IRIS Università Commerciale Luigi Bocconihttps://iris.unibocconi.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Tue, 25 Jan 2022 00:58:54 GMT2022-01-25T00:58:54Z10131- Mean-field optimal control as Gamma-limit of finite agent controlshttp://hdl.handle.net/11565/4032536Titolo: Mean-field optimal control as Gamma-limit of finite agent controls
Abstract: This paper focuses on the role of a government of a large population of interacting agents as a meanfield optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a Partial Differential Equation of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a Γ-convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11565/40325362019-01-01T00:00:00Z
- Diffusion, optimal transport and Ricci curvaturehttp://hdl.handle.net/11565/4032547Titolo: Diffusion, optimal transport and Ricci curvature
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11565/40325472018-01-01T00:00:00Z
- Global existence results for viscoplasticity at finite strainhttp://hdl.handle.net/11565/4032545Titolo: Global existence results for viscoplasticity at finite strain
Abstract: We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate, and thus depends on the plastic state variable. The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance and energy-dissipation-inequality solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11565/40325452018-01-01T00:00:00Z
- Viscous corrections of the time incremental minimization scheme and Visco-Energetic solutions to rate-independent evolution problemshttp://hdl.handle.net/11565/4032549Titolo: Viscous corrections of the time incremental minimization scheme and Visco-Energetic solutions to rate-independent evolution problems
Abstract: We propose the new notion of Visco-Energetic solutions to rate-independent systems (X, E, d) driven by a time dependent energy E and a dissipation quasi-distance d in a general metric-topological space X. As for the classic Energetic approach, solutions can be obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation quasi-distance d is incremented by a viscous correction δ (for example proportional to the square of the distance d), which penalizes far distance jumps by inducing a localized version of the stability condition. We prove a general convergence result and a typical characterization by Stability and Energy Balance in a setting comparable to the standard energetic one, thus capable of covering a wide range of applications. The new refined Energy Balance condition compensates for the localized stability and provides a careful description of the jump behavior: at every jump the solution follows an optimal transition, which resembles in a suitable variational sense the discrete scheme that has been implemented for the whole construction.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11565/40325492018-01-01T00:00:00Z
- Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measureshttp://hdl.handle.net/11565/4031109Titolo: Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures
Abstract: We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger–Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger–Kakutani and Kantorovich–Wasserstein distances.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11565/40311092018-01-01T00:00:00Z
- A variational approach to the mean field planning problemhttp://hdl.handle.net/11565/4032534Titolo: A variational approach to the mean field planning problem
Abstract: We investigate a first-order mean field planning problem of the form {−∂tu+H(x,Du)=f(x,m)in (0,T)×Rd,∂tm−∇⋅(mHp(x,Du))=0in (0,T)×Rd,m(0,⋅)=m0,m(T,⋅)=mTin Rd, associated to a convex Hamiltonian H with quadratic growth and a monotone interaction term f with polynomial growth. We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density m and of its dual, involving the maximization of an integral functional among all the subsolutions u of an Hamilton-Jacobi equation. Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution (m,u). A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form −∂tu+H(x,Du)≤α, under minimal summability conditions on α, and to a measure-theoretic description of the optimality via a suitable contact-defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11565/40325342019-01-01T00:00:00Z
- Nonlinear diffusion equations and curvature conditions in metric measure spaceshttp://hdl.handle.net/11565/4032543Titolo: Nonlinear diffusion equations and curvature conditions in metric measure spaces
Abstract: The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11565/40325432019-01-01T00:00:00Z
- Reverse approximation of gradient flows as Minimizing Movements: a conjecture by De Giorgihttp://hdl.handle.net/11565/4032532Titolo: Reverse approximation of gradient flows as Minimizing Movements: a conjecture by De Giorgi
Abstract: We consider the Cauchy problem for a gradient flow generated by a continuously differentiable function phi in a Hilbert space H and study the reverse approximation of solutions to by the De Giorgi Minimizing Movement approach.
We prove that if H has finite dimension and phi is quadratically bounded from below (in particular if phi is Lipschitz) then for every solution u there exist perturbations converging to phi in the Lipschitz norm such that u can be approximated by the Minimizing Movement scheme generated by the recursive minimization associated to the perturbations.
We show that the piecewise constant interpolations will converge to u as the time step vanishes. This result solves a question raised by Ennio De Giorgi
We also show that even if H has infinite dimension the above approximation holds for the distinguished class of minimal solutions, that generate all the other solutions to by time reparametrization.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11565/40325322020-01-01T00:00:00Z
- Gradient flows and Evolution Variational Inequalities in metric spaces. I: structural propertieshttp://hdl.handle.net/11565/4032123Titolo: Gradient flows and Evolution Variational Inequalities in metric spaces. I: structural properties
Abstract: This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space (X,d) that can be characterized by Evolution Variational Inequalities (EVI). We present new results concerning the structural properties of solutions to the EVI formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behavior and their link with the geodesic convexity of the driving functional. Under the crucial assumption of the existence of an EVI gradient flow, we will also prove two main results: – the equivalence with the De Giorgi variational characterization of curves of maximal slope; – the convergence of the Minimizing Movement-JKO scheme to the EVI gradient flow, with an explicit and uniform error estimate of order 1/2 with respect to the step size, independent of any geometric hypothesis (as upper or lower curvature bounds) on d. In order to avoid any compactness assumption, we will also introduce a suitable relaxation of the Minimizing Movement algorithm obtained by the Ekeland variational principle, and we will prove its uniform convergence as well.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/11565/40321232020-01-01T00:00:00Z
- Spatially inhomogeneous evolutionary gameshttp://hdl.handle.net/11565/4044253Titolo: Spatially inhomogeneous evolutionary games
Abstract: We introduce and study a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. One of the main novelties of our approach concerns the description of the whole system, which can be represent-dimensional state space (pairs (x, σ) of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution. As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean-field formulation, when the number N of the players goes to infinity, and the initial discrete distribution of positions and strategies converge. To this aim we develop some basic functional analytic tools to deal with interaction dynamics and continuity equations in Banach spaces that could be of independent interest. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/11565/40442532021-01-01T00:00:00Z