arXiv subjects

Cristian Mendico

Publications and source records attributed to Cristian Mendico.

At least 19 recordsLinked to original sources

arXiv2026

A Fully Discrete Variational Approximation of Mather Measures and Sets

We introduce a fully--discrete variational approximation of Mather measures and sets for Tonelli Lagrangians on the flat torus, together with a numerical procedure for approximating the entire Mather set. The scheme is based on a fully--discrete Lax--Oleinik operator with integer winding labels. We prove an $O(τ+h/τ)$ error estimate for the critical value, convergence of critical solutions, and a finite-dimensional characterization of fully--discrete Mather measures. Accumulation points of the reconstructed minimizing measures are continuous Mather measures, while the supports satisfy complementary upper and lower convergence results involving the Mañé and Mather sets. To avoid the selection of only some minimizing components by exact discrete minimizers, we introduce a mass-threshold approximation based on almost-minimizing holonomic measures. This yields a finite-dimensional constrained optimization procedure designed to recover the whole Mather set.

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arXiv2026

Long-time behavior of generalized gradient flows of solutions to Hamilton-Jacobi equations

We study the long-time behavior of the generalized gradient flow associated with solutions of the critical Hamilton-Jacobi equation for mechanical Hamiltonians on the flat torus. For any semiconcave function, we show that its critical set -- points whose superdifferential contains the zero vector -- acts as an approximate attractor for the flow. When the function is a solution of the critical equation, the critical set decomposes into regular and singular parts, and we establish a dichotomy describing which part trajectories approach as $t \to \infty$. Our analysis uses limiting occupational measures, a class of invariant measures capturing the asymptotic distribution of the flow. An essential ingredient is a complete proof of the global invariance of the singular set, a result previously announced by Albano (2016) but not fully established.

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arXiv2026

Semi-Discrete Approximation of Aubry and Mather sets

We study the semi-discrete approximation of Aubry and Mather sets for Tonelli Lagrangians on the flat torus. Starting from the discrete Lax--Oleinik equation, we introduce natural discrete analogues of these sets and analyze their convergence, as the time step tends to zero, in the sense of Kuratowski. Our results show that the semi-discrete variational framework captures not only the ergodic constant, but also the minimizing invariant geometry of the continuous dynamics. In full generality, we prove upper Kuratowski limit inclusions for both the Aubry and Mather sets. For the Aubry set, we establish full convergence under a hyperbolicity assumption on the continuous Aubry set. For the Mather set, we prove full convergence under a genericity assumption ensuring that the Lagrangian admits finitely many ergodic Mather measures. This provides a first rigorous step toward a structure-preserving approximation theory for Aubry and Mather sets in the Tonelli setting, and clarifies how discrete variational models recover the central geometric objects of weak KAM and Aubry--Mather theory.

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arXiv2026

Multimodal branched transport infers anatomically aligned brain reaction maps

How external stimulation is transformed into distributed reaction patterns remains unresolved at the level of propagation architecture. Existing large-scale control models quantify transition costs on prescribed networks but do not infer the routing map itself from source and target activity. Here we combine task-related blood-oxygen-level-dependent responses, source-reconstructed electrophysiology and tractography-derived anisotropy to estimate stimulation and reaction measures, define an anatomical transport cost, and infer a branched propagation architecture by variational optimisation. Unlike standard transport formulations, branched transport favours aggregation of signal into shared neural highways before redistribution. We further attach a stochastic graph-induced dynamics to the inferred map and quantify the trade-off between geometric efficiency and dynamical controllability. We show that multimodal data generate anatomically aligned brain reaction maps, that anisotropic costs qualitatively reshape routing backbones relative to isotropic baselines, and that hybrid geometric--dynamical optimisation reveals non-trivial rank reversals across branching regimes.

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arXiv2026

Branched Optimal Transport for Stimulus to Reaction Brain Mapping

A central problem in systems neuroscience is to determine how an external stimulation is propagated through the brain so as to produce a reaction. Current deterministic and stochastic control models quantify transition costs between brain states on a prescribed network, but do not treat the transport network itself as an unknown. Here we propose a variational framework in which the inferred object is a graph/current connecting a stimulation source measure to a reaction target measure. The model is posed as an anisotropic branched optimal transport problem, where concavity of the flux cost promotes aggregation and branching. The support of an optimal current defines a stimulus-to-reaction routing architecture, interpreted as a brain reaction map. We prove existence of minimizers in discrete and continuous formulations and introduce a hybrid stochastic extension combining ramified transport with a path-space Kullback--Leibler control cost on the induced graph dynamics. This approach provides a mathematical mechanism for inferring propagation architectures rather than controlling trajectories on fixed substrates.

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arXiv2025

Periodic limit for non-autonomous Lagrangian systems and applications to a Kuramoto type model

This paper explores the asymptotic properties of non-autonomous Lagrangian systems, assuming that the associated Tonelli Lagrangian converges to a time-periodic function. Specifically, given a continuous initial condition, we provide a suitable construction of a Lax-Oleinik semigroup such that it converges toward a periodic solution of the equation. Moreover, the graph of its gradient converges as time tends to infinity to the graph of the gradient of the periodic limit function with respect to the Hausdorff distance. Finally, we apply this result to a Kuramoto-type model, proving the existence of an invariant torus given by the graph of the gradient of the limiting periodic solution of the Hamilton-Jacobi equation.

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arXiv2025

Discretization and Vanishing Discount Problems for First-order Mean Field Games

This article focuses two issues related to the first-order discounted mean field games system. The first is the time discretization problem. The time discretization approach enables us to prove the existence of solutions (u,m) of the system, where u is a viscosity solution of the discounted Hamilton-Jacobi equation and m is a projected minimizing measure satisfying the continuity equation in the sense of distributions. The second is the vanishing discount problems for both the discounted mean field games system and its discretized system. The methods we use primarily derive from weak KAM theory. Moreover, we provide an example demonstrating the non-uniqueness of solutions to the discounted mean field games system.

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arXiv2025

Relaxed Lagrangian Approach to First-Order Non-Convex Mean Field Type Control Problem

This paper addresses the existence of equilibria for Mean Field type Control problems of first-order with non-convex action functional. Introducing a relaxed Lagrangian approach on the Wasserstein space to handle the lack of convexity. we prove the existence of new relaxed Nash equilibria and we show that our existence result encompasses the classical Mean Field Control problem's existence result under convex data conditions.

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arXiv2024

Quantitative and qualitative properties for Hamilton-Jacobi PDEs via the nonlinear adjoint method

We provide some new integral estimates for solutions to Hamilton-Jacobi equations and we discuss several consequences, ranging from $L^p$-rates of convergence for the vanishing viscosity approximation to regularizing effects for the Cauchy problem in the whole Euclidean space and Liouville-type theorems. Our approach is based on duality techniques à la Evans and a careful study of advection-diffusion equations. The optimality of the results is discussed by several examples.

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arXiv2024

Fokker-Planck equations on homogeneous Lie groups and probabilistic counterparts

We address the well-posedness of subelliptic Fokker-Planck equations arising from stochastic control problems, as well as the properties of the associated diffusion processes. Here, the main difficulty arises from the possible polynomial growth of the coefficients, which is related to the growth of the family of vector fields generating the first layer of the associated Lie algebra. We prove the existence and uniqueness of the energy solution, and its representation as the transition density of the underlying subelliptic diffusion process. Moreover, we show its Holder continuity in time w.r.t.the Fortet-Mourier distance, where the Holder seminorm depends on the degree of homogeneity of the vector fields. Finally, we provide a probabilistic proof of the Feyman-Kac formula, as a consequence of the uniform boundedness in finite time intervals of all moments.

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arXiv2024

Aubry-Mather theory for optimal control systems with nonholonomic constraints

In this work, we extend Aubry-Mather theory to the case of control systems with nonholonomic constraints. In this framework, we consider an optimal control problem where admissible trajectories are solutions of a control-affine equation. Such an equation is associated with a family of smooth vector fields that satisfy the Hormander condition, which implies the controllability of the system. In this case, the Hamiltonian fails to be coercive, so results for Tonelli Hamiltonians cannot be applied. To overcome these obstacles, we develop an intrinsic approach based on the metric properties of the geometry induced on the state space by the sub-Riemannian structure.

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arXiv2024

A note on first order quasi-stationary Mean Field Games

Quasi-stationary Mean Field Games models consider agents who base their strategies on current information without forecasting future states. In this paper we address the first-order quasi-stationary Mean Field Games system, which involves an ergodic Hamilton-Jacobi equation and an evolutive continuity equation. Our approach relies on weak KAM theory. We introduce assumptions on the Hamiltonian and coupling cost to ensure continuity of the Peierls barrier and the Aubry set over time. These assumptions, though restrictive, cover interesting cases such as perturbed mechanical Hamiltonians.

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arXiv2024

Rate of convergence for first-order singular perturbation problems: Hamilton-Jacobi-Isaacs equations and mean field games of acceleration

This work focuses on the rate of convergence for singular perturbation problems for first-order Hamilton-Jacobi equations. As an application we derive the rate of convergence for singularly perturbed two-players zero-sum deterministic differential games (i.e., leading to Hamilton-Jacobi-Isaacs equations) and, subsequently, in case of singularly perturbed mean field games of acceleration. Namely, we show that in both the models the rate of convergence is $\varepsilon$.

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arXiv2024

Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time

A classical problem in ergodic continuous time control consists of studying the limit behavior of the optimal value of a discounted cost functional with infinite horizon as the discount factor $λ$ tends to zero. In the literature, this problem has been addressed under various controllability or ergodicity conditions ensuring that the rescaled value function converges uniformly to a constant limit. In this case the limit can be characterized as the unique constant such that a suitable Hamilton-Jacobi equation has at least one continuous viscosity solution. In this paper, we study this problem without such conditions, so that the aforementioned limit needs not be constant. Our main result characterizes the uniform limit (when it exists) as the maximal subsolution of a system of Hamilton-Jacobi equations. Moreover, when such a subsolution is a viscosity solution, we obtain the convergence of optimal values as well as a rate of convergence. This mirrors the analysis of the discrete time case, where we characterize the uniform limit as the supremum over a set of sub-invariant half-lines of the dynamic programming operator. The emerging structure in both discrete and continuous time models shows that the supremum over sub-invariato half-lines with respect to the Lax-Oleinik semigroup/dynamic programming operator, captures the behavior of the limit cost as discount vanishes.

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arXiv2023

Semi-linear parabolic equations on homogeneous Lie groups arising from mean field games

The existence and the uniqueness of solutions to some semilinear parabolic equations on homogeneous Lie groups, namely, the Fokker-Planck equation and the Hamilton-Jacobi equation, are addressed. The anisotropic geometry of the state space plays a crucial role in our analysis and creates several issues that need to be overcome. Indeed, the ellipticity directions span, at any point, subspaces of dimension strictly less than the dimension of the state space. Finally, the above results are used to obtain the short-time existence of classical solutions to the mean field games system defined on an homogenous Lie group.

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arXiv2023

A singular perturbation problem for mean field games of acceleration: application to mean field games of control

We study the singular perturbation problem for mean field game systems with control of acceleration. For such a problem we analyze the behavior of solutions as the acceleration costs vanishes. In this setting the Hamiltonian fails to be strictly convex and coercive w.r.t. the momentum variable and this creates new issues in the analysis of the problem. We show that the limit system is of MFG type: we first study the convergence to the classical MFG system and, then, by a finer analysis of the Euler-Lagrange flow associated with the control of acceleration we show the convergence to a class of, so-called, MFG of control problems.

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arXiv2023

Differential N-players game: Nash equilibria and Mather measures

We study Nash equilibria for the deterministic ergodic N-players game. We introduce pure strategies, mixed strategies and Nash equilibria associated with those. We show that a Nash equilibrium in mixed strategies exists and it is a Mather measure for the Lagrangian system defined by the cost functional. In conclusion, we show that the mean field limit of the N-players game is described by the ergodic PDE's system for a continuum of players.

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arXiv2022

Aubry set for sub-Riemannian control systems

In the paper [P. Cannarsa, C. Mendico, Asymptotic analysis for Hamilton-Jacobi- Bellman equations on Euclidean space, (2021) Arxiv], we proved the existence of the limit as the time horizon goes to infinity of the averaged value function of an optimal control problem. For the classical Tonelli case such a limit is called the critical constant of the problem. In the special case of sub-Riemannian control systems, we also proved the existence of a critical solution, that is, a continuous solution to the Hamilton-Jacobi equation associated with such a constant, which also coincides with its Lax-Oleinik evolution. Here, we focus our attention on the sub- Riemannian case providing a variational representation formula for the critical constant which uses an adapted notion of closed measures. Having such a formula at our disposal, we define and study the Aubry set. First, we investigate dynamical and topological properties of such a set w.r.t. a suitable class of minimizing trajectories of the Lagrangian action. Then, we show that critical solutions to the Hamilton-Jacobi equation are horizontally differentiable and satisfy the equation in classical sense on the Aubry set.

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