Rossana Capuani

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Scholarly Contributions

Chapters

• Cannarsa, P., & Capuani, R. (2018). Existence and Uniqueness for Mean Field Games with State Constraints. In Springer INdAM Series. doi:10.1007/978-3-030-01947-1_3
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  In this paper, we study deterministic mean field games for agents who operate in a bounded domain. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of the solution to the associated minimization problem is no longer guaranteed. We attack the problem by interpreting equilibria as measures in a space of arcs. In such a relaxed environment the existence of solutions follows by set-valued fixed point arguments. Then, we give a uniqueness result for such equilibria under a classical monotonicity assumption.

Journals/Publications

• Bagagiolo, F., Capuani, R., & Marzufero, L. (2025). A zero-sum differential game for two opponent masses. Partial Differential Equations and Applications, 6(Issue 3). doi:10.1007/s42985-025-00322-5
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  We investigate an infinite dimensional partial differential equation of Isaacs’ type, which arises from a zero-sum differential game between two masses. The evolution of the two masses is described by a controlled transport/continuity equation, where the control is given by the velocity vector field. Our study is set in the framework of the viscosity solutions theory in Hilbert spaces, and we prove the uniqueness of the value functions as solutions of the Isaacs equation.
• Bagagiolo, F., Capuani, R., & Marzufero, L. (2024). A single player and a mass of agents: A pursuit evasion-like game. ESAIM - Control, Optimisation and Calculus of Variations, 30. doi:10.1051/cocv/2024009
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  We study a finite-horizon differential game of pursuit-evasion like, between a single player and a mass of agents. The player and the mass directly control their own evolution, which for the mass is given by a first order PDE of transport equation type. Using also an adapted concept of non-anticipating strategies, we derive an infinite dimensional Isaacs equation, and by dynamic programming techniques we prove that the value function is the unique viscosity solution on a suitable invariant subset of a Hilbert space.
• Capuani, R., Marigonda, A., & Quincampoix, M. (2024). Set-Driven Evolution for Multiagent System. Journal of Optimization Theory and Applications, 200(1). doi:10.1007/s10957-023-02344-8
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  We consider the deterministic evolution in the Euclidean space of a multiagent system with a large number of agents (possibly infinitely many). At each instant of time, besides from time and its current position, the set of velocities available to each agent is influenced by the set described by the current position of all the other agents. The latter is in turn determined by the overall motion of the crowd of all the agents. The interplay to the microscopical point of view of each single agent, and the macroscopical one of the set-evolution yields a non-trivial dynamical system. This two-level multiagent system can be described either by the evolution of a probability measure—describing the instantaneous density of the crowd—or by the evolution of a set—describing the positions where there is at least one agent. In this paper, we precise the links between the two descriptions, providing also some quantitative estimates on the macroscopical admissible evolutions.
• Capuani, R., Marigonda, A., & Ricciardi, M. (2023). Random Lift of Set Valued Maps and Applications to Multiagent Dynamics. Set-Valued and Variational Analysis, 31(3). doi:10.1007/s11228-023-00693-0
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  We introduce an abstract framework for the study of general mean field games and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set valued map expressing the admissible trajectories for the microscopical agents. The techniques used can be applied to consider a broad class of dependence between the trajectories of the single agent and the state of the system. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system.
• Cannarsa, P., Capuani, R., & Cardaliaguet, P. (2021). Mean field games with state constraints: from mild to pointwise solutions of the PDE system. Calculus of Variations and Partial Differential Equations, 60(3). doi:10.1007/s00526-021-01936-4
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  Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we show a global semiconvavity property of the value function associated with optimal control problems with state constraints.
• Capuani, R., Dutta, P., & Nguyen, K. (2021). Metric entropy for functions of bounded total generalized variation. SIAM Journal on Mathematical Analysis, 53(1). doi:10.1137/20M1310953
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  We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space (E, ρ ) up to an accuracy of ∊ > 0 with respect to the L1-distance. Such an estimate is explicitly computed in terms of doubling and packing dimensions of (E, ρ ). The obtained result is applied to provide an upper bound on the metric entropy for a set of entropy admissible weak solutions to scalar conservation laws in one-dimensional space with weakly genuinely nonlinear fluxes.
• Capuani, R., Nguyen, K., & Gilmore, S. (2020). A model of debt with bankruptcy risk and currency devaluation. Minimax Theory and its Applications, 5(2).
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  The paper studies a system of Hamilton-Jacobi equations, arising from a stochastic optimal debt management problem in an infinite time horizon with exponential discount, modeled as a noncooperative interaction between a borrower and a pool of risk-neutral lenders. In this model, the borrower is a sovereign state that can decide how much to devaluate its currency and which fraction of its income should be used to repay the debt. Moreover, the borrower has the possibility of going bankrupt at a random time and must declare bankruptcy if the debt reaches a threshold x∗. When bankruptcy occurs, the lenders only recover a fraction of their capital. To offset the possible loss of part of their investment, the lenders buy bonds at a discounted price which is not given a priori. This leads to a nonstandard optimal control problem. We establish an existence result of solutions to this system and in turn recover optimal feedback payment strategy u∗(x) and currency devaluation v∗(x). In addition, the behavior of (u∗, v∗) near 0 and x∗ is studied.
• Cannarsa, P., Capuani, R., & Cardaliaguet, P. (2019). C1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games. Mathematics In Engineering, 1(1). doi:10.3934/Mine.2018.1.174
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  We derive necessary optimality conditions for minimizers of regular functionals in the calculus of variations under smooth state constraints. In the literature, this classical problem is widely investigated. The novelty of our result lies in the fact that the presence of state constraints enters the Euler-Lagrange equations as a local feedback, which allows to derive the C1;1-smoothness of solutions. As an application, we discuss a constrained Mean Field Games problem, for which our optimality conditions allow to construct Lipschitz relaxed solutions, thus improving an existence result due to the first two authors.

Proceedings Publications

• Capuani, R., & Marigonda, A. (2024). A Mean Field Model for Counter CBRN Threats. In Large-Scale Scientific Computations, LSSC 2023.
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  In this paper we propose a model to optimize the allocation of resources by a decontamination team facing a CBRN emergency. Due to the possible diffusion of the hazardous or noxious substances in the environment, the team must implement a dynamic strategy which can be seen as a coupling between the countermeasures and the concentration of the threats (which can be seen as a measure of the risk). The aim is to prove the existence of an optimal strategy, and some necessary conditions.
• Capuani, R., & Marigonda, A. (2022). Constrained Mean Field Games Equilibria as Fixed Point of Random Lifting of Set-Valued Maps. In 25th IFAC Symposium on Mathematical Theory of Networks and Systems.
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  We introduce an abstract framework for the study of general mean field game and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set-valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set-valued map expressing the admissible trajectories for the microscopical agents. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system.
• Capuani, R., Marigonda, A., & Mogentale, M. (2022). Random Lifting of Set-Valued Maps. In 13th International Conference on Large-Scale Scientific Computations, LSSC 2021.
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  In this paper we discuss the properties of particular set-valued maps in the space of probability measures on a finite-dimensional space that are constructed by mean of a suitable lift of set-valued map in the underlying space. In particular, we are interested to establish under which conditions some good regularity properties of the original set-valued map are inherited by the lifted one. The main motivation for the study is represented by multi-agent systems, i.e., finite-dimensional systems where the number of (microscopic) agents is so large that only macroscopical description are actually available. The macroscopical behaviour is thus expressed by the superposition of the behaviours of the microscopic agents. Using the common description of the state of a multi-agent system by mean of a time-dependent probability measure, expressing the fraction of agents contained in a region at a given time moment, the results of this paper yield regularity results for the macroscopical behaviour of the system.