## 4 Distributed Power Control for Spectrum Sensor Nodes
### Noncooperative Optimal Solution  
We formulate a dynamic optimization problem \(\mathcal{P}1\) to derive the optimal solution to the differential game model \(\mathcal{G}\) with the objective of the utility function maximization problem under the linear differential equation constraint of the energy consumption for the \(m\)-th SSN. From Equations (21) and (24), the problem \(\mathcal{P}1\) can be formulated as:  
\[ \begin{array}{cc}\mathcal{P}1:&\max\limits_{p_{m}(t),m\in\mathcal{N}}&\int_{I_{ 0}}^{t_{0}^{\prime}+\tau_{tp}}\big{(}\varphi_{m}^{R}(P_{m}^{\max}-p_{m}(t))- \varphi_{m}^{C}E_{m}(t)\big{)}e^{-r(t-t_{0}^{\prime})}dt\\ &\text{s.t.}&\frac{dE_{m}(t)}{dt}=E_{A}(t)-\tau_{rp}p_{m}(t)-E_{m}(t)\end{array}  \tag{25}\]  
For the noncooperation scenario if all the players play noncooperatively, we aim at deriving an optimal solution to the problem \(\mathcal{P}1\) in the distributed noncooperative power control (NoCoPC) problem for all IT and MT SSNs by employing the theory of dynamic programming developed by [PERSON]. Note that the players can abandon the cooperation due to their selfishness and own interests in the NoCoPC problem, e.g., the selfish behavior in forwarding the data sequence of the SSR to bottleneck SU. Specifically, we employ \(p_{m}^{NC}(t)\) to represent the noncooperative optimal solution to the problem \(\mathcal{P}1\), and assume that there exists a continuously differentiable function \(\mathcal{V}_{m}^{NC}(p_{m},E_{m})\) satisfying the following partial differential equation:  
\[r\mathcal{V}_{m}^{NC}(p_{m},E_{m})=\max\Big{(}\varphi_{m}^{R}(P_{m}^{\max}-p_{ m}(t))-\varphi_{m}^{C}E_{m}(t)+\frac{\partial\mathcal{V}_{m}^{NC}(p_{m},E_{m})}{ \partial E_{m}}\big{(}E_{A}(t)-\tau_{rp}p_{m}(t)-E_{m}(t)\big{)}\Big{)} \tag{26}\]  
**Proposition 1**. : _The function \(\mathcal{V}_{m}^{NC}(p_{m},E_{m})\) for the \(m\)-th SSN in Equation (26) should be subject to the partial differential equation constraint as follows:_  
\[\frac{\partial\mathcal{V}_{m}^{NC}(p_{m},E_{m})}{\partial E_{m}}=-\frac{ \varphi_{m}^{C}}{r+1} \tag{27}\]  
**Proof. ** See Appendix A. \(\square\)  
**Proposition 2**. : _The noncooperative optimal solution \(p_{m}^{NC}(t)\) constitutes a Nash equilibrium solution to the problem \(\mathcal{P}1\) if and only if the optimal transmit power for the \(m\)-th SSN can be expressed as:_  
\[p_{m}^{NC}(t)=P_{m}^{\max}\left(1-\frac{H_{m}\Big{(}\Upsilon_{d}(m)\|\Upsilon _{f}(m)\Big{)}\tau_{rp}}{2(r+1)e^{\delta_{i}D_{m}(\mathcal{P}_{d}\|\mathcal{ P}_{f})}}\right) \tag{28}\]  
**Proof. ** Recall the following expression of the noncooperative optimal solution in Equation (A1) on the basis of Proposition 1. By substituting the expression of function \(\mathcal{V}_{m}^{NC}(p_{m},E_{m})\) in Equation (27) into Equation (A1), we can easily obtain the noncooperative optimal transmit power \(p_{m}^{NC}(t)\) which constitutes a Nash equilibrium solution to the problem \(\mathcal{P}1\) can be formulated by Equation (28). \(\square\)  
By observing Proposition 2, it is clear that an increased discount factor \(r\) will enhance the noncooperative optimal transmit power \(p_{m}^{NC}(t)\) for the \(m\)-th SSN.