arXiv subjects

Saeed Sadeghi Arjmand

Publications and source records attributed to Saeed Sadeghi Arjmand.

6 recordsLinked to original sources

arXiv2026

Mean field game model for human capital accumulation under budget constraints

We study a deterministic first-order mean field game model for human capital accumulation in which a continuum of agents invest in education and training to reach a certain qualification level in minimum time. The productivity of learning depends on both the individual's current level of human capital and the distribution of the population, modeling knowledge spillovers and social interactions. Learning is financed by the income generated by existing human capital, leading to a state constraint requiring the financial resources to remain nonnegative throughout the evolution. For a fixed population flow, we formulate the individual problem as a state-constrained minimum-time optimal control problem with time-dependent dynamics. We establish the existence of optimal controls and trajectories, analyze the associated value function through the Dynamic Programming Principle and a Hamilton--Jacobi equation, and characterize the optimal learning effort under suitable regularity and convexity assumptions. The optimal feedback induces a continuity equation describing the evolution of the population distribution. The resulting mean field game is formulated as a fixed-point problem for the population flow. By constructing a compact invariant set and applying Schauder's fixed-point theorem, we prove the existence of at least one equilibrium on every finite population horizon, while the individual minimum-time problem remains formulated on the infinite time horizon.

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arXiv2026

A variational mean field game of controls with free final time and pairwise interactions

This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.

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arXiv2026

Dynamic resource allocation in eukaryotic Resource Balance Analysis

Resource Balance Analysis (RBA) is a framework for predicting steady-state cellular growth under resource constraints. However, classical RBA formulations are static and do not capture the dynamic regulation of biosynthetic resources or macromolecular turnover, which is particularly important in eukaryotic cells. In this work, we propose a dynamic extension of eukaryotic RBA based on an optimal control formulation. Cellular growth is modeled as the result of a time-dependent allocation of translational capacity between metabolic enzymes and macromolecular machinery, aimed at maximizing biomass accumulation over a finite time horizon. Using Pontryagin's Maximum Principle, we characterize optimal allocation strategies and show that steady-state RBA solutions arise as limiting regimes of the dynamic problem.

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arXiv2022

Nonsmooth mean field games with state constraints

In this paper, we consider a mean field game model inspired by crowd motion where agents aim to reach a closed set, called target set, in minimal time. Congestion phenomena are modeled through a constraint on the velocity of an agent that depends on the average density of agents around their position. The model is considered in the presence of state constraints: roughly speaking, these constraints may model walls, columns, fences, hedges, or other kinds of obstacles at the boundary of the domain which agents cannot cross. After providing a more detailed description of the model, the paper recalls some previous results on the existence of equilibria for such games and presents the main difficulties that arise due to the presence of state constraints. Our main contribution is to show that equilibria of the game satisfy a system of coupled partial differential equations, known mean field game system, thanks to recent techniques to characterize optimal controls in the presence of state constraints. These techniques not only allow to deal with state constraints but also require very few regularity assumptions on the dynamics of the agents.

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arXiv2021

Multi-population minimal-time mean field games

In this paper, we consider a mean field game model inspired by crowd motion in which several interacting populations evolving in $\mathbb R^d$ aim at reaching given target sets in minimal time. The movement of each agent is described by a control system depending on their position, the distribution of other agents in the same population, and the distribution of agents on other populations. Thus, interactions between agents occur through their dynamics. We consider in this paper the existence of Lagrangian equilibria to this mean field game, their asymptotic behavior, and their characterization as solutions of a mean field game system, under few regularity assumptions on agents' dynamics. In particular, the mean field game system is established without relying on semiconcavity properties of the value function.

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arXiv2021

On the characterization of equilibria of nonsmooth minimal-time mean field games with state constraints

In this paper, we consider a first-order deterministic mean field game model inspired by crowd motion in which agents moving in a given domain aim to reach a given target set in minimal time. To model interaction between agents, we assume that the maximal speed of an agent is bounded as a function of their position and the distribution of other agents. Moreover, we assume that the state of each agent is subject to the constraint of remaining inside the domain of movement at all times, a natural constraint to model walls, columns, fences, hedges, or other kinds of physical barriers at the boundary of the domain. After recalling results on the existence of Lagrangian equilibria for these mean field games and the main difficulties in their analysis due to the presence of state constraints, we show how recent techniques allow us to characterize optimal controls and deduce that equilibria of the game satisfy a system of partial differential equations, known as the mean field game system.

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