## 4 Distributed Power Control for Spectrum Sensor Nodes
### Cooperative Optimal Solution  
In this subsection, we move on to explore the distributed cooperative power control (CoPC) problem for all IT and MT SSNs by building up the cooperation scenario if all the players play cooperatively via the differential game model \(\mathcal{G}\). Note that this is a natural idea for the players aiming to achieve an optimal power allocation through full cooperation for their common interests. Under this scenario, our optimization objective is to maximize the sum of the utility functions of all players (i.e., \(\sum_{m\in\mathcal{N}}U_{m}\)) while the linear differential equation constraint of the energy consumption should also be satisfied for the \(m\)-th SSN. To be specific, we formulate a dynamic optimization problem \(\mathcal{P}2\) as follows to attain the objective of maximizing the sum of the utility functions of all players:  
\[ \begin{array}{ll}\mathcal{P}2:&\max_{p_{1}(t),\cdots,p_{m}(t),\cdots,p_{| \mathcal{N}|}(t),m\in\mathcal{N}}&\sum\limits_{m\in\mathcal{N}}\int_{t_{0}^{ \prime}}^{t_{0}^{\prime}+\tau_{rp}}\big{(}\wp_{m}^{R}(p_{m}^{\max}-p_{m}(t))- \wp_{m}^{C}E_{m}(t)\big{)}e^{-r(t-t_{0}^{\prime})}dt\\ &\text{s.t.}&\frac{dE_{m}(t)}{dt}=E_{A}(t)-\tau_{rp}p_{m}(t)-E_{m}(t)\end{array}  \tag{31}\]  
Under the cooperation scenario, we use \(p_{m}^{C}(t)\) to represent the cooperative optimal solution to the problem \(\mathcal{P}2\), and assume that there exists a continuously differentiable function \(\mathcal{W}_{m}^{C}(p_{m},E_{m})\) satisfying the following partial differential equation:  
\[r\mathcal{W}_{m}^{C}(p_{m},E_{m})=\max\bigg{(}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
Similar to Proposition 2, it is also revealed that an increased discount factor \(r\) will enhance the cooperative optimal transmit power \(p_{m}^{C}(t)\) for the \(m\)-th SSN. According to Equation (34), the cooperative optimal transmit power vector \(\mathbf{P}^{C}\) for all IT and MT SSNs in the CoPC problem can be also combined as follows:  
\[\mathbf{P}^{C}=\left\{\underbrace{p_{1}^{C}(t),\cdots,p_{|\mathcal{N}_{IT}|}^{C }(t)}_{\text{for }n_{IT}\text{IT}\text{TSSNs}},\overbrace{p_{|\mathcal{N}_{IT}|+1}^{C}(t), \cdots,p_{|\mathcal{N}_{IT}|+|\mathcal{N}_{MT}|}^{\text{for }n_{MT}\text{MT SSNs}}}^{\text{for }n_{MT}\text{MT SSNs}}}_{ \text{for }n_{MT}\text{IT}+|\mathcal{N}_{MT}|}(t)\right\} \tag{35}\]  
Define the notation \(\Psi_{m}=P_{m}^{\max}H_{m}\Big{(}\Upsilon_{d}(m)\|\Upsilon_{f}(m)\Big{)}\) for simplicity in the following. Given the cooperative optimal solution \(p_{m}^{C}(t)\), by plugging \(\mathcal{W}_{m}^{C}(p_{m},E_{m})\) and \(p_{m}^{C}(t)\) into Equation (32), the function \(\mathcal{W}_{m}^{C}\big{(}p_{m}^{C}(t),E_{m}^{C}(t)\big{)}\) for the \(m\)-th SSN is further given by:  
\[ \begin{array}{ll}\mathcal{W}_{m}^{C}\big{(}p_{m}^{C}(t),E_{m}^{C}(t)\big{)}& =\frac{\tau_{tp}^{2}}{4r(r+1)^{2}}\left(\sum\limits_{m\in\mathcal{N}}\left( \Psi_{m}\left(\sum\limits_{m\in\mathcal{N}}\wp_{m}^{C}\right)^{2}\right)-2 \Psi_{m}\left(\sum\limits_{m\in\mathcal{N}}\wp_{m}^{C}\right)^{2}\right)\\ &+\frac{\left(E_{m}^{C}(t)-E_{A}(t)\right)\sum\limits_{m\in\mathcal{N}}\wp_{ m}^{C}}{r(r+1)}+\frac{\tau_{tp}p_{m}^{\max}\sum\limits_{m\in\mathcal{N}}\wp_{ m}^{C}}{r}-\frac{1}{r}\sum\limits_{m\in\mathcal{N}}\left(\wp_{m}^{C}E_{m}^{C}(t) \right)\end{array}  \tag{36}\]  
where \(E_{m}^{C}(t)\) is the cooperative optimal energy consumption value of the \(m\)-th SSN at time \(t\). By using \(\mathbf{P}^{NC}\) of the NoCoPC problem and \(\mathbf{P}^{C}\) of the CoPC problem, we then design a distributed optimal transmit power adjustment (OTPA) algorithm as stated in Algorithm 2. Recall that the pricing factor \(\wp_{m}^{R}\) is employed to characterize the efficiency-to-stability ratio for the \(m\)-th SSN, for \(m\in\mathcal{N}\), aiming to describe the power reduction efficiency under the stability of local spectrum sensing.