arXiv subjects

Cristian Mendico

Publications and source records attributed to Cristian Mendico.

24 records · Page 2Linked to original sources

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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arXiv2021

Asymptotic analysis for Hamilton-Jacobi-Bellman equations on Euclidean space

The long-time average behavior of the value function in the calculus of variations is known to be connected to the existence of the limit of the corresponding Abel means. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical Hamilton-Jacobi equation. The goal of this paper is to address similar issues when set on the whole Euclidean space and the Hamiltonian fails to be Tonelli. We first study the convergence of the time-averaged value function as the time horizon goes to infinity, proving the existence of the critical constant for a general control system. Then, we show that the ergodic equation admits solutions for systems associated with a family of vector fields which satisfies the Lie Algebra rank condition. Finally, we construct a critical solution of the HJB equation on the whole space which coincides with its Lax-Oleinik evolution.

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arXiv2020

Ergodic behavior of control and mean field games problems depending on acceleration

The goal of this paper is to study the long time behavior of solutions of the first-order mean field game (MFG) systems with a control on the acceleration. The main issue for this is the lack of small time controllability of the problem, which prevents to define the associated ergodic mean field game problem in the standard way. To overcome this issue, we first study the long-time average of optimal control problems with control on the acceleration: we prove that the time average of the value function converges to an ergodic constant and represent this ergodic constant as a minimum of a Lagrangian over a suitable class of closed probability measure. This characterization leads us to define the ergodic MFG problem as a fixed-point problem on the set of closed probability measures. Then we also show that this MFG ergodic problem has at least one solution, that the associated ergodic constant is unique under the standard mono-tonicity assumption and that the time-average of the value function of the time-dependent MFG problem with control of acceleration converges to this ergodic constant.

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arXiv2020

Weak KAM approach to first-order Mean Field Games with state constraints

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon $T$ goes to infinity. For this purpose, we analyze first Hamilton-Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on $[0,T]$ converges to the solution of the ergodic system as $T \to +\infty$.

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arXiv2019

Mild and weak solutions of Mean Field Games problem for linear control systems

The aim of this paper is to study first order Mean field games subject to a linear controlled dynamics on $\mathbb R^{d}$. For this kind of problems, we define Nash equilibria (called Mean Field Games equilibria), as Borel probability measures on the space of admissible trajectories, and mild solutions as solutions associated with such equilibria. Moreover, we prove the existence and uniqueness of mild solutions and we study their regularity: we prove Hölder regularity of Mean Field Games equilibria and fractional semiconcavity for the value function of the underlying optimal control problem. Finally, we address the PDEs system associated with the Mean Field Games problem and we prove that the class of mild solutions coincides with a suitable class of weak solutions.

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arXiv2018

Long Time Behavior of First Order Mean Field Games on Euclidean Space

The aim of this paper is to study the long time behavior of solutions to deterministic mean field games systems on Euclidean space. This problem was addressed on the torus ${\mathbb T}^n$ in [P. Cardaliaguet, {\it Long time average of first order mean field games and weak KAM theory}, Dyn. Games Appl. 3 (2013), 473-488], where solutions are shown to converge to the solution of a certain ergodic mean field games system on ${\mathbb T}^n$. By adapting the approach in [A. Fathi, E. Maderna, {\it Weak KAM theorem on non compact manifolds}, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 1-27], we identify structural conditions on the Lagrangian, under which the corresponding ergodic system can be solved in $\mathbb{R}^{n}$. Then we show that time dependent solutions converge to the solution of such a stationary system on all compact subsets of the whole space.

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