Differential Games for Compositional Handling of Competing Control Tasks

Computer Science > Multiagent Systems
[Submitted on 1 Sep 2026]
Title:Differential Games for Compositional Handling of Competing Control Tasks
View PDF HTML (experimental)Abstract:We introduce a novel Divide and Conquer control design methodology leveraging differential games in single-agent, multi-objective dynamical systems. The proposed framework associates each control objective with a virtual input and establishes a non-cooperative, finite or infinite horizon differential game among representative players. Each player optimizes a distinct virtual cost function tailored to its specific goal, the full system state, and the other virtual inputs, while accounting for the remaining players' optimal policies. By establishing a Nash Equilibrium for this game, we synthesize a composite controller that achieves a stable balance across competing objectives, providing control engineers with an intuitive and modular framework for parameter re-tuning throughout the design cycle. We provide formal mathematical derivations for both continuous-time and discrete-time dynamical systems, targeting large-scale single-agent applications where complex, dynamically conflicting control objectives make global weighting intractable. To demonstrate the methodology, we developed an open-source Python package implementing a novel numerical algorithm for solving Coupled Algebraic Riccati Equations arising in infinite-horizon differential games. We evaluate the approach on two benchmark case studies: an inverted pendulum on a cart and a non-linear hierarchically controlled quadrotor. The resulting closed-loop performance is compared against the classical Linear Quadratic Regulator (LQR) across various transient and steady-state control metrics, demonstrating superior trajectory tracking and robust multi-objective regulation.
Submission history
From: Joshua Shay Kricheli [view email][v1] Tue, 1 Sep 2026 20:14:40 UTC (3,536 KB)
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Facts Only

* The methodology uses differential games for single-agent, multi-objective dynamical systems.
* Control objectives are associated with virtual inputs in the framework.
* A non-cooperative differential game is established among representative players.
* Each player optimizes a distinct virtual cost function based on its goal, system state, and other virtual inputs.
* The objective is to find a Nash Equilibrium for the game to synthesize a composite controller.
* Formal mathematical derivations are provided for continuous-time and discrete-time systems.
* A numerical algorithm for solving Coupled Algebraic Riccati Equations in infinite-horizon differential games was implemented in Python.
* Benchmarks included an inverted pendulum on a cart and a non-linear hierarchically controlled quadrotor.
* Closed-loop performance was compared against the Linear Quadratic Regulator (LQR).

Executive Summary

A novel control design methodology is introduced using differential games for handling competing control tasks in single-agent, multi-objective dynamical systems. The framework associates each control objective with a virtual input and sets up a non-cooperative differential game among representative players. Each player optimizes a unique cost function based on its goal, the full system state, and the other players' inputs, considering the optimal policies of all other players. A composite controller is synthesized by finding the Nash Equilibrium of this game to achieve a balanced control strategy across conflicting objectives. The methodology has formal mathematical derivations for both continuous-time and discrete-time systems, aimed at large-scale applications where global weighting of conflicting objectives is difficult. The approach includes an open-source Python package implementing a numerical algorithm for solving Coupled Algebraic Riccati Equations arising in infinite-horizon differential games. Evaluation was performed on an inverted pendulum on a cart and a non-linear hierarchically controlled quadrotor, demonstrating superior trajectory tracking and robust multi-objective regulation compared to the LQR method.

Full Take

The work proposes shifting the problem of multi-objective control from an intractable global weighting issue to a game-theoretic structure, specifically leveraging Nash Equilibrium. This frames control synthesis not as optimization under constraints but as strategic interaction among competing goals. The novelty rests in making this framework modular; control engineers can re-tune parameters intuitively during the design cycle by adjusting the virtual inputs or cost functions. The successful application to physical systems like the inverted pendulum and quadrotor suggests a path toward designing inherently robust, trade-off-aware controllers rather than relying on post-hoc weighting adjustments. The demonstration via solving Coupled Algebraic Riccati Equations highlights the computational tractability of applying advanced game theory methods to complex control problems. A critical implication is that complex, conflicting objectives in large systems can be managed by decomposing the problem into a competitive, equilibrium-seeking process, suggesting a paradigm where competing demands are resolved through strategic negotiation rather than simple scalar trade-offs. Future inquiry should explore how this framework extends to stochastic systems or scenarios involving partially observable states to assess its resilience outside of deterministic benchmarks.

Sentinel — Human

Confidence

The text exhibits the highly structured and formal characteristics of a peer-reviewed academic abstract, strongly suggesting human authorship within the field of control theory.

Signals Detected

Slightly formal and dense academic tone; moderate sentence variance.

High internal coherence, typical of technical research communication.

Standard citation format and structure; the submission history looks like a genuine arXiv filing.

The text reads like an abstract/summary of a genuine, specialized machine learning/control theory paper.

Human Indicators

Presence of specific technical terminology (e.g., Coupled Algebraic Riccati Equations, LQR) and direct citation trails suggest an original academic submission structure.