Reach-avoid games for players with damped double integrator dynamics

This paper investigates a reach-avoid game between two players with damped double integrator dynamics. An optimal state-feedback strategy is derived using a differential game framework combined with geometric analysis. To facilitate the analysis, we introduce the concept of multiple reachable region by characterizing the motion of players with damped double integrator dynamics. Based on this, a new type of the attackers dominance region is introduced. We show that distinct strategies are required depending on the location of the terminal position within different areas of the attacker’s dominance region. Furthermore, we prove that the proposed strategies satisfy the necessary condition for optimality. Numerical simulations are provided to illustrate the conclusions.

💡 Research Summary

This paper studies a reach‑avoid differential game in which both the attacker and the defender are modeled by damped double‑integrator dynamics. The state of each player consists of a planar position and velocity, and the control inputs are a bounded acceleration magnitude and a heading angle. A constant positive damping coefficient μ acts on both agents, and the attacker’s maximum acceleration is strictly smaller than the defender’s ( u_A^max < u_D^max ). The attacker’s objective is to reach a fixed target at the origin while avoiding capture; the defender seeks to capture the attacker as far from the target as possible.

The authors first formulate the game in the Hamilton‑Jacobi‑Isaacs (HJI) framework. By constructing the Hamiltonian and solving the co‑state dynamics, they show that the optimal control for each player is always saturated at the maximum admissible acceleration and that the optimal heading is constant in time. This class of controls is termed the “Normal strategy”. Under a Normal strategy the closed‑form trajectories of position and velocity are obtained (Eqs. 12‑13), which consist of an exponentially decaying component (due to damping) plus a term proportional to the integrated acceleration.

A central contribution is the introduction of the Multiple Reachable Region (MRR). For a given time t, the set of points that a player can reach with a Normal strategy is an isochrone I_i(t), which is a circle with time‑varying centre x_ic(t) and radius r_ic(t). Because the radius grows non‑monotonically (the term t – (1–e^{–μt})/μ appears), the isochrones of the attacker and defender can intersect up to three times. The authors prove (Theorem 1) that external tangency, internal tangency, and up to three intersections are guaranteed, and consequently any point belonging to the MRR can be reached by the same player via at least two distinct Normal strategies (Corollary 1).

The paper defines reaching times t_{ij}(x) for each point x and each of the possible strategies (j = 1,2,3). Lemmas 2‑4 characterize the intervals in which a point is reachable (between the first and second reaching times) and the sign of the scalar product v·