Terminal Goal Guarding (TGG)

• Terminal Goal Guarding (TGG) is a framework for preventing adversaries from reaching secured regions through pursuit–evasion, differential games, and reinforcement learning.
• It employs mathematical constructs like Apollonius circles and Hamilton–Jacobi PDEs to design barrier surfaces and optimized interception strategies under kinematic constraints.
• TGG systems integrate continuous, discrete, and cooperative models to deliver explicit policies and coordinated defense for heterogeneous agent teams in dynamic security scenarios.

Terminal Goal Guarding (TGG) denotes a class of pursuit–evasion, guarding, and reinforcement-learning problems wherein agents (defenders or “guards”) act to prevent adversaries (attackers or intruders) from reaching protected terminal regions or “goals.” TGG frameworks are prominent in continuous-time differential games, combinatorial security on graphs, stochastic boundary protection, and goal-reachability augmentation for learning agents. Research on TGG formulates adversarial and cooperative multi-agent dynamics, rigorously characterizes barrier surfaces or policies guaranteeing terminal defense or goal achievement, and provides explicit strategies for heterogeneous agent teams under both single-shot and persistent attack scenarios.

1. Mathematical Formulations and Archetypal Models

TGG problems admit diverse but tightly characterized mathematical formulations, typically as zero-sum or dynamic games. In the canonical single-attacker case, the state $x$ collects agent positions (e.g., $x_A, x_D$ in $\mathbb R^2$ for attacker and defender) and possibly additional configuration variables (e.g., turret angle $\theta_T$ for a fixed defender) (Moll et al., 11 Sep 2025). Each agent adheres to simple-motion or kinematic constraints: $$\begin{aligned} \dot{x}_A &= v\,[\cos u_A,\,\sin u_A]^\top, \ \dot{x}_D &= p\,[\cos u_D,\,\sin u_D]^\top, \ \dot{\theta}_T &= w\,u_T, \end{aligned}$$ with $v < p < w$, and bounded control inputs. Capture and protection are formally defined as reachability or contact in the state space, often requiring that the attacker is intercepted prior to entry into the goal region (e.g., within a unit ball centered at the origin for turret defense (Moll et al., 11 Sep 2025), or a line segment for translating line targets (Das et al., 2022, Das et al., 2022)).

Graph-based TGG introduces the network abstraction $G=(V,E)$, where heterogeneous mobile guards of varying sensing/range $r_i$ move among vertices to ensure eternal security, i.e., indefinite coverage against an infinite sequence of adversarial intrusions at the nodes (Abbas et al., 2015).

In the continuous goal-reachability context, TGG describes the algebraic and algorithmic enforcement of convergence to a target region $\mathcal U\subset\mathcal S$ in Markov decision processes through policy design subject to formal mean-convergence criteria (Osinenko et al., 2024).

2. Barrier Analysis and Solution via Geometric/Dynamical Constructs

Central to continuous TGG (in pursuit-evasion games) is the construction of barrier surfaces—partitionings of the joint state space into defender-win and attacker-win regions—using Apollonius circles (or spheres in higher dimensions) and time-to-go inequalities (Moll et al., 11 Sep 2025, Das et al., 2022, Das et al., 2022, Lee et al., 2024). For two-player planar games:

• The Defender–Attacker barrier is the Apollonius locus: points $y$ for which $\|y - x_A\|/v = \|y - x_D\|/p$.
• The Fixed Guard (Turret)–Attacker barrier is the level set where attacker’s travel time to $(r,\theta)$ equals turret’s time to align and fire:

$$t_A = \frac{\|y - x_A\|}{v}, \quad t_T = \frac{|\theta - \theta_T|}{w}.$$

The intersection and relative inclusion of these boundaries yield conditions for which defense regime is dominant: solo mobile interception, solo fixed-guard interception, or simultaneous capture (Moll et al., 11 Sep 2025).

In line-guarding games with moving targets, explicit value functions $V(x)$ and feedback controls are derived by solving the stationary Hamilton–Jacobi–Isaacs PDE, with the barrier surface $\mathcal{B} = \{x: V(x) = 0\}$ dividing the state space into regions of successful and failed defense (Das et al., 2022, Das et al., 2022).

Stochastic dynamic-geometry TGG employs reachability graph constructions (directed acyclic graphs for guarding boundaries against streaming targets) and minimum-Hamiltonian-path algorithms adapted by linear dilation to handle translating targets (0908.3929).

3. Algorithmic and Policy Structures

TGG solutions exhibit structure across continuous, discrete, and reinforcement-learning regimes:

• Differential Games: Equilibrium is achieved via straight-line pursuit (steering to a capture point) and, for angularly constrained defenses, bang–bang control of guard orientation (Moll et al., 11 Sep 2025, Lee et al., 2024). When multiple attackers are present and heterogeneous, a sequence of joint programs is solved to determine capture points $(p_1^*,...,p_N^*)$, supporting feedback laws for defender and attackers (Lee et al., 2024).
• Graph Guarding: Algorithmic security is realized through the decomposition of $G$ into clusters of bounded diameter (cliques in $G^{r_i}$), greedy coverage approximations (guaranteeing at least $(1-1/e)$-fraction of optimal security), and constant-time movement protocols, with exactly one guard responding per incident (Abbas et al., 2015).
• Boundary Defense under Arrival Streams: Path-planning is based on DAG longest-path scheduling (for slow vehicles) or translational minimum Hamiltonian paths (for fast chasers), with mathematically proven competitive ratios approaching optimality in limiting regimes (0908.3929).
• Reinforcement Learning with Terminal Goal Guarding: Policy design inserts a fallback mechanism, ensuring that whenever a learning agent’s critic does not sufficiently improve a stored value-certificate, it “falls back” to a basis policy $\pi_0$ known to guarantee goal-reaching. This wrapping ensures every produced policy retains the goal-reachability property in mean, regardless of critic error or exploration (Osinenko et al., 2024).

4. Cooperative and Heterogeneous Agent Dynamics

Multi-agent TGG games with heterogeneous teams are characterized by cooperative behaviors among attackers (e.g., sacrificing to enable more critical teammates to penetrate closer to the goal) and by the assignment of mobile guards of distinct ranges to network or spatial clusters to guarantee coverage (Abbas et al., 2015, Lee et al., 2024). In continuous settings, cooperative strategies are engineered by joint minimization of aggregate proximity-to-goal metrics, with rigorous verification via sensitivity analysis and parametric programming techniques validating the sufficiency of the derived equilibria (Lee et al., 2024). In graph guarding, cluster formation is tailored such that high-range guards maximize coverage through larger clusters, while lower-range guards secure remaining small clusters, establishing both optimality bounds and operational simplicity (Abbas et al., 2015).

5. Explicit Value Characterization and Theoretical Guarantees

TGG research provides closed-form expressions for game values and critical boundaries:

• For the turret and mobile defender system, the game’s value (minimal guaranteed terminal distance $d^*$) is given explicitly for each capture regime:
  • $d^* = \|c\| - \rho - 1$ (solo defender), $d^* = r^* - 1$ (solo turret), or via polar equality in the intersection case (Moll et al., 11 Sep 2025).
• For translating line targets, the value function admits a piecewise definition depending on whether the infinite-barrier interception point lies within the target segment: $V(X, Y) = \operatorname{sign}(X)\,[X - Y/m(X, Y)]$, with $V=0$ as the critical barrier (Das et al., 2022, Das et al., 2022).
• In stochastic boundary defense, for Poisson arrival settings, the capture fraction using the LP or TMHP-fraction policy is lower-bounded, approaching optimality in high target speed or arrival rate regimes (0908.3929).
• Formal convergence to the terminal set $\mathcal U$ is proven in the mean, with associated Lyapunov-type certification, for the TGG wrapper in RL (Osinenko et al., 2024).

Empirically, TGG-augmented learning agents outperform standard baselines across a variety of continuous control problems, and graph-based clustering leverages range heterogeneity to maximize secured nodes and minimize response times (Osinenko et al., 2024, Abbas et al., 2015).

6. Extensions and Generalizations

TGG frameworks are extended to accommodate:

• Multiple defenders or attackers with differentiated motion models;
• Targets with arbitrary convex (or even nonconvex) geometries, and time-varying movement;
• Decentralized or repeated-attack settings (eternal security), admitting coverage theorems and online response rules (Abbas et al., 2015);
• Adaptation to reinforcement learning and control synthesis via safe certificate maintenance, hybrid fallback, and penalty-enforced critic updates (Osinenko et al., 2024).

Barrier construction, control synthesis, and equilibrium certification are established through a combination of geometric, optimization, and dynamic-programming tools, making TGG a foundational methodology for adversarial protection, persistent monitoring, and safe exploration in multi-agent and learning environments.