,i}(t)-\mu_{L}x(t)\\ +V_{x}^{L}(t,x)\left[\sum\limits_{k=\{H,L\}}\eta_{k}p_{k}(t)+\sum\limits_{i=1 }^{N_{ll}}p_{H,i}(t)+\sum\limits_{i=1}^{N_{ll}}p_{L,i}(t)+\delta x(t)\right] \end{array} \right\} \tag{9}\]  
\[V^{L}(T,x)=s_{L}x(T)e^{-rT}, \tag{10}\]  
_where,_  
\[V^{H}(t)=\max\!\!\int_{0}^{T}\left[\sum\limits_{i=1}^{N_{ll}}\!\!\left[\left[ \gamma_{L,i}(p_{H,i}(t))-\gamma_{th}\right]^{2}-\pi_{H}(t)p_{H,i}(t)\right]-c _{H}p_{H}^{2}(t)\\ -\varepsilon_{H}^{H}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!the two categories of wireless nodes based on the objective functions during the observation time \([0,T]\). Substituting Formulas (1) and (3) into Formulas (7) and (9), we can obtain the following differential equations:  
\[-V_{t}^{H}(t,x)=\max_{\pi_{H}(t),p_{H}(t)}\left\{\begin{array}{l}\left[ \begin{array} []{l}\sum\limits_{i=1}^{N_{H}}\biggl{[}-\frac{1}{4}\pi_{L}^{2}(t)\frac{l_{H_{ i}^{2}(t)}^{2}}{\delta_{H_{i}^{2}(t)}^{2}}-\pi_{L}(t)\gamma_{th}\frac{l_{H_{ i}^{2}(t)}}{\delta_{H_{i}^{2}(t)}}\biggr{]}-c_{L}p_{L}^{2}(t)\\ -c_{L}^{H}\sum\limits_{i=1}^{N_{H}}\biggl{[}\frac{1}{2}\pi_{H}(t)\frac{l_{H_{ i}^{2}(t)}^{2}}{\delta_{H_{i}^{2}(t)}^{2}}+\gamma_{th}\frac{l_{H_{i}^{2}(t)}}{ \delta_{H_{i}^{2}(t)}}\biggr{]}-\mu_{L}x(t)\\ -c_{L}^{L}\sum\limits_{i=1}^{N_{L}}\biggl{[}\frac{1}{2}\pi_{L}(t)\frac{l_{L}^{2 }(t)}{\delta_{L_{i}^{2}(t)}^{2}}+\gamma_{th}\frac{l_{L_{i}^{2}(t)}}{\delta_{L_ {i}^{2}(t)}}\biggr{]}-\mu_{H}x(t)\\ +V_{x}^{H}(t,x)\end{array} \right\}, \tag{13}\]  
\[-V_{t}^{L}(t,x)=\max_{\pi_{L}(t),p_{L}(t)}\left\{