where $\mathrm{t}$ is the time when the pursuit mission is carried out, $x(t)$ represents the current state of the pursuing UAV, and $u_{\mathrm{p}}(t)$ and $u_e(t)$ are the control vectors of the pursuit UAV and evader UAV at time $\mathrm{t}$.

(1)

The game is classified as a complete information game, meaning that both parties are aware of the necessary information regarding each other’s relative states, with no limitations on observing communication or other conditions.

(2)

The system state is assumed to be accurate, disregarding the effects of sensor errors and delays on information accuracy during operation.

(3)

The maximum acceleration (in absolute value) of the UAV is constrained. The acceleration range is softly constrained using weights in the evaluation function, and the change in acceleration varies gently near the boundary.

where $(S_1,S_2,S_3)$ represents the weight matrix of the final state; $(R_p,R_e)$ denotes the acceleration weight matrix; $\left({d_i}^{\mathrm{T}}{S}_1{d}_i\right)$ is the weighted square of the relative distance between the pursuit UAV and evader UAV; the optimization objectives of these terms are to minimize the relative distance and relative velocity, ensuring that the pursuer approaches the evader more effectively, thereby enhancing the tracking accuracy of the UAV.$\left({d_i}^{\mathrm{T}}{S}_2{u}_{pi}\right)$ is the weighted projection of the relative velocity over the relative distance; $\left({u_{pi}}^{\mathrm{T}}{S}_3{u}_{pi}\right)$ is the weighted square of the relative velocity; $\left(a_{pi}^{ \mathrm{T}}{R}_{\mathrm{p}}{a}_{\mathrm{p}i}\right)$ represents the weighted square of pursuit UAV’s acceleration; $\left(a_{ei}^{ \mathrm{T}}{R}_{\mathrm{e}}{a}_{\mathrm{ei}}\right)$ is the weighted square of the evader UAV’s acceleration; the square of acceleration is closely related to energy consumption, and this part aims to constrain acceleration to reduce energy consumption. $t_f$ represents the end time of the game. The design of the payoff function balances static terminal performance and dynamic performance. The specific values of the weight matrices $(S_1,S_2,S_3)$ and $(R_p,R_e)$ determine the performance optimization priorities. For instance, a higher weight on $S_1$ prioritizes the optimization of tracking accuracy, while higher weights on $(R_p,R_e)$ emphasize energy efficiency.

where ${u_{\mathrm{p}}}^{*}$ and ${u_e}^{*}$ represent the Nash equilibrium solutions of the pursuit UAV and evader UAV, respectively. When the pursuit–evasion system adopts this combination of equilibrium solutions as the pursuit strategy, the system achieves a dynamic equilibrium state. In the differential game-based pursuit–evasion model, it is assumed that both the pursuit UAV and the evader UAV have independent decision-making systems and are unaware of each other’s current strategy choices. When the pursuit UAV selects a control input ${u_{\mathrm{p}}}^{*}$, regardless of the control input $u_e$ chosen by the evader, the pursuit UAV’s payoff is always greater than or equal to that achieved by any other strategy. Similarly, when the evader UAV selects the control quantity ${u_e}^{*}$, its payoff is always greater than or equal to that achieved by any other strategy, regardless of the pursuer’s choice of $u_{\mathrm{p}}$. At this point, the system is considered to have reached the saddle point $\left({u_{\mathrm{p}}}^{*},{u_e}^{*}\right)$ of the differential game, and $J\left({u_{\mathrm{p}}}^{*},{u_e}^{*}\right)$ is the value of the differential game. If the pursuit VAV fails to capture the evader UAV within the specified time frame, the pursuit mission is deemed a failure. The real-time dynamic target point distribution diagram of the pursuit UAVs and evader UAV based on the differential game algorithm is shown in Figure 4.

where ${\alpha }^k$, ${\beta }^k$ represent the iteration steps.

where ${\mathrm{k}}_1,{\mathrm{k}}_2,{\mathrm{k}}_3,{\mathrm{k}}_4>0$, $v_{\max ,i}$ > 0, represents the maximum allowable speed of the i-th UAV, and the saturation function is formulated to ensure that $\mathbf{v}$ does not exceed $v_{\max ,i}$.

where $v_e$ is the desired speed of the UAV capture.

where ${\mathcal{N}}_{\mathrm{m},i}$ is the neighborhood of the i-th UAV, which is the set of other UAVs that could potentially collide with it. ${\mathrm{r}}_{\mathrm{s}}$ ($r_{\mathrm{a}}>r_{\mathrm{s}}$) is defined as the UAV safety radius, and collision prevention begins when the distance between UAVs exceeds $r_{\mathrm{s}}+r_{\mathrm{a}}$. The function value reaches its maximum when the UAVs distance equals $2r_{\mathrm{s}}$. The objective of the speed command ${\mathbf{u}}_{2,i}$ is to minimize its speed value, ideally approaching zero, which implies that the i-th UAV will not collide with the j-th UAV.

where $A=-2/{\left(d_1-d_2\right)}^3$, $B=3\left(d_1+d_2\right)/{\left(d_1-d_2\right)}^3$, $C=-6d_1{d}_2/{\left(d_1-d_2\right)}^3$, and $D=d_2^2\left(3d_1-d_2\right)/{\left(d_1-d_2\right)}^3$. This design is suitable for avoiding drastic changes in speed or acceleration near the critical distance, ensuring smoother and more stable flight trajectories for the UAV.

where ${\sigma }_{\mathrm{t}}(x)\triangleq \sigma \left(x,r_{\mathrm{s}},r_{\mathrm{a}}\right)$, and $x_e$ is the center point for the closed circular pipeline. $d_{\mathrm{t},i}$ represents the distance between the UAV and the closed circular pipe wall. No force is applied when the distance exceeds $r_{\mathrm{a}}$; the force begins to act when the distance is less than $r_{\mathrm{a}}$, reaching its maximum when the distance is exactly $r_{\mathrm{s}}$. As the pipe width remains constant, the velocity command is designed to minimize the obstacle avoidance function, ideally reducing it to zero. This ensures that the i-th UAV and its safety zone stay within the closed circular pipe.

where ${\sigma }_{\mathrm{e}}(x)\triangleq \sigma \left(x,2l_{\mathrm{e}},2\alpha l_{\mathrm{e}}\right)$, $\alpha >1$, and $l_{\mathrm{e}}={\displaystyle \frac{L}{2N}}$. To achieve an approximately uniform capture, the circumference of the centerline of the closed circular pipeline is calculated as $L=2\pi r_l$. The relative positions of the two UAVs are represented by the length of the corresponding curve inside the pipeline $l_{\mathrm{ij}}=L{\displaystyle \frac{\theta }{2\pi }}$;$\theta <\pi$ is the angle between the lines connecting two adjacent capture UAVs and the target UAV. When the distance $l_{\mathrm{ij}}$ is greater than $2\alpha l_{\mathrm{e}}$, no force is applied; when the distance is less than $2\alpha l_{\mathrm{e}}$, force starts to be applied; and when the distance equals $2l_{\mathrm{e}}$, the force is maximized. The velocity command ${\mathbf{u}}_{4,i}$ is designed to approach zero or be as small as possible to ensure $l_{ij}=2l_e$. Uniform target capture is achieved when all UAVs satisfy these conditions.

where $V_1={\displaystyle \frac{1}{2}}{{\displaystyle \sum _{\mathrm{i}=1}^N‖v_i-v_{\mathrm{e}}‖}}^2$, $V_2={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i,j\in {\mathcal{N}}_i}{\sigma }_{\mathrm{m}}\left(‖x_i-x_j‖\right)}$, $V_3={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^N{\sigma }_{\mathrm{t}}\left(\left|d_{\mathrm{t},i}\right|\right)}$, and $V_4={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1,j\ne i}^N{\sigma }_{\mathrm{e}}\left(\left|l_{ij}\right|\right)}}$. By using the velocity command (16) for all UAVs, $\stackrel{•}{V}$ satisfies $\stackrel{•}{V}\le 0$, and the system is stable.