arXiv subjects

Tigran Bakaryan

Publications and source records attributed to Tigran Bakaryan.

14 recordsLinked to original sources

arXiv2026

A Mean-Field Game Model For Large-Scale Attrition in Attacker-Defender Systems

This paper proposes a novel Mean-Field Game (MFG) framework for large-scale attacker-defender systems aimed at protecting one or multiple High-Value Units (HVUs). Motivated by classical agent-wise attrition models, we introduce a population-wise attrition mechanism formulated by statistical distance between populations, enabling a macroscopic description of weapon-based interactions between large populations. Leveraging this and Lions derivative on the space of probability measures, we derive the associated MFG system, which characterizes optimal strategies and the evolution of population distributions in attacker-defender interactions. We analyze the model by establishing upper and lower bounds on the defender density, ensuring physical consistency by preventing concentration and depletion. For numerical investigation, we develop a numerical scheme combining physics-informed neural networks with Sinkhorn method to solve attacker-defender MFG system. Simulations confirm the effectiveness of the framework and reveal key insights, including sensitivity to weapon strengths and population dispersion.

math.AP↗

arXiv2026

Distributed MPC For Coordinated Path-Following

In this paper, we consider a distributed model predictive control (MPC) algorithm for coordinated path-following. Relying on the time-critical cooperative path-following framework, which decouples space and time and reduces the coordination problem to a one-dimensional setting, we formulate a distributed MPC scheme for time coordination. Leveraging properties of the normalized Laplacian, we decouple the MPC dynamics into independent modes and derive a recursive relation linking current and predicted states. Using this structure, we prove that, for prediction horizon $K=1$ and a fixed connected communication network, the system is exponentially stable even in the presence of path-following errors. This provides a first result on the convergence analysis of discrete-time distributed MPC schemes within this framework. The proposed approach enables scalable and efficient real-time implementation with low communication overhead. Moreover, in contrast to the time-critical cooperative path-following framework, the optimization-based structure relaxes the reliance on preplanning by allowing the incorporation of mission-specific requirements, such as vehicle limitations, collision avoidance, and conflict resolution. Simulation results demonstrate applicability to complex scenarios, highlighting agility and exponential convergence under communication failures.

math.DS↗

arXiv2026

Game-Theoretic Coordination for Time-Critical Missions of UAV Systems

Coordinated missions involving Unmanned Aerial Vehicles (UAVs) in dynamic environments pose significant challenges in maintaining both coordination and agility. In this paper, relying on the cooperative path following framework and using a game-theoretic formulation, we introduce a novel and scalable approach in which each UAV acts autonomously in different mission conditions. This formulation naturally accommodates heterogeneous and time-varying objectives across the system. In our setting, each UAV optimizes a cost function that incorporates temporal and mission-specific constraints. The optimization is performed within a one-dimensional domain, significantly reducing the computational cost and enabling real-time application to complex and dynamic scenarios. The framework is distributed in structure, enabling global, system-wide coordination (a Nash equilibrium) by using only local information. For ideal systems, we prove the existence and the Nash equilibrium exhibits exponential convergence. Furthermore, we invoke model predictive control (MPC) for non-ideal scenarios. In particular, we propose a discrete-time optimization approach that tackles path-following errors and communication failures, ensuring reliable and agile performance in dynamic and uncertain environments. Simulation results demonstrate the effectiveness and agility of the approach in ensuring successful mission execution across diverse realistic scenarios.

cs.MA↗

arXiv2026

Ranking Mean-Field Planning Games

This paper studies a one-dimensional Mean-Field Planning (MFP) system with a non-local, rank-based coupling. Using a potential formulation, we rewrite the system as an associated scalar partial differential equation. We prove an equivalence between classical solutions to the ranking MFP system with positive density and classical solutions to the associated potential problem, and we derive explicit reconstruction formulas. We then identify a monotonicity structure in the associated operator, which, under strict convexity assumptions, yields uniqueness of classical solutions to the associated problem and, hence, uniqueness of the ranking MFP system up to an additive constant in the value function. Finally, under superlinear growth assumptions, we exploit monotonicity to address existence in a low-regularity setting. By formulating a variational inequality for a q-Laplacian regularized operator, we apply Minty's method to establish the existence of weak solutions in the space of functions of bounded variation for a relaxed potential formulation.

math.AP↗

arXiv2025

Hessian Riemannian Flow For Multi-Population Wardrop Equilibrium

In this paper, we address the problem of optimizing flows on generalized graphs that feature multiple entry points and multiple populations, each with varying cost structures. We tackle this problem by considering the multi-population Wardrop equilibrium, defined through variational inequalities. We rigorously analyze the existence and uniqueness of the Wardrop equilibrium. Furthermore, we introduce an efficient numerical method to find the solution. In particular, we reformulate the equilibrium problem as a distributed optimization problem over subgraphs and introduce a novel Hessian Riemannian flow method, a Riemannian-manifold-projected Hessian flow, to efficiently compute a solution. Finally, we demonstrate the effectiveness of our approach through examples in urban traffic management, including routing for diverse vehicle types and strategies for minimizing emissions in congested environments.

eess.SY↗

arXiv2023

Multi-population opinion dynamics model

We introduce multi-population opinion dynamics models linked to the bounded confidence model, aiming to explore how interactions between individuals contribute to the emergence of consensus, polarization, or fragmentation. Existing models either neglect agent similarities, sacrificing accuracy for scalability, or prioritize accuracy by introducing agent-wise connections, constraining scalability. Our proposed model captures similarities between agents in scalable matter. In our setting, agents similarities are defined by their group affiliations. Specifically, each sub-population is characterized by its distribution, and the closeness between two sub-populations is measured by the Wasserstein distance of their corresponding distributions. This leads to two mutually connected dynamics: micro, the individual-based dynamics, and the macro, the distribution-based one. The individual-wise interactions take into account the population-wise interactions (similarities), and the population-wise interactions are updated based on the individual-wise interactions. We have proven the well-posedness of our models. Additionally, we conducted several simulations to mimic certain complex social phenomena.

math.DS↗

arXiv2023

$C^{1,α}$ Regularity For Stationary Mean-Field Games With Logarithmic Coupling

This paper investigates stationary mean-field games (MFGs) on the torus with Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The primary objective is to understand the existence of $C^{1,α}$ solutions to address the research gap between low-regularity results for bounded and measurable diffusions and the smooth results modeled by the Laplacian. We use the Hopf--Cole transformation to convert the MFG system into a scalar elliptic equation. Then, we apply Morrey space methods to establish the existence and regularity of solutions. The introduction of Morrey space methods offers a novel approach to address regularity issues in the context of MFGs.

math.AP↗

arXiv2022

First-order mean-field games on networks and Wardrop equilibrium

Here, we examine the Wardrop equilibrium model on networks with flow-dependent costs and its connection with stationary mean-field games (MFG). In the first part of this paper, we present the Wardrop and the first-order MFG models on networks. Then, we show how to reformulate the MFG problem into a Wardrop problem and prove that the MFG solution is the Wardrop equilibrium for the corresponding Wardrop problem. Moreover, we prove that the solution of the MFG problem can be recovered using the solution to the associated Wardrop problem. Finally, we study the cost properties and the calibration of MFG with Wardrop travel cost problems. We describe a novel approach to the calibration of MFGs. Further, we show that even simple travel costs can give rise to non-monotone MFGs.

math.AP↗

arXiv2022

A Variational Approach For Price Formation Models In One Dimension

In this paper, we study a class of first-order mean-field games (MFGs) that model price formation. Using Poincar{é} Lemma, we eliminate one of the equations and obtain a variational problem for a single function. This variational problem offers an alternative approach for the numerical solution of the original MFGs system. We show a correspondence between solutions of the MFGs system and the variational problem. Moreover, we address the existence of solutions for the variational problem using the direct method in the calculus of variations. We end the paper with numerical results for a linear-quadratic model.

math.AP↗

arXiv2022

The Potential Method For Price-Formation Models

We consider the mean-field game price formation model introduced by Gomes and Saúde. In this MFG model, agents trade a commodity whose supply can be deterministic or stochastic. Agents maximize profit, taking into account current and future prices. The balance between supply and demand determines the price. We introduce a potential function that converts the MFG into a convex variational problem. This variational formulation is particularly suitable for machine learning approaches. Here, we use a recurrent neural network to solve this problem. In the last section of the paper, we compare our results with known analytical solutions.

math.NA↗

arXiv2021

Discrete Approximation Of Stationary Mean Field Games

In this paper, we focus on stationary (ergodic) mean-field games (MFGs). These games arise in the study of the long-time behavior of finite-horizon MFGs. Motivated by a prior scheme for Hamilton-Jacobi equations introduced in Aubry-Mather's theory, we introduce a discrete approximation to stationary MFGs. Relying on Kakutani's fixed-point theorem, we prove the existence and uniqueness (up to additive constant) of solutions to the discrete problem. Moreover, we show that the solutions to the discrete problem converge, uniformly in the nonlocal case and weakly in the local case, to the classical solutions of the stationary problem.

math.AP↗

arXiv2021

A duality approach to a price formation MFG model

We study the connection between the Aubry-Mather theory and a mean-field game (MFG) price-formation model. We introduce a framework for Mather measures that is suited for constrained time-dependent problems in R. Then, we propose a variational problem on a space of measures, from which we obtain a duality relation involving the MFG problem examined in [36].

math.AP↗

arXiv2021

A Potential Approach for Planning Mean-Field Games in One Dimension

This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton-Jacobi equation coupled with a Fokker-Planck equation. Applying Poincaré's Lemma to the Fokker-Planck equation, we deduce the existence of a potential. Rewriting the Hamilton-Jacobi equation in terms of the potential, we obtain a system of Euler-Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation. We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the one-dimensional Hughes' model.

math.AP↗

arXiv2020

Uniform estimates for the planning problem with potential

In this paper, we study a priori estimates for a first-order mean-field planning problem with a potential. In the theory of mean-field games (MFGs), a priori estimates play a crucial role to prove the existence of classical solutions. In particular, uniform bounds for the density of players' distribution and its inverse are of utmost importance. Here, we investigate a priori bounds for those quantities for a planning problem with a non-vanishing potential. The presence of a potential raises non-trivial difficulties, which we overcome by exploring a displacement-convexity property for the mean-field planning problem with a potential together with Moser's iteration method. We show that if the potential satisfies a certain smallness condition, then a displacement-convexity property holds. This property enables $L^q$ bounds for the density. In the one-dimensional case, the displacement-convexity property also gives $L^q$ bounds for the inverse of the density. Finally, using these $L^q$ estimates and Moser's iteration method, we obtain $L^\infty$ estimates for the density of the distribution of the players and its inverse.

math.AP↗