# Precise Mission Process Control Based on a Novel Dual-Code Group Network Plan Diagram
## 5 Mission Process Control Strategy
### Control Strategy Solution
#### 5.2.2 Controllable Mission Validity  
(43) and (44) determine the controllable missions at the current moment. However, in order to realize efficient and precise control, we need to determine the validity of each controllable task, i.e., for any moment \(t\), only the tasks that are controlled must change the output, and only those missions must have no effect on the output. We give definitions of two types of controllable mission. **Definition 7**. : _(Valid Controllable Mission). A controllable mission \(m_{i}\) is a valid controllable mission if, at a given moment \(t\), the output \(y(t)\) decreases with the varying mission duration \(x_{m_{i}}(t)\) in the case of only varying the mission duration \(x_{m_{i}}(t)\), i.e.,_  
\[x_{m_{i}}(t)\propto y(t) \tag{49}\]  
**Definition 8**. : _(Invalid Controllable Mission). A controllable mission \(m_{i}\) is an invalid controllable mission, if at moment \(t\), the mission duration \((x_{m_{i}}(t))\) decreases but the output \(y(t)\) remains unchanged in the case of only varying the mission duration \(x_{m_{i}}(t)\)._  
Before we give the discriminative conditions for the effectiveness of controllable missions, we first define the concepts of core critical path and critical path. **Definition 9**. : _(Core Critical Path). The core critical path is a specified critical path in the set of critical paths \(\tilde{\mathcal{K}}(t)\) obtained by Algorithm 2 that contains the constant critical mission \(m^{*}\) at the current moment \(t\). We denote it by \(\mathcal{K}^{*}(t)\), where \(\mathcal{K}^{*}(t)\subseteq\tilde{\mathcal{K}}(t)\) and \(m^{*}\in\mathcal{K}^{*}(t)\)._  
**Definition 10**. : _(Critical Path). The critical path is the part of the core critical path from the starting mission to the constant critical mission \(m^{*}\), denoted by \(\hat{\mathcal{K}}^{*}(t)=\{m_{1}^{*},m_{2}^{*},\ldots,m_{n}^{*},m^{*}\}\), where for \(m_{i}^{*}\in\hat{\mathcal{K}}^{*}(t)\), all have \(T_{PF}^{m_{i}^{*}}(t)\leq T_{PF}^{m^{*}}(t)\)._  
As shown in Figure 11, there are two critical paths in the set of critical paths \(\tilde{\mathcal{K}}(t)\), which are \(\mathcal{K}_{1}(t)=\{m_{1}^{*},m_{2}^{*},m_{3}^{*},m_{4}^{*},m^{*},m_{7}\}\) and \(\mathcal{K}_{2}(t)=\{m_{1}^{*},m_{6},m_{4}^{*},m^{*},m_{7}\}\). Both critical paths contain \(m^{*}\), and we arbitrarily specify one of the critical paths as the core critical path \(\hat{\mathcal{K}}^{*}(t)\), such that \(\mathcal{K}_{1}(t)=\mathcal{K}^{*}(t)\). Furthermore, the path consisting of \(\{m_{1}^{*},m_{2}^{*},m_{3}^{*},m_{4}^{*},m^{*}\}\) is the core path \(\hat{\mathcal{K}}^{*}(t)\). According to the conclusion of Theorem 2, the core critical paths and core paths always exist since constant critical missions are always located on the critical paths. Based on the concept of the critical path, the output \(y(t_{c})\) of any controllable moment \(t_{c}\) is equal to the difference between the length of the critical path \(\hat{\mathcal{K}}^{*}(t_{c})\) and the length of the constant critical mission at the moment \(t_{c}\). In accordance with the control condition illustrated in (48), at the current controllable moment \(t_{c}\), there is no unfinished delayed mission, and the entire system is deterministic. According to (42), the output \(y(t_{c})\) can be expressed as  
\[ \begin{split} y(t_{c})&=\sum_{m_{1}^{*}\in \mathcal{K}(t_{c}),T_{PS}^{m_{i}^{*}}(t_{c})<T_{PS}^{m_{i}^{*}}(t_{c})}x_{m_{i} ^{*}}(t_{c})\\ &=\sum_{m_{1}^{*}\in\hat{\mathcal{K}}^{*}(t_{c})}x_{m_{i}^{*}}(t_ {c})-x_{m^{*}}(t_{c})\end{split}  \tag{50}\]  
In this paper, a discriminant theorem for the validity of controllable missions based on the concept of critical paths is presented. **Theorem 5**.