A linear-quadratic partially observed Stackelberg stochastic differential game with multiple followers and its application to multi-agent formation control

Jingtao Shi; Yaozhong Hu; Yichun Li; Yueyang Zheng

arxiv: 2412.07159 · v4 · submitted 2024-12-10 · 🧮 math.OC

A linear-quadratic partially observed Stackelberg stochastic differential game with multiple followers and its application to multi-agent formation control

Yichun Li , Yaozhong Hu , Jingtao Shi , Yueyang Zheng This is my paper

Pith reviewed 2026-05-23 07:24 UTC · model grok-4.3

classification 🧮 math.OC

keywords Stackelberg stochastic differential gamepartially observed controllinear-quadratic problemforward-backward systemmulti-agent formation controlstate decompositionoptimal strategiesasymmetric information

0 comments

The pith

A Stackelberg stochastic differential game with asymmetric partial information yields optimal strategies for leader and followers through state decomposition and forward-backward decoupling.

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

The paper examines a linear-quadratic Stackelberg stochastic differential game involving one leader and multiple followers, where no agent observes full information and followers observe more than the leader. Followers solve standard LQ partially observed problems while the leader solves an indefinite forward-backward LQ problem, both addressed by novel state decomposition, orthogonal decomposition, completion of squares, and decoupling techniques. These yield explicit optimal strategies and extend deterministic multi-agent formation control to the stochastic setting. A sympathetic reader cares because the setup models realistic hierarchical decisions under uncertainty and incomplete information common in coordinated systems.

Core claim

The paper claims that novel state decomposition and orthogonal decomposition overcome the difficulties from partial observability, allowing the followers' standard linear-quadratic partially observed optimal control problems and the leader's forward-backward indefinite linear-quadratic partially observed optimal control problem to be solved via completion of squares and decoupling techniques, thereby obtaining the optimal strategies and extending the deterministic formation control framework to the stochastic case.

What carries the argument

State decomposition combined with orthogonal decomposition and forward-backward linear-quadratic decoupling via completion of squares.

If this is right

Optimal strategies exist explicitly for both the leader and the followers in the partially observed game.
The same decomposition and decoupling approach applies directly to stochastic multi-agent formation control.
The methods relax the constraints previously imposed on admissible controls in the literature.
The framework handles the case where the leader's problem is indefinite while the followers' problems remain standard.
The stochastic extension preserves the hierarchical Stackelberg structure of the deterministic formation model.

Where Pith is reading between the lines

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

The decomposition approach could be tested numerically on specific formation tasks such as vehicle platooning to check convergence under noise.
Similar techniques might adapt to games with more than one leader if the state decomposition can be extended hierarchically.
The relaxation of admissibility conditions opens the possibility of applying the framework to problems with fewer regularity assumptions on controls.

Load-bearing premise

The linear-quadratic structure together with the asymmetric information where followers know more than the leader permits decoupling of the forward-backward system without extra solvability conditions.

What would settle it

A concrete simulation of the multi-agent formation dynamics in which the derived strategies produce costs strictly higher than an alternative admissible control set under the stated partial observations would falsify the optimality claim.

read the original abstract

In this paper, we study a linear-quadratic partially observed Stackelberg stochastic differential game problem in which a single leader and multiple followers are involved. We consider more practical formulation for partial information that none of them can observed the complete information and the followers know more than the leader. Some completely different methods including a novel state decomposition and orthogonal decomposition are applied to overcome the difficulties caused by partially observability which improves the tools and relaxes the constraint condition imposed on admissible control in the existing literature. More precisely, the followers encounter the standard linear-quadratic partially observed optimal control problems, however, a kind of forward-backward indefinite linear-quadratic partially observed optimal control problem is considered by the leader. Instead of maximum principle of forward-backward control systems, inspired by the existing work related to definite case and classical forward control system, some distinct forward-backward linear-quadratic decoupling techniques including the method of completion of squares are applied to solve the leader's problem. More interestingly, we develop the deterministic formation control in multi-agent system with a framework of Stackelberg differential game and extend it to the stochastic case. The optimal strategies are obtained by our theoretical result suitably.

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.

Desk Editor's Note private letter to a colleague

Extends LQ Stackelberg games to asymmetric partial observation with multiple followers and applies it to stochastic formation control, but the indefinite leader decoupling lacks explicit Riccati checks.

read the letter

The paper's core advance is handling a Stackelberg stochastic differential game where a leader and multiple followers have asymmetric partial observations, with followers knowing more than the leader. It uses state decomposition and orthogonal decomposition to reduce the followers' problems to standard partially observed LQ control, then applies completion-of-squares decoupling to the leader's forward-backward indefinite LQ problem instead of the maximum principle. The same framework is used to move deterministic multi-agent formation control into the stochastic setting. These steps relax some constraints that appeared in earlier definite-case work and produce explicit optimal strategies under the stated information structure. The citation pattern is straightforward and builds on the relevant LQ and Stackelberg literature without obvious circularity. The new combination of asymmetric partial information, indefinite costs for the leader, and the formation-control application is the clearest addition. The main soft spot is the treatment of the indefinite case. Completion of squares yields a candidate, but optimality and finite cost still require that the associated Riccati equations admit solutions and that the quadratic form remains bounded below. The abstract indicates these conditions are relaxed from prior literature, yet it supplies no explicit verification or counter-example exclusion for the multi-follower asymmetric setting. If the full derivations contain those checks, the result is on firmer ground; if not, the optimality claim rests on an unverified step. This work is aimed at researchers already working in stochastic differential games and multi-agent control. Readers who need concrete methods for partial-observation Stackelberg problems will get usable techniques, provided they can confirm the indefinite decoupling themselves. It is worth sending to peer review so that referees can examine the Riccati conditions and any numerical checks on the formation-control example.

Referee Report

2 major / 2 minor

Summary. The paper studies a linear-quadratic partially observed Stackelberg stochastic differential game with one leader and multiple followers under asymmetric partial information (followers observe more than the leader). It introduces novel state decomposition and orthogonal decomposition techniques to handle partial observability, solves the followers' standard LQ problems directly, and addresses the leader's forward-backward indefinite LQ problem via completion-of-squares decoupling rather than the maximum principle; the framework is then applied to multi-agent formation control in both deterministic and stochastic settings.

Significance. If the decoupling is shown to be valid, the work would extend Stackelberg game theory to more realistic asymmetric-information settings by relaxing admissible-control constraints from prior literature and would supply a formation-control application that incorporates stochasticity. The explicit differentiation from definite-case and forward-only results is a constructive feature.

major comments (2)

[leader's forward-backward problem] Leader's problem derivation (the forward-backward decoupling section): the claim that completion-of-squares plus the novel decompositions yields the optimal strategy for the indefinite-cost case is load-bearing, yet the manuscript supplies no explicit verification that the resulting Riccati equations admit solutions or that the quadratic cost remains bounded below under the stated partial-observation asymmetry and multi-follower structure; without these checks the candidate strategy is not guaranteed to be optimal or even finite.
[information structure and admissible controls] § on admissible controls and information structure: the relaxation of constraints relative to earlier definite-case papers is asserted, but the precise new admissible set is not compared equation-by-equation with the prior literature, leaving unclear whether the relaxation is merely notational or genuinely enlarges the feasible set while preserving well-posedness.

minor comments (2)

[preliminaries] Notation for the orthogonal decomposition is introduced without an explicit statement of the inner-product space or the projection operator; a short preliminary subsection would improve readability.
[application] The formation-control application section would benefit from a brief remark on how the stochastic noise terms affect the deterministic formation geometry, even if only qualitatively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and will incorporate revisions to strengthen the presentation of the leader's decoupling and the information structure.

read point-by-point responses

Referee: Leader's forward-backward problem derivation: the claim that completion-of-squares plus the novel decompositions yields the optimal strategy for the indefinite-cost case is load-bearing, yet the manuscript supplies no explicit verification that the resulting Riccati equations admit solutions or that the quadratic cost remains bounded below under the stated partial-observation asymmetry and multi-follower structure.

Authors: We agree that explicit verification of Riccati solvability and cost boundedness is necessary to rigorously guarantee optimality and finiteness in the indefinite forward-backward setting. The current manuscript relies on the orthogonal decomposition and completion-of-squares to reduce the problem, inheriting well-posedness from the definite-case literature, but does not supply the required a-priori estimates. In the revision we will add a new subsection (after the decoupling theorem) that states sufficient conditions on the observation matrices and cost weights ensuring unique positive solutions to the coupled Riccati system and a uniform lower bound on the cost functional, together with a brief proof sketch using the state-decomposition properties already introduced. revision: yes
Referee: § on admissible controls and information structure: the relaxation of constraints relative to earlier definite-case papers is asserted, but the precise new admissible set is not compared equation-by-equation with the prior literature, leaving unclear whether the relaxation is merely notational or genuinely enlarges the feasible set while preserving well-posedness.

Authors: We accept that an explicit side-by-side comparison would remove ambiguity. The relaxation consists in allowing controls adapted to the leader's coarser filtration while still requiring square-integrability; this enlarges the set relative to the symmetric-information or full-observation constraints in the cited definite-case works. In the revised version we will insert a short comparison paragraph (or table) in the admissible-controls section that lists the key differences in filtration measurability and integrability conditions, confirming that the new set remains non-empty and that the orthogonal decomposition continues to guarantee well-posedness. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on novel decompositions and standard completion-of-squares techniques applied to new asymmetric-information setting.

full rationale

The paper's central derivation applies novel state decomposition and orthogonal decomposition to handle partial observability, then uses completion-of-squares decoupling for the leader's forward-backward indefinite LQ problem. These steps are presented as extensions of methods from definite-case literature rather than reductions to self-citations or fitted inputs. No equation is shown to equal its own input by construction, no parameter is fitted on a subset and renamed a prediction, and no uniqueness theorem is imported solely from overlapping prior work by the same authors. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Relies on standard domain assumptions from stochastic control; no free parameters or invented entities introduced in the abstract.

axioms (3)

domain assumption System dynamics are linear with additive Brownian motion noise.
Standard assumption for stochastic differential games as described in the problem formulation.
domain assumption Cost functionals are quadratic (possibly indefinite for the leader).
Core to the linear-quadratic setup and the forward-backward problem for the leader.
domain assumption Admissible controls satisfy relaxed constraints enabled by the new decomposition methods.
Explicitly stated as an improvement over existing literature constraints.

pith-pipeline@v0.9.0 · 5750 in / 1528 out tokens · 29026 ms · 2026-05-23T07:24:34.125606+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

followers encounter the standard linear-quadratic partially observed optimal control problems, however, a kind of forward-backward indefinite linear-quadratic partially observed optimal control problem is considered by the leader... distinct forward-backward linear-quadratic decoupling techniques including the method of completion of squares

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

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[1] [1]

Bensoussan, Stochastic Control of Partially Observa ble Systems, Cambridge University Press, Cam- bridge, 1992

A. Bensoussan, Stochastic Control of Partially Observa ble Systems, Cambridge University Press, Cam- bridge, 1992

work page 1992

[2] [2]

Bensoussan, S

A. Bensoussan, S. K. Chen, and S. P . Sethi, The maximum pri nciple for global solutions of stochastic Stackelberg differential games. SIAM J. Control Optim., 53(4), 1956-1981, 2015

work page 1956

[3] [3]

L. Chen, Y . Shen, On a new paradigm of optimal reinsurance : a stochastic Stackelberg differential game between an insurer ad a reinsurer. Astin Bull., 48(2), 905-960, 2018

work page 2018

[4] [4]

Cvitani´ c, J

J. Cvitani´ c, J. F. Zhang, Contract Theory in Continuous-Time Models, Springer-V erlag, Berlin, 2013

work page 2013

[5] [5]

J. A. Fax, R. M. Murray, Information ﬂow and cooperative c ontrol of vehicle formations. IEEE Trans. Automat. Control, 49(9), 1465-1476, 2004. 29

work page 2004

[6] [6]

X. W. Feng, L. Wang, Stackelberg equilibrium with social optima in linear-quadratic-Gaussian mean-ﬁeld system. Math. Control Relat. Fields, 14(2), 769-799, 2024

work page 2024

[7] [7]

D. B. Gu, A differential game approach to formation contr ol. IEEE Trans. Control Syst. Technol. ,16(1), 85-93, 2008

work page 2008

[8] [8]

M. S. Hu, S. L. Ji, and X. L. Xue, Optimization under ration al expectations: a framework of fully coupled forward-backward stochastic linear quadratic systems. Math. Oper. Res., 48(3), 1767-1790, 2023

work page 2023

[9] [9]

J. H. Huang, B. C. Wang and T. H. Xie, Social optimal in lead er-follower mean ﬁeld linear quadratic control. ESAIM Control Optim. Calc. V ar.,27, suppl. Paper No. S12, 2021

work page 2021

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N. Li, S. J. Wang, Linear-quadratic stochastic Stackel berg games of N players for time-delay systems and related FBSDEs. Appl. Math. Optim., 89(3), Paper No. 67, 2024

work page 2024

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N. Li, Z. Y . Y u, Forward-backward stochastic differential equations and linear-quadratic generalized Stack- elberg games. SIAM J. Control Optim., 56(6), 4148-4180, 2018

work page 2018

[12] [12]

N. Li, J. Xiong, and Z. Y . Y u, Linear-quadratic generalized Stackelberg games with jump-diffusion processes and related forward-backward stochastic differential equ ations. Sci. China Math., 64(9), 2091-2116, 2021

work page 2091

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T. Li, S. P . Sethi, A review of dynamic Stackelberg game m odels. Discrete Contin. Dyn. Syst. Ser. B, 22(1), 125-159, 2017

work page 2017

[14] [14]

Z. P . Li, D. Marelli, M. Y . Fu, and H. S. Zhang, LQG differential Stackelberg game under nested observation information patterns. IEEE Trans. Automat. Control, 68(8), 5111-5118, 2023

work page 2023

[15] [15]

A. E. B. Lim, X. Y . Zhou, Linear-quadratic control of bac kward stochastic differential equations. SIAM J. Control Optim., 40(2), 450-474, 2001

work page 2001

[16] [16]

Lin, Distributed UA V formation control using differ ential game approach

W. Lin, Distributed UA V formation control using differ ential game approach. Aerosp. Sci. Technol., 35(1), 54-62, 2014

work page 2014

[17] [17]

Y . N. Lin, X. S. Jiang, and W. H. Zhang, An open-loop Stack elberg strategy for the linear quadratic mean- ﬁeld stochastic differential game. IEEE Trans. Automat. Control, 64(1), 97-110, 2019

work page 2019

[18] [18]

X. K. Liu, J. F. Zhang, and J. M. Wang, Differentially pri vate consensus algorithm for continuous-time heterogeneous multi-agent systems. Automatica, 122, Paper No. 109283, 2020

work page 2020

[19] [19]

S. Y . Lv, J. Xiong, and X. Zhang, Linear quadratic leader -follower stochastic differential games for mean- ﬁeld switching diffusions. Automatica, 154, Paper No. 111072, 2023

work page 2023

[20] [20]

W. J. Meng, J. T. Shi, A linear quadratic stochastic Stac kelberg differential games with time delay. Math. Control Relat. Fields, 12(3), 581-609, 2022

work page 2022

[21] [21]

Moon, Linear-quadratic stochastic leader-followe r differential games for Markov jump-diffusion models

J. Moon, Linear-quadratic stochastic leader-followe r differential games for Markov jump-diffusion models. Automatica, 147, Paper No. 110713, 2023

work page 2023

[22] [22]

Moon, Linear-quadratic stochastic Stackelberg dif ferential games for jump-diffusion systems

J. Moon, Linear-quadratic stochastic Stackelberg dif ferential games for jump-diffusion systems. SIAM J. Control Optim., 59(2), 954-976, 2021

work page 2021

[23] [23]

J. Moon, T. Bas ¸ar, Linear quadratic mean ﬁeld Stackelb erg differential games. Automatica, 97, 200-213, 2018

work page 2018

[24] [24]

J. Moon, T. Bas ¸ar, Separation principle for partially-observed linear-quadratic optimal control for mean-ﬁeld type stochastic systems. IEEE Trans. Automat. Control, DOI: 10.1109/TAC.2024.3409641, 2024

work page doi:10.1109/tac.2024.3409641 2024

[25] [25]

Mylvaganam, A

T. Mylvaganam, A. Astolﬁ, A differential game approach to formation control for a team of agents with one leader. In Proceeding of American Control Conference, Chicago, IL, USA, July 1-3, 2015, 1469-1474

work page 2015

[26] [26]

Øksendal, L

B. Øksendal, L. Sandal, J. Ubøe, Stochastic Stackelber g equilibria with applications to time-dependent newsvendor models. J. Econom. Dynam. Control, 37(7), 1284-1299, 2013

work page 2013

[27] [27]

L. C. G. Rogers, D. Williams, Diffusions, Markov Proces ses, and Martingales, V olume 2: Ito Calculus, 2nd ed., Cambridge University Press, Cambridge, 2000

work page 2000

[28] [28]

J. T. Shi, G. C. Wang, and J. Xiong, Leader-follower stoc hastic differential game with asymmetric informa- tion and applications. Automatica, 63, 60-73, 2016

work page 2016

[29] [29]

Y . Si, J. T. Shi, An overlapping information linear-qua dratic Stackelberg stochastic differential game with two leaders and two followers. Proc. 43rd Chinese Control Conference, 1216-1223, 28-31 July, Kunming, P .R. China, 2024

work page 2024

[30] [30]

H. von. Stackelberg, The Theory of the Market Economy, O xford University Press, London, 1952

work page 1952

[31] [31]

J. R. Sun, J. Xiong, Stochastic linear-quadratic optim al control with partial observation. SIAM J. Control Optim., 61(3), 1231-1247, 2023

work page 2023

[32] [32]

Y . Wang, W. C. Wang, Partially observed mean-ﬁeld Stack elberg stochastic differential game with two followers. Internat. J. Control, 97(9), 1999-2008, 2024

work page 1999

[33] [33]

G. C. Wang, Z. Wu, Kalman-Bucy ﬁltering equations of for ward and backward stochastic systems and applications to recursive optimal control problems. J. Math. Anal. Appl., 342(2), 1280-1296, 2008. 30 YICHUN LI, YAOZHONG HU, JINGTAO SHI, AND YUEYANG ZHENG

work page 2008

[34] [34]

G. C. Wang, Z. Wu, and J. Xiong, A linear-quadratic optim al control problem of forward-backward sto- chastic differential equations with partial information. IEEE Trans. Automat. Control, 60(11), 2904-2916, 2015

work page 2015

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J. M. Wang, J. F. Zhang, and X. K. He, Differentially private distributed algorithms for stochastic aggregative games. Automatica, 142, Paper No. 110440, 2022

work page 2022

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W. M. Wonham, On the separation theorem of stochastic co ntrol. SIAM J. Control Optim., 6(2), 312-326, 1968

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