This article studies a multiplayer capture-the-flag (CTF) differential game, where multiple pursuers try to intercept evaders whose objectives are to first reach a flag and then reach a return region. The critical point is that the flag and the return region are half-planes. Our goal is to address the problem of determining the game winner and computing the pursuit winning strategies. By decomposing the complex multiplayer game into many manageable subgames involving multiple pursuers and one evader, we present the strategies under which the pursuers guarantee to win against the evader, regardless of the evader's strategy, with the necessary and sufficient conditions to determine the game winner. We then extend the results to the cases of the flag-staying time and the minimum safe flag position. To reduce the computational burdens, we prove that if multiple pursuers can ensure the pursuit winning against an evader, then at most two pursuers in this coalition are required. Finally, we solve the multiplayer game by evaluating pairwise subgame outcomes for pursuer-evader matchings. Numerical and experimental results are presented to illustrate the theoretical conclusions.

Abstract Stochastic games are a well established model for multi-agent sequential decision making under uncertainty. In practical applications, though, agents often have only partial observability of their environment. Furthermore, agents increasingly perceive their environment using data-driven approaches such as neural networks trained on continuous data. We propose the model of neuro-symbolic partially-observable stochastic games (NS-POSGs), a variant of continuous-space concurrent stochastic games that explicitly incorporates neural perception mechanisms. We focus on a one-sided setting with a partially-informed agent using discrete, data-driven observations and another, fully-informed agent. We present a new method, called one-sided NS-HSVI, for approximate solution of one-sided NS-POSGs, which exploits the piecewise constant structure of the model. Using neural network pre-image analysis to construct finite polyhedral representations and particle-based representations for beliefs, we implement our approach and illustrate its practical applicability to the analysis of pedestrian-vehicle and pursuit-evasion scenarios.

This paper studies attack-defense differential games for multiple 3D defenders and 2D or 3D attackers, where the attackers aim to attack a target region protected by the defenders with non-negative capture radii. Since the direct analysis of many-to-many games is difficult, the problem is decomposed into multiple many-to-one subgames. For the subgame between multiple 3D defenders and a 2D attacker, this paper presents a concept of attack region and constructs a defense winning strategy based on the attack region. Based on this strategy, it is proved that if a defense coalition can win, there are no more than two members that can guarantee this winning. The analysis is extended to 3D attackers with a lowest flight altitude constraint. Under this defense strategy, the defenders move towards an estimated interception point, and the optimality of the strategy is shown. The bipartite matching with constraints is adopted to resolve many-to-many games based on subgame outcomes. Simulations are provided to validate the theoretical results.

Neuro-symbolic approaches to artificial intelligence, which combine neural networks with classical symbolic techniques, are growing in prominence, necessitating formal approaches to reason about their correctness. We propose a novel modelling formalism called neuro-symbolic concurrent stochastic games (NS-CSGs), which comprise two probabilistic finite-state agents interacting in a shared continuous-state environment. Each agent observes the environment using a neural perception mechanism, which converts inputs such as images into symbolic percepts, and makes decisions symbolically. We focus on the class of NS-CSGs with Borel state spaces and prove the existence and measurability of the value function for zero-sum discounted cumulative rewards under piecewise-constant restrictions. To compute values and synthesise strategies, we first introduce a Borel measurable piecewise-constant (B-PWC) representation of value functions and propose a B-PWC value iteration. Second, we introduce two novel representations for the value functions and strategies, and propose a minimax-action-free policy iteration based on alternating player choices.

This study addresses the task assignment problem with robot coalitions, as encountered in practical scenarios, such as multiplayer reach-avoid games. Unlike the classical assignment problem where a single robot performs each task, the problem considered here involves tasks that require execution by a robot coalition consisting of two robots. This task assignment problem is a special instance of 3-set packing problem, which is known to be nondeterministic polynomial time (NP)-hard. We introduce the concept of$\epsilon$-coalition-competitive equilibrium ($\epsilon$-CCE) to characterize a kind of approximate solution that offers guaranteed performance. A distributed auction algorithm is developed to find an$\epsilon$-CCE within a finite number of iterations. In addition, several enhancements have been implemented to adapt the auction algorithm for practical applications where the task assignment problem may vary over time. Numerical simulations demonstrate that the distributed algorithm achieves satisfactory approximation quality.

Deep neural networks can be trained to be efficient and effective controllers for dynamical systems; however, the mechanics of deep neural networks are complex and difficult to guarantee. This work presents a general approach for providing guarantees for deep neural network controllers over multiple time steps using a combination of reachability methods and open source neural network verification tools. By bounding the system dynamics and neural network outputs, the set of reachable states can be over-approximated to provide a guarantee that the system will never reach states outside the set. The method is demonstrated on the mountain car problem as well as an aircraft collision avoidance problem. Results show that this approach can provide neural network guarantees given a bounded dynamic model.

We present novel techniques for neuro-symbolic concurrent stochastic games, a recently proposed modelling formalism to represent a set of probabilistic agents operating in a continuous-space environment using a combination of neural network based perception mechanisms and traditional symbolic methods. To date, only zero-sum variants of the model were studied, which is too restrictive when agents have distinct objectives. We formalise notions of equilibria for these models and present algorithms to synthesise them. Focusing on the finite-horizon setting, and (global) social welfare subgame-perfect optimality, we consider two distinct types: Nash equilibria and correlated equilibria. We first show that an exact solution based on backward induction may yield arbitrarily bad equilibria. We then propose an approximation algorithm called frozen subgame improvement, which proceeds through iterative solution of nonlinear programs. We develop a prototype implementation and demonstrate the benefits of our approach on two case studies: an automated car-parking system and an aircraft collision avoidance system.

This article considers a subspace guarding game in high-dimensional space which consists of a play subspace and a target subspace. Two faster defenders as a team cooperate to protect the target subspace by capturing an attacker which strives to enter the target subspace from the play subspace without being captured. A closed-form solution is provided from the perspectives of kind and degree. Contributions of the work include the use of the attack subspace (AS) method to construct the barrier, by which the game winner can be perfectly predicted before the game starts. In addition to this inclusion, with the priori information about the game result, a critical payoff function is designed when the defenders can win the game. Then, the optimal strategy for each player is explicitly reformulated as a saddle-point equilibrium. Finally, we apply these theoretical results to two half-space and half-plane guarding games in 3-D space and 2-D plane, respectively. Since the entire achieved developments are analytical, they require a little memory without the computational burden and allow for real-time updates, beyond the capacity of the traditional Hamilton-Jacobi-Isaacs method. It is worth noting that this is the first time in the current work to consider the target guarding games for arbitrary high-dimensional space and in a fully analytical form.

This article considers a multiplayer reach-avoid game between two adversarial teams in a general convex domain which consists of a target region and a play region. The evasion team, initially lying in the play region, aims to send as many team members into the target region as possible, while the pursuit team with its team members initially distributed in both play region and target region, strives to prevent that by capturing the evaders. We aim at investigating a task assignment about the pursuer-evader matching, which can maximize the number of the evaders who can be captured before reaching the target region safely when both teams play optimally. To address this, two winning regions for a group of pursuers to intercept an evader are determined by constructing an analytical barrier which divides these two parts. Then, a task assignment to guarantee the most evaders intercepted is provided by solving a simplified 0-1 integer programming instead of a nondeterministic polynomial problem, easing the computation burden dramatically. It is worth noting that except the task assignment, the whole analysis is analytical. Finally, simulation results are also presented.

This paper considers a reach-avoid game on a rectangular domain with two defenders and one attacker. The attacker aims to reach a specified edge of the game domain boundary, while the defenders strive to prevent that by capturing the attacker. First, we are concerned with the barrier, which is the boundary of the reach-avoid set, splitting the state space into two disjoint parts: 1) defender dominance region (DDR) and 2) attacker dominance region (ADR). For the initial states lying in the DDR, there exists a strategy for the defenders to intercept the attacker regardless of the attacker's best effort, while for the initial states lying in the ADR, the attacker can always find a successful attack strategy. We propose an attack region method to construct the barrier analytically by employing Voronoi diagram and Apollonius circle for two kinds of speed ratios. Then, by taking practical payoff functions into considerations, we present optimal strategies for the players when their initial states lie in their winning regions, and show that the ADR is divided into several parts corresponding to different strategies for the players. Numerical approaches, which suffer from inherent inaccuracy, have already been utilized for multiplayer reach-avoid games, but computational complexity complicates solving such games and consequently hinders efficient on-line applications. However, this method can obtain the exact formulation of the barrier and is applicable for real-time updates.

Formation control problems for second-order unmanned aerial vehicle (UAV) swarm systems with switching topologies to achieve the time-varying formation and the predefined time-varying formation reference are investigated. The second-order UAV swarm systems considered are not only required to achieve the specified time-varying formation but also track the predefined time-varying formation reference, and the switching topologies are also taken into consideration. To achieve the predefined time-varying formation reference, a new time-varying formation protocol is proposed, and the formation problem is transformed into the consensus problem. Sufficient and necessary conditions for second-order UAV swarm systems modeled by double integrator to achieve the time-varying formation and the predefined time-varying formation reference are presented. Finally, a quadrotor formation platform consisting of three quadrotors is introduced. The simulation and experiment to demonstrate the efficiency of the proposed formation tracking theory are performed.

Using a constructive function approximation network, an adaptive learning control (ALC) approach is proposed for finite interval tracking problems. The constructive function approximation network consists of a set of bases, and the number of bases can evolve when learning repeats. The nature of the basis allows the continuous adaptive learning of parameters when the network undergoes any structural changes, and consequently offers the flexibility in tuning the network structure. The expandability of the bases guarantees precision of the function approximation and avoids the trial-and-error procedure in structure selection for any fixed structure network. Two classes of unknown nonlinear functions, namely, either global L(2) or local L(2) with a known bounding function, are taken into consideration. Using the Lyapunov method, the existence of solution and the convergence property of the proposed ALC system are discussed in a rigorous manner. By virtue of the celebrated orthonormal and multiresolution properties, wavelet network is used as the universal function approximator, with the weights tuned by the proposed adaptive learning mechanism.