Consider a multiplayer reach–avoid differential game in the 3D Euclidean space ℝ³ with Np pursuers and Ne evaders. The target region is a plane Π_tar, and the pursuers strive to capture evaders to prevent them from approaching the target plane while the evaders aim to reach it as far as possible. The group of pursuers is denoted by {P_i, i = 1, ⋯, Np}, and the group of evaders is denoted by {E_j, j = 1, ⋯, Ne}. The states of P_i and E_j at time t are given by the Cartesian coordinates x_Pi(t) = [x_Pi(t), y_Pi(t), z_Pi(t)] ∈ ℝ³ and x_Ej(t) = [x_Ej(t), y_Ej(t), z_Ej(t)] ∈ ℝ³, respectively. The full state set of the multiplayer differential game is defined as x(t) ≔ [x_Pi(t), y_Pi(t), z_Pi(t), x_Ej(t), y_Ej(t), z_Ej(t)] ∈ ℝ^{3(Np + Ne)} for i = 1, ⋯, Np and j = 1, ⋯, Ne. The initial state set is denoted as x₀ ≔ x(0). The control inputs of P_i and E_j at time t are the unit direction vectors of their instantaneous velocities denoted by u_Pi(t) = [u_Pi^x(t), u_Pi^y(t), u_Pi^z(t)] ∈ ℝ³ and u_Ej(t) = [u_Ej^x(t), u_Ej^y(t), u_Ej^z(t)] ∈ ℝ³, respectively, where ‖u_Pi(t)‖₂ = 1 and ‖u_Ej(t)‖₂ = 1. u_P(t) = {u_Pi(t), i = 1, ⋯, Np} denotes the control set of pursuers. u_E(t) = {u_Ej(t), j = 1, ⋯, Ne} denotes the control set of evaders. The players have simple motion as Isaacs [11] introduces, i.e., they are permitted to instantaneously adjust their orientations. The dynamics \stackrel{·}{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t),{\mathbf{u}}_P(t),{\mathbf{u}}_E(t))of the multiplayer differential game are described by the system of 3(Np + Ne) equations for t ≥ 0:where V_Pi > 0 and V_Ej > 0 are the speeds of P_i and E_j, respectively. For the sake of symbolic brevity, the time t will be omitted hereafter.

The point capture is considered in this paper. Without loss of generality, we assume that the target plane Π_tar is delineated by {x ∈ ℝ³∣z = 0}, and all the players located in the region {x ∈ ℝ³∣z > 0} and their initial positions do not coincide. The game terminates with the pursuers winning if the pursuers capture all evaders, and with the evaders winning if any evader reaches Π_tar.

Consistent with current research on the differential game, we also consider players with constant speed. However, the difference lies in our support for heterogeneous players, that is, both the pursuers and evaders can have different speeds. With respect to the speed ratio, in combination with the capture way, there are several cases. For the point capture, if V_Ej > V_Pi, the evader can always escape without being captured; if V_Ej ≤ V_Pi, the capture of the evader by the pursuer is a feasible outcome, which is the focus of most recent work. In the case of the radius capture, the speed ratio will not have a decisive impact on the outcome, and the analysis is relatively simple, which is not within the scope of consideration in this paper, so the following assumption is given.

Assumption 1. The speed ratio of evader E_j and pursuer P_i is assumed to be V_Ej/V_Pi = w_ij, w_ij < 1, for b_ij = 1, ∀ i = 1, ⋯, Np, ∀ j = 1, ⋯, Ne.

The differential game is usually divided into the game of kind and the game of degree. The game of kind determines who will win given the initial states of players by constructing the barrier, which separates the game space into 2 disjoint parts: pursuers’ winning region {\mathcal{W}}_P and evaders’ winning region {\mathcal{W}}_P. The game of degree gives the optimal strategies for pursuers and evaders in the specified winning region, which is transformed into the Value of the game by maximizing and minimizing the payoff with the property of a continuum of numerical values. In this paper, we focus on the game of degree in the pursuers’ winning region {\mathcal{W}}_P, where the pursuers are guaranteed to capture all evaders and win the game, to solve the optimal strategies for players. Moreover, the game of kind is also discussed in the “Apollonius Sphere” section.

The information pattern is essential for players in the differential game. The non-anticipative information pattern is a commonly used one where the player has access to all states and the opponent’s current control inputs. Stackelberg game employs this information pattern. However, considering the rationality of the adversarial game, the opponent’s control inputs should be unavailable; thus, the state feedback information pattern is employed in this paper, where players are only privy to the state information, and not the control inputs of their opponents. The objective of this paper is to determine the optimal strategy based on the state feedback information, which is also referred to as the optimal state feedback strategy.

For the multiplayer differential game, we define a binary matching matrix for pursuers and evaders: B = [b_ij]_Np×Ne, and b_ij = 1 if P_i is assigned to E_j; otherwise, b_ij = 0. Note that one pursuer is only assigned to one evader, one evader is assigned to one pursuer when Np = Ne, and one evader is assigned to multiple pursuers when Np > Ne, and the case with Np < Ne is not considered. The matching matrix does not change once the assignment is determined throughout the game.

In the pursuers’ winning region {\mathcal{W}}_P, the game terminates when P_i captures E_j for all matching pairs P_i − E_j. Therefore, the terminal set \mathcal{D} of pursuers is described by

With respect to the target plane {x ∈ ℝ³∣z = 0}, the purpose of evaders is to make z-coordinates as small as possible when captured. The pursuers’ objective is just the opposite. Hence, the terminal payoff function J can be constructed using the terminal state, which is of Mayer-type and defined aswhere t_f is the terminal time of the game. Then, the objective of pursuers and evaders yields the Value of the game V described as

The generic theorem for the differential game is given as follows, which is a primary cornerstone for subsequent analysis in this paper.

Theorem 1 (Constant optimal strategy). For the differential game wherein the players have simple motion described as Eq. 1 and the payoff function is of Mayer-type, all players’ optimal strategies are constant, and their optimal trajectories are straight lines.

Proof. Let λ = [λ_Pi^x, λ_Pi^y, λ_Pi^z, λ_Ej^x, λ_Ej^y, λ_Ej^z]^T ∈ ℝ^{3(Np + Ne)} be the costate variable, for i = 1, ⋯, Np,j = 1, ⋯, Ne. The Hamiltonian of the multiplayer differential game is

The optimal strategies of all the players in terms of the costate variables can be obtained from \underset{{\mathbf{u}}_E}{\text{min}} \underset{{\mathbf{u}}_P}{\text{max}}\mathcal{H}, and they are characterized by

Because the costate variable dynamics satisfy \stackrel{·}{\mathbf{\lambda }}=-\frac{\partial \mathcal{H}}{\partial \mathbf{x}}=\mathbf{0}, λ is constant, and thus u_Pi^x*, u_Pi^y*, u_Pi^z* for i = 1, ⋯, Np and u_Ej^x*, u_Ej^y*, u_Ej^z* for j = 1, ⋯, Ne are constant. Consequently, the optimal trajectories of all the players are straight lines.

Harris’ Hawk (Parabuteo unicinctus) is a social medium-large bird of prey that is native to the Americas and found throughout the United States in Arizona, New Mexico, and Texas, usually living in the deserts in the presence of high cacti, holes, and woody shrubs, etc. Raptors often hunt alone, while Harris’ Hawk’s special living environment with the scarcity of food makes for their unique cooperative hunting behaviors. Harris’ Hawks cooperatively hunt in groups of 2 to 6 using different group sizes and cooperative tactics depending on the prey's size, escape patterns, hiding place, and other characteristics. As Bednarz [30] studied with Harris’ Hawks cooperative hunting, hawks conduct team foraging in an organized and strategic manner. Hunts begin early in the morning, when the Harris’ Hawks communicate with each other through chirping, and gather on tall cacti or power poles, and then the hunting team splits into several groups (usually 1 to 3 hawks) and takes turns to fly short distances in search of prey. Some members, responsible for lookout search, often perch on huge trees, cacti, or power poles to find potential prey, and when the prey is once found, other members will be notified quickly to attack the prey. Some members constantly “leapfrog” throughout the group's range to actively seek prey, occasionally regrouping or splitting again, while moving in a general direction, and this “leapfrog” will continue until they can catch prey or it is dark.

Bednarz describes Harris’ Hawk’s most commonly used hunting tactic as a “surprise pounce,” with several hawks flying toward their prey from different directions, and captures can occur instantaneously or may require multiple quick dives to capture prey. The surprise pounce described by Bednarz includes the flush-and-ambush tactic and relay attack tactic; the former is very effective in situations where the prey is hidden by the cover, one hawk flushes into the cover to shock the prey out, and other hawks ambush at the exit of the prey to hunt. The latter refers to the continuous alternation of the leader hawk when chasing prey to ensure that the team’s hunting energy utilization is maximized and the hunting success rate is improved. From a more specific perspective, Dawson [31] summarized 3 Harris’ Hawks cooperative hunting tactics: dual-attack, ascend–attack, and intercept–attack. These tactics can also be classified as a form of “surprise pounce” in certain scenarios. In the dual-attack tactic, 2 hawks fly in the same direction within 1 m of each other, trying to approach the prey at the same time. Two hawks descend together, and the leader hawk tries to touch the prey; if the leader hawk fails, the other will continue to hover above the leader hawk and wait for the opportunity to dive sharply to catch the prey. The ascend–attack tactic is employed when the hawks and prey move in a nearly parallel path. The leader hawk first accelerates and flaps its wings to descend sharply in the direction of the prey, and the other hawk begins to ascend and accelerate. Once the prey escapes, they immediately fly above the prey area and quickly dive to pounce on the prey. When the hawk and the prey move in a nonparallel path, the hunting team will use the intercept–attack tactic. The leader hawk flies directly to the prey and tries to approach and contact the prey, and the second or third hawk flies to the interception path in the area ahead of the prey. The unusual distinction of this tactic is that other hawks do not directly chase the prey but follow the leader hawk to change the flight path and fly to the front of the prey to intercept it. The illustration of the Harris’ Hawks’ intercept–attack tactic is shown in Fig. 2.

Harris’s Hawk cooperative hunting tactics offer many benefits and advantages by improving hunting efficiency, expanding the hunting range, and facilitating social learning. This cooperative behavior enables them to better adapt to the wild environment and increases their chances of successful predation. Inspired by Harris’ Hawk’s cooperative hunting tactics and considering the cooperative problem of the multiplayer reach–avoid game, this paper applies Harris’ Hawk’s intercept–attack tactic to the case of multiple pursuers and one evader, which is further elaborated in the following subsections.

In this subsection, the optimal strategies for 3D multiplayer reach–avoid differential games in the case of Np = Ne with one pursuer and one evader pairings will be provided. In Harris’ Hawk’s surprise pounce tactic, each hawk in a hunting group selects a potentially captured prey that gives the team the most energy and launches a hunt toward the prey. Based on Corollaries 1 and 2, when dealing with a pairing of one pursuer and one evader, utilizing the Apollonius sphere technique can determine whether the pursuer is capable of capturing the evader. Furthermore, it facilitates the selection of optimal pairings of pursuers and evaders in feasible assignments. Simulating the hawks’ intelligence, the group of pursuers aims to make optimal interception points of each pursuer with respect to the corresponding evader as far away from the target plane as possible, and the group of evaders has the opposite purpose. Definewhere g denotes the gth assignment. b_{\mathit{ij}}^g = 1 if the pursuer P_i can capture the evader E_j in assignment g; otherwise, b_{\mathit{ij}}^g = 0.

Consequently, the following theorem is formulated that gives the solution of the multiplayer differential game in 3D, including both the optimal assignment and the optimal state feedback strategies of all players.

Theorem 4 (Optimal strategies). Consider the 3D multiplayer differential game described in Eqs. 1 to 4 with Np pursuers and Ne evaders (Np = Ne) under Assumption 1, where \mathbf{x}\in {\mathcal{W}}_P. The Value of the game is V(\mathbf{x})={\text{max}}_g{\mathcal{K}}_g(\mathbf{x}), which is continuously differentiable on the nonsingular surface and satisfies the HJI equation. The optimal assignment of the game is g^{*}=\text{arg}{\text{max}}_g{\mathcal{K}}_g(\mathbf{x}). The optimal strategies of all players for i = 1, ⋯, Np and j = 1, ⋯, Ne are given byfor the pairing P_i − E_j subjecting to b_{\mathit{ij}}^{g*} = 1, where (x_{I_{ij}}^{*},y_{I_{ij}}^{*},z_{I_{ij}}^{*}) is shown in Eq. 8.

Proof. The following proof is based on b_{\mathit{ij}}^{g*} = 1. First, we prove that the Value function V(\mathbf{x})={\text{max}}_g{\mathcal{K}}_g(\mathbf{x}) is continuously differentiable on the nonsingular surface. The partial derivatives of V(x) with respect to each state variable are as follows.where d_{\mathit{ij}}=\sqrt{{(x_{P_i}-x_{E_j})}^2+{(y_{P_i}-y_{E_j})}^2+{(z_{P_i}-z_{E_j})}^2}.

Second, we prove that the Value function V(\mathbf{x})={\text{max}}_g{\mathcal{K}}_g(\mathbf{x}) satisfies the HJI equation. The conventional HJI equation is -\frac{\partial V}{\partial t}=\frac{\partial V}{\partial \mathbf{x}}\mathbf{f}(\mathbf{x},{\mathbf{u}}_P^{*},{\mathbf{u}}_E^{*})+g(t,\mathbf{x},{\mathbf{u}}_P^{*},{\mathbf{u}}_E^{*}), but the payoff is of Mayer-type in this problem, so \frac{\partial V}{\partial t}=0 and gg\left(t,\mathbf{x},{\mathbf{u}}_p^{*},{\mathbf{u}}_E^{*}\right)=0 = 0. Therefore, to prove that the Value function satisfies the HJI equation, it is only necessary to prove \frac{\partial V}{\partial \mathbf{x}}\mathbf{f}(\mathbf{x},{\mathbf{u}}_P^{*},{\mathbf{u}}_E^{*})=0.

To obtain it, it is needed to compute the following. According to Eqs. 8 and 14, the following equations hold.

Substituting Eq. 16 into Eq. 13 gives

Substituting Eqs. 17 and 18 into Eq. 15 yieldswhere according to Eq. 14 we have that

Substituting Eqs. 20 and 21 into Eq. 19 gives

Thus, the Value function V(\mathbf{x})={\text{max}}_g{\mathcal{K}}_g(\mathbf{x}) satisfies the HJI equation.

On the singular surface, there is {\mathcal{K}}_{g_1^{*}}(\mathbf{x})={\mathcal{K}}_{g_2^{*}}(\mathbf{x}), that is, there are 2 or more optimal assignments; hence, the Value of the game is equal on the singular surface and is continuous, but it is not continuously differentiable. For example, suppose P_i is assigned to E_j1 in the optimal assignment g_1^{*}, and P_i is assigned to E_j2 in the optimal assignment g_2^{*}, then the following inequality holds.

The remaining corresponding partial derivatives also hold, so it can be proved that the Value of the game on the singular surface is not continuously differentiable.

Remark 1. On the singular surface, the pursuers can select any one among the equally optimal assignments without affecting their payoff. However, to avoid conflicts, one pursuer is given priority and, after determining the optimal assignment, broadcasts the assignment information to other pursuers, thus avoiding the situation where multiple pursuers choose the same evader, and some evaders are missed.

Remark 2. The optimal strategies are the saddle-point strategies of the game, and any player who deviates from the saddle-point strategy will achieve a reduced payoff. The optimal strategies in Eq. 13 are state feedback strategies; thus, players can react in real time to the opponent’s varying strategies. For instance, if the evader employs a non-optimal strategy, the pursuer will continuously recalculate its optimal strategy as given by Eq. 13 and obtain greater benefits. Meanwhile, the evader will not only be captured but also incur higher terminal costs.

In this subsection, the cooperative tactic for the case of Np > Ne with multiple pursuers and one evader assignment is proposed, inspired by Harris’ Hawk’s intercept–attack tactic. In the intercept–attack tactic, the leader hawk is responsible for chasing the prey, and the secondary hawk is designed to intercept the prey on the intended path of the prey. Similarly, in multiple pursuers against one evader game, the pursuers are divided into the leader pursuer and the secondary pursuer. With respect to Harris’ Hawk’s pursuit strategy, the study of Brighton and Taylor [32] demonstrates that Harris’ Hawks use a mixed guidance law to pursue the prey, combining low-gain proportional navigation (PN) with a low-gain pure pursuit (PP). As shown in Fig. 3, the mixed guidance law is given by

where \stackrel{·}{\theta } is the turning rate, \stackrel{·}{\gamma } is the line-of-sight angle rate, and φ is the deviation angle between the hawk’s velocity and the line of sight. K_PN is a PN guidance constant, and K_PP is a PP guidance constant. Refer to [32], K_PN = 0.8, K_PP = 1.2.

The Harris’ Hawks’ intercept–attack tactic is used as the cooperative tactic for the pursuers, the leader pursuer adopts Harris’ Hawk’s mixed guidance law to chase the evader, and the secondary pursuer adopts the optimal saddle-point strategy given in Theorem 4 to intercept the evader. With regard to the evaders, their optimal strategies still follow Theorem 4.

In conclusion, the matching algorithm for multiple pursuers and multiple evaders and the cooperative tactic of the pursuers are presented as follows. Additionally, Fig. 4 displays the flowchart of the proposed multiplayer 3D reach–avoid differential games via Harris' Hawks cooperative hunting tactics.

Step 1: Construct a bipartite graph with the pursuers {P_i, i = 1, ⋯, Np} and evaders {E_j, j = 1, ⋯, Ne} as 2 node sets, and each player is regarded as a vertex.

Step 2: For any P_i, E_j, according to Corollaries 1 and 2, calculate the optimal interception point for P_i to intercept E_j, and an edge is connected to nodes P_i and E_j if P_i can capture E_j. Let the distance between the optimal interception point and the target plane be the weight of this edge.

Step 3: Based on the node set of the evaders, find the maximum matching in the bipartite graph using any matching algorithm to maximize the total weights. Hungarian algorithm is employed in this paper.

Step 4: The remaining unmatched pursuers, regarded as the leader pursuer, select the nearest evader as the target evader, and the previous pursuer paired with the target evader is regarded as the secondary pursuer. The leader pursuer and the secondary pursuer form a pursuit subgroup, and they attack the target evader adopting the Harris’ Hawks’ intercept–attack cooperative tactic.

Remark 3. The solution obtained for the 3D multiplayer reach–avoid differential game in this paper exhibits excellent scalability with respect to the number of players. As the number of players increases, the solution's performance remains unaffected, and the computation is only slightly increased in the initial multiplayer assignment phase. Furthermore, the proposed method can be implemented in real time, supporting online decision-making in large-scale multiplayer games.

Considering the case with 2 pursuers and 2 evaders, the initial positions of players are x_P1(0) = [5,10,10], x_P2(0) = [8,25,15], x_E1(0) = [10,10,15], and x_E2(0) = [15,20,18]. The velocities of players are V_P1 = V_P2 = 1 and V_E1 = V_E2 = 0.8. For simplicity, the simulation assumes that the pursuers have identical speeds and the evaders also have identical speeds. However, as long as Assumption 1 is satisfied, both pursuers and evaders are allowed to have different speeds. This same premise applies to the subsequent simulation settings.

For 2 pursuers P₁, P₂ and 2 evaders E₁, E₂, the Apollonius sphere of P₁E₁ and the Apollonius sphere of P₂E₂ are shown in Fig. 5, in which the optimal interception point of P₁E₁ is I_{11}^{*}(18.89,10,8.18), and the optimal interception point of P₂E₂ is I_{22}^{*}(27.44,11.11,3.09). Both Apollonius spheres have no intersection with the target plane, indicating that P₁E₁ and P₂E₂ are feasible assignments, and the pursuers are guaranteed to win with the optimal strategies. The Apollonius sphere of P₁E₂ and the Apollonius sphere of P₂E₁ are shown in Fig. 6, in which the optimal interception point of P₁E₂ is I_{12}^{*}(32.78,37.78, − 3.88) and the optimal interception point of P₂E₁ is I_{21}^{*}(13.56, − 16.67, − 18.63), while both Apollonius spheres intersect the target plane, indicating that P₁E₂ and P₂E₁ are invalid assignments, and the evaders can win the game if they adopt the optimal strategies. According to Theorem 3, the optimal assignment of the game is P₁E₁ and P₂E₂. If P₁E₂ and P₂E₁ are also valid assignments, the optimal assignment is the assignment corresponding to the maximum values in z_{I_{11}}^{*}+z_{I_{22}}^{*} and z_{I_{12}}^{*}+z_{I_{21}}^{*}. Three examples are conducted when players adopt optimal or non-optimal strategies to indicate the effectiveness of the proposed solution to the reach–avoid differential game as follows.

Example 1: Pursuers and evaders adopt optimal strategies

According to Theorem 3, the pursuers and the evaders adopt the optimal strategies shown in Eq. 10, the optimal trajectories are shown in Fig. 7, as the theorem states, the optimal interception point is time invariant, and the optimal trajectory is a straight line. The evader E₁ is captured by the pursuer P₁ at the optimal interception point I_{11}^{*}, and the evader E₂ is captured by the pursuer P₂ at the optimal interception point I_{22}^{*}, as expected. The Value of the game is V=z_{I_{11}}^{*}+z_{I_{22}}^{*}=11.283.

Example 2: Pursuers adopt optimal strategies, and evaders adopt non-optimal strategies

In this example, the evaders adopt non-optimal strategies, while the pursuers do not know the evaders’ strategies and execute the optimal strategy and the optimal assignment (P₁E₁ and P₂E₂). As the evaders do not execute the optimal strategy, the time-varying optimal interception point will be generated. The pursuers need to update the optimal strategies by continuously calculating Eq. 10 to react to the evaders' non-optimal strategy. We consider 2 types of non-optimal strategies for the evaders: the first one is a greedy strategy, in which the evaders move directly toward the target plane; the second one is to execute the corresponding optimal strategy based on the non-optimal assignment (P₁E₂ and P₂E₁) with the pursuers.

The game result in the first case is shown in Fig. 8A, where the evaders are captured faster at a farther distance from the target plane. The terminal payoff (the distance between the evader and the target plane) is D_{\mathit{tf}}^1 = 20.6 > V = 10.283, indicating that the pursuers win the game with greater profits, while the evader incurs more losses. The game result in the second case is shown in Fig. 8B; the evaders adopt the non-optimal assignment and the optimal strategy, where the evader E₁ incorrectly matches itself with the pursuer P₂ and executes the optimal strategy given by Eq. 11 according to the optimal interception point on the Apollonius sphere of P₁E₂, and the evader E₂ follows the same manner. Finally, the evaders are captured at locations further away from the target plane, resulting in the terminal payoff D_{\mathit{tf}}^2 = 21.1744 > V = 10.283. These results confirm the robustness of the pursuers’ saddle-point strategy against any strategies of the evaders, as well as the fact that the evaders deviating from the saddle-point strategy led to higher costs.

Example 3: Evaders adopt optimal strategies and pursuers adopt non-optimal strategies

In this example, the evaders adopt optimal strategies, while the pursuers do not. Two cases are considered: In the first case, the pursuer follows optimal assignment (P₁E₁ and P₂E₂) and executes the PP guidance law to pursue its assigned evader; in the second case, the pursuer adopts non-optimal assignment (P₁E₂ and P₂E₁) and executes the optimal strategy given by Eq. 11 on its chosen evader. The evaders do not know the pursuers’ strategies, and they adopt optimal assignment (P₁E₁ and P₂E₂) and the optimal strategies given by Eq. 10. Since the pursuers adopt non-optimal strategies, the optimal interception point is also time varying, and the evaders will constantly update their optimal strategies according to Eq. 10.

The game result in the first case is shown in Fig. 9A and B, although E₁ is captured by P₁ and its terminal distance to the target plane is smaller, z_E1(t_f) = 2.4689. Worse still, E₂ successfully reaches the target plane without being captured by P₂. This shows that even under the optimal assignment, failure to execute the optimal strategy can still degrade the performance of the pursuers. However, due to the non-optimal strategy of the pursuers, the evader implementing the optimal strategy will gain a greater benefit, making it closer to the target plane or even successfully reaching the target plane before being captured. The game result in the second case is shown in Fig. 9C and D, where both E₁ and E₂ are not captured, and E₁ successfully reaches the target plane with the game ends. The terminal distance between E₂ and the target plane is z_E2(t_f) = 3.1659. Under the non-optimal assignment, P₁ selects E₂, and P₂ selects E₁. Although the pursuers adopt the optimal strategy for the assigned evaders, both pursuers failed to capture the corresponding evader. It seemed manifest that the non-optimal strategies result in inferior performance for the pursuers. Only by combining the optimal assignment and optimal strategy can the maximum benefits be achieved.

Based on the findings from the 3 examples, it is evident that the proposed optimal state feedback strategies in this paper exhibit effectiveness and robustness against any unknown opponent strategy, allowing for adaptive responses. Since the optimal state feedback strategy is the saddle-point strategy in the differential game, any deviation from this strategy by a player will inevitably result in inferior outcomes.

This subsection carries out a simulation to demonstrate the effectiveness of the proposed cooperative tactic and the assignment method in the case of the pursuers outnumbered the evaders (Np > Ne). Considering the case with 5 pursuers and 3 evaders, the initial positions of players are x_P1(0) = [5,10,10], x_P2(0) = [8,25,15], x_P3(0) = [15,15,16], x_P4(0) = [18,22,14], x_P5(0) = [13,6,12], x_E1(0) = [10,10,15], x_E2(0) = [15,20,18], and x_E3(0) = [10,18,20]. The velocities of players are the same as in Case 1.

According to the matching algorithm described in the “Cooperative pursuit tactic inspired by Harris’ Hawk for multiple pursuers one evader assignments” section, the bipartite graph for 5 pursuers and 3 evaders was constructed, the optimal interception point I_{ii}^{*} for each pursuer P_i and evader E_j pairing was calculated, and then the matching weights can be obtained by z_{I_{\mathit{ij}}}^{*}, shown in Table. Each value in Table represents the distance between the corresponding optimal interception point and the target plane, and the negative value means that the corresponding pursuer cannot capture the evader; hence, the corresponding matching is invalid. Then, the optimal assignment is obtained using the Hungarian algorithm: P₁E₁, P₃E₃, P₄E₂. According to the cooperative tactics presented in the “Cooperative pursuit tactic inspired by Harris’ Hawk for multiple pursuers one evader assignments” section, P₁, P₃, and P₄ serve as the secondary pursuers and adopt the optimal strategies. Furthermore, the remainder pursuers P₂ and P₅ select the nearest evader and serve as the leader pursuer employing Harris’ Hawk’s mixed guidance law, forming the pursuit subgroup of P₁ and P₅, and the pursuit subgroup of P₂ and P₃. The final assignment results are shown in Fig. 10.

The trajectories of players are shown in Fig. 11. This multiplayer reach–avoid game terminated with the winning of the pursuers. The whole game is divided into 3 subgames, namely, P₁P₅ − E₁, P₂P₃ − E₃, and P₄ − E₂. For the one-on-one paring P₄ − E₂, the pursuer P₄ and the evader E₂ follow the optimal strategies, and the evader is captured at the optimal interception point I_{42}^{*}. For the multiple pursuers one evader assignment P₁P₅ − E₁, P₁ is the secondary pursuer adopting the optimal strategy and P₅ is the leader pursuer adopting Harris’ Hawk’s mixed guidance law. Finally, the evader E₁ is captured by the leader pursuer P₅ before reaching the optimal interception point I_{11}^{*}, which indicates that the cooperative tactic of P₁ and P₅ is effective and enhances the pursuers group payoff. The scenario with multiple pursuers and one evader assignment P₂P₃ − E₃ resembles the P₁P₅ − E₁ assignment scenario, where the evader E₃ is captured by the leader pursuer P₂ before being intercepted by the secondary pursuer P₃, which demonstrates the effectiveness of Harris’ Hawk’s mixed guidance law.