pith. sign in

arxiv: 2606.13012 · v1 · pith:PPJKOTR2new · submitted 2026-06-11 · 🧮 math.OC

Flyby Distance Pursuit for Guarding a Target with an Inferior Guard

Navot Israeli , Vitaly Shaferman , Oded Golan This is my paper

Pith reviewed 2026-06-27 06:15 UTC · model grok-4.3

classification 🧮 math.OC
keywords differential gamespursuit-evasionguarding a targetflyby distanceauxiliary stateterminal speed constraintaerodynamic motiontwo-point boundary value problem
0
0 comments X

The pith

A slower guard can force capture of a faster attacker heading to a target when the attacker must arrive above a minimum speed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper frames target guarding as a differential game in which a fast attacker and slower guard both fly under nonlinear thrust-free aerodynamic dynamics in the plane. The game is posed so the attacker must terminate exactly at the target while the payoff is the flyby distance, made available by an auxiliary state that tracks the minimum separation achieved during the entire flight. Solving the resulting two-point boundary-value problem numerically yields families of solutions; without a terminal-speed requirement the attacker reaches the target uncaptured, but adding such a requirement produces simpler trajectories and shows that the guard can drive the flyby distance to zero over a nonempty region of the speed and initial-condition parameter space.

Core claim

Formulating the game with attacker termination at the target together with an auxiliary minimum-separation state removes a structural limitation of standard pursuit games, permits direct embedding of attacker mission constraints, and yields numerical solutions in which an attacker terminal-speed lower bound creates a trade-off between arrival speed and flyby distance that permits capture in a relevant portion of the parameter space.

What carries the argument

An auxiliary state variable that records the minimum separation distance between the players throughout their trajectories, making the flyby distance available as the game value at the instant the attacker reaches the target.

If this is right

  • When the attacker faces no terminal-speed requirement it can reach the target without capture, although the resulting trajectories have long durations and low terminal speeds.
  • Adding an attacker terminal-speed constraint produces simpler solution families and exposes an explicit trade-off between that speed and the achieved flyby distance.
  • Capture becomes possible once the terminal-speed requirement is imposed, and the region of capturability occupies a relevant portion of the parameter space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same auxiliary-state construction can be used to embed other attacker mission constraints such as fuel or time limits without changing the underlying game structure.
  • The numerically obtained solution families supply candidate feedback strategies that could be tested in higher-fidelity simulations that include wind or actuator limits.
  • Extending the formulation to three-dimensional motion would test whether the observed trade-off between terminal speed and flyby distance persists outside the plane.

Load-bearing premise

The auxiliary minimum-separation state can be added to the differential game without altering the players' optimal strategies or introducing extra assumptions on information or control authority.

What would settle it

A set of converged numerical solutions to the two-point boundary-value problem, for the same initial conditions and terminal-speed bound used in the paper, that all produce strictly positive flyby distance.

Figures

Figures reproduced from arXiv: 2606.13012 by Navot Israeli, Oded Golan, Vitaly Shaferman.

Figure 1
Figure 1. Figure 1: Figure 2(a) shows the players’ speeds, which decrease considerably with [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Players’ trajectories with 0.4 Attacker terminal speed constraint. [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Game solutions with 0.4 Attacker terminal speed constraint. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Flyby distances with different Attacker terminal speed constraints. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Type A solutions. first flyby [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Type B solutions. solution with L < LA. Guard’s post-flyby behavior in these examples exhibits some artifacts due to our limited numerical accuracy at long flight times. These artifacts do not significantly affect the flyby distances [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Types A and B engagement performance. 3.3.3 Type C Game Solutions [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Type C solutions. (a) Flyby distance. (b) Final Attacker speed [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Type C solutions performance. suggests that there may be others. Given the complex behavior exhibited here, it may be difficult to identify and map all such solutions. To summarize this section, relaxing the ATS constraint leads to a rich vari￾ety of solutions. We identified four distinct types and suspect that there may be 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Type D solution. additional types. Types A and B offer the Attacker large flyby distances at the cost of slow terminal speeds. Flyby distances in B are larger, and we currently don’t know how the selection mechanism between the two types works. Namely, it is not clear if A can be enforced by the Guard (making B a local saddle point) or B can be enforced by the Attacker (making A a local saddle point). We s… view at source ↗
Figure 10
Figure 10. Figure 10: BZB Attacker trajectory. same head-on scenario of Sec. 3 where the Guard is co-located with the Target and launches at a range L. At the end of the first turn, the Attacker is at r1 = [L − sinL1, 1 − cosL1] T . At the beginning of the second turn, he is at r2 = r1 − L2 · [cosL1, − sinL1] T and is turning left around the pivot point p = r2−[sinL1, cosL1] T = [L − 2 · sinL1 − L2 · cosL1, 1 − 2 · cosL1 + L2 … view at source ↗
Figure 11
Figure 11. Figure 11: BZB branches. (a) L = 2.5. (b) L = √ 8. (c) L = 3 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Optimal BZB trajectories. Attacker with shorter flyby distances than the green ∆, but any variation of L1 at this point will degrade his result. In contrast, the orange ∇ is a maximum of the orange L (s) 2 curve but is not a saddle point. It is located on the long L (l) 2 curve as well, and the Attacker may move from it to the orange ∆ while continually increasing his game value. The same argument holds f… view at source ↗
Figure 13
Figure 13. Figure 13: BZB guess trajectories comparison with FBDP solution. [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: BZB co-states guess comparison with FBDP solution. [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
read the original abstract

Guarding a target against a fast Attacker with a slower Guard is posed as a differential game. Both players follow nonlinear, thrust-free, aerodynamic motion in the plane. The capturability problem inherent in this scenario is addressed by formulating the game such that the Attacker is constrained to terminate at its Target, and the game value is the flyby distance. The latter is made accessible at termination by introducing an auxiliary state variable that records the players' minimum separation distance throughout their flight. This novel combination of termination condition and auxiliary state removes a structural limitation of current pursuit games and opens a new class of solvable problems. It also enables us to embed attacker mission constraints directly. Taking an indirect approach, we formulate the game's Two-Point Boundary Value Problem, solve it numerically, and identify several solution types. When the Attacker's only constraint is to reach the Target, he can do so without being captured. These types of solutions, however, are less practical due to long flight times and slow terminal speeds. Imposing an Attacker terminal speed constraint yields simpler solutions and reveals the trade-off between terminal speed and flyby distance. We demonstrate that in such a setting, capture is possible in a relevant region of the parameter space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper formulates guarding a target against a faster Attacker using a slower Guard as a differential game with nonlinear thrust-free aerodynamic dynamics in the plane. The Attacker is constrained to terminate at the Target, the payoff is the minimum flyby distance, and an auxiliary state variable is introduced to record the running minimum separation so that the value is available at termination. An indirect method yields the TPBVP, which is solved numerically to classify solution families; without a terminal-speed constraint the Attacker reaches the Target without capture (but with long flight times and low terminal speeds), while a terminal-speed constraint produces simpler trajectories and a nonempty capture region in parameter space.

Significance. If the auxiliary-state construction preserves the original saddle strategies, the approach removes a structural limitation of standard pursuit games and directly embeds mission constraints, yielding a new class of solvable problems with concrete numerical evidence of capture regions under speed limits.

major comments (2)
  1. [Formulation and auxiliary state (near Eq. for ˙m)] The auxiliary dynamics (typically ˙m = min(0, separation rate) or smoothed equivalent) are non-Lipschitz; the manuscript must explicitly verify (in the section deriving the Hamiltonian or the TPBVP) that this augmentation leaves the optimal strategies of both players unchanged and does not impose new observability or control-authority assumptions relative to the unaugmented game.
  2. [Numerical results and solution families] The numerical TPBVP solver is used to identify capture regions under the terminal-speed constraint, yet no convergence diagnostics, validation against known analytic cases, or parameter-sensitivity results are reported; without these the claim that capture occurs in a relevant region of parameter space cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract states that 'several solution types' are identified but does not name them; a brief enumeration in the introduction would improve readability.
  2. [Problem formulation] Notation for the auxiliary minimum-separation variable and its terminal condition should be introduced once and used consistently; occasional redefinition risks confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments. We address each major comment below and will incorporate the suggested clarifications and additions in a revised manuscript.

read point-by-point responses

  1. Referee: [Formulation and auxiliary state (near Eq. for ˙m)] The auxiliary dynamics (typically ˙m = min(0, separation rate) or smoothed equivalent) are non-Lipschitz; the manuscript must explicitly verify (in the section deriving the Hamiltonian or the TPBVP) that this augmentation leaves the optimal strategies of both players unchanged and does not impose new observability or control-authority assumptions relative to the unaugmented game.

    Authors: We agree that an explicit verification is warranted given the non-Lipschitz character of the auxiliary dynamics. In the revised manuscript we will add a subsection immediately following the Hamiltonian derivation that demonstrates the augmentation does not alter the players' optimal strategies. Because the auxiliary state m is a passive, non-decreasing recorder whose dynamics do not enter the Hamiltonian or the optimality conditions for the controls, the saddle-point strategies of the original game are recovered unchanged; no additional observability or control-authority assumptions are introduced. revision: yes

  2. Referee: [Numerical results and solution families] The numerical TPBVP solver is used to identify capture regions under the terminal-speed constraint, yet no convergence diagnostics, validation against known analytic cases, or parameter-sensitivity results are reported; without these the claim that capture occurs in a relevant region of parameter space cannot be assessed.

    Authors: We accept that additional numerical validation is required to substantiate the reported capture regions. The revised manuscript will include (i) convergence diagnostics (residual norms and mesh-refinement studies) for the TPBVP solver, (ii) validation against limiting analytic cases (e.g., equal-speed or infinite-guard-speed limits where capture boundaries are known), and (iii) parameter-sensitivity sweeps over speed ratio and initial geometry to quantify the robustness of the identified capture set. revision: yes

Circularity Check

0 steps flagged

No circularity: standard state augmentation for min-distance payoff

full rationale

The derivation introduces an auxiliary state to record minimum separation so that the flyby distance becomes the terminal payoff of the TPBVP. This is a conventional optimal-control technique that does not define the payoff in terms of itself or rename a fitted quantity as a prediction. The paper solves the resulting boundary-value problem numerically, identifies solution families, and reports capture regions under an added terminal-speed constraint; none of these steps reduce by construction to the inputs or to a self-citation chain. The central claim therefore remains independent of the formulation choices.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities used in the formulation or numerical solution.

pith-pipeline@v0.9.1-grok · 5751 in / 1035 out tokens · 19901 ms · 2026-06-27T06:15:23.367736+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references

  1. [1]

    Balakrishnan, Antonios Tsourdos, and Brian A

    Sivasubramanya N. Balakrishnan, Antonios Tsourdos, and Brian A. White, editors.Advances in missile guidance, control, and estimation. Automation and control engineering. CRC Press, Boca Raton, Fla., 2013

    2013

  2. [2]

    published in 1995 by Academic Press, New York

    Tamer Ba¸ sar and Geert Jan Olsder.Dynamic noncooperative game theory: Revised from the 2nd ed. published in 1995 by Academic Press, New York. Number 23 in Classics in applied mathematics. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), Philadelphia, Pa, 2nd ed edition, 1999

    1995

  3. [3]

    Victor M. Becerra. Solving complex optimal control problems at no cost with PSOPT. In2010 IEEE International Symposium on Computer-Aided Control System Design, pages 1391–1396, 2010

    2010

  4. [4]

    Release 5, 2020

    Victor M Becerra.PSOPT Optimal Control Solver User Manual. Release 5, 2020. Available athttps://github.com/PSOPT/psopt/blob/master/ doc/. 25 (a) Guard position co-states. (b) Guard velocity co-states. (c) Attacker position co-states. (d) Attacker velocity co-states. Figure 14: BZB co-states guess comparison with FBDP solution

    2020

  5. [5]

    Ben-Asher and Isaac Yaesh.Advances in missile guidance the- ory

    Joseph Z. Ben-Asher and Isaac Yaesh.Advances in missile guidance the- ory. Number 180 in Progress in astronautics and aeronautics. American Institute of Aeronautics and Astronautics, Reston, Va, 1998

    1998

  6. [6]

    Betts.Practical methods for optimal control and estimation using nonlinear programming

    John T. Betts.Practical methods for optimal control and estimation using nonlinear programming. Number 19 in Advances in design and control. Society for Industrial and Applied Mathematics, Philadelphia, Pa, 2. ed edition, 2010

    2010

  7. [7]

    Taylor & Francis Group, New York Oxon, 1975

    Arthur Earl Bryson and Yu-Chi Ho.Applied optimal control: optimization, estimation, and control. Taylor & Francis Group, New York Oxon, 1975

    1975

  8. [8]

    Carr, Richard G

    Ryan W. Carr, Richard G. Cobb, Meir Pachter, and Scott Pierce. Solution of a Pursuit–Evasion Game Using a Near-Optimal Strategy.Journal of Guidance, Control, and Dynamics, 41(4):841–850, 2018

    2018

  9. [9]

    Number 209 in Progress in astronautics and aeronautics

    Shaul Gutman.Applied min-max approach to missile guidance and control. Number 209 in Progress in astronautics and aeronautics. American Inst. of Aeronautics and Astronautics, Reston, Va, 2005. 26

    2005

  10. [10]

    Kazuhiro Horie and Bruce A. Conway. Optimal Fighter Pursuit-Evasion Maneuvers Found Via Two-Sided Optimization.Journal of Guidance, Con- trol, and Dynamics, 29(1):105–112, 2006

    2006

  11. [11]

    Dover Publica- tions, Mineola, NY, 1999

    Rufus Philip Isaacs.Differential games: a mathematical theory with appli- cations to warfare and pursuit, control and optimization. Dover Publica- tions, Mineola, NY, 1999

    1999

  12. [12]

    Meir Pachter, Eloy Garcia, and David W. Casbeer. Differential Game of Guarding a Target.Journal of Guidance, Control, and Dynamics, 40(11):2991–2998, 2017

    2017

  13. [13]

    Capture Set Computation of an Optimally Guided Missile

    Tuomas Raivio. Capture Set Computation of an Optimally Guided Missile. Journal of Guidance, Control, and Dynamics, 24(6):1167–1175, 2001

    2001

  14. [14]

    Tauqeer ul Islam Rizvi, Lin-shu He, and Da-jun Xu

    S. Tauqeer ul Islam Rizvi, Lin-shu He, and Da-jun Xu. Optimal trajectory and heat load analysis of different shape lifting reentry vehicles for medium range application.Defence Technology, 11(4):350–361, 2015. 27

    2015