Martino Bardi - Academia.edu (original) (raw)

Papers by Martino Bardi

Lecture Notes in Control and Information Sciences, 1991

... is only one controller (A1) is equivalent to the local controllability of the continuous-time... more ... is only one controller (A1) is equivalent to the local controllability of the continuous-time system around the whole target, for smooth targets; (A2) is a sort of local controllability assumption on the discrete-time system for all small time-step h. For a true pursuit-evasion game we ...

Discrete and Continuous Dynamical Systems, 2000

X (t)= f (X (t), α (t)), Y (t)= g (Y (t), β (t)) with the control constraints α (t)∈ A, β (t)∈ B ... more X (t)= f (X (t), α (t)), Y (t)= g (Y (t), β (t)) with the control constraints α (t)∈ A, β (t)∈ B and the state constraints X (t)∈ Ω1, Y (t)∈ Ω2, where A and B are compact subsets of R m while Ω1⊂ R n1 and Ω2⊂ R n2 are bounded open sets. We are also given a closed target set T ...

Lecture Notes in Control and Information Sciences, 1989

... [13] O. Hajek: Pursuit Games, Academic Press, New York 1975. [141 H. Hermes: Feedback sgnthes... more ... [13] O. Hajek: Pursuit Games, Academic Press, New York 1975. [141 H. Hermes: Feedback sgnthesis and positive, local solutions to Hamilton-Jacobi-Bellman equations, Proc. ... [19] EY Rodin: A pursuit-evasion bibliographg-version I, Comput. Math. Applic. 15 (1987), 275-340.

SIAM Journal on Mathematical Analysis, 1991

Journal of Differential Equations, 2016

Lecture Notes in Mathematics, 2000

ABSTRACT Without Abstract

Networks and Heterogeneous Media, 2012

ABSTRACT We consider N -person differential games involving linear systems affected by white nois... more ABSTRACT We consider N -person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Plank equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number N of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [19]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.

European Journal of Control, Dec 31, 2011

We model the parameters of a control problem as an ergodic diffusion process evolving at a faster... more We model the parameters of a control problem as an ergodic diffusion process evolving at a faster time scale than the state variables. We study the asymptotics as the speed of the parameters gets large. We prove the convergence of the value function to the solution of a limit Cauchy problem for a Hamilton-Jacobi equation whose Hamiltonian is a suitable average of the initial one. We give several examples where the effective Hamiltonian allows to define a limit (deterministic) control problem whose dynamics and payoff are linear or nonlinear averages of the initial data. This is therefore a constant-parameter approximation of the control problem with random entries.

Applied Mathematics and Optimization, Jun 30, 1991

We present two convergence theorems for Hamilton Jacobi equations and we apply them to the conver... more We present two convergence theorems for Hamilton Jacobi equations and we apply them to the convergence of approximations and perturbations of optimal control problems and of two-players zero-sum differential games. One of our results is, for instance, the following. Let T and T h be the minimal time functions to reach the origin of two control systems y' = f(y, a) and y' = fh(Y, a), both locally controllable in the origin, and let ~ be any compact set of points controllable to the origin. If 11 fhf II ~ < Ch, then IT(x)-Th(x)l < Cgh", for all x ~ o,~, where ~ is the exponent of H61der continuity of T(x).

Modern Birkhauser Classics Many of the original research and survey monographs in pure and applie... more Modern Birkhauser Classics Many of the original research and survey monographs in pure and applied mathematics published by Birkhauser in recent decades have been groundbreaking and have come to be regarded as foun-dational to the subject. Through the MBC Series, ...

We consider the short time behaviour of stochastic systems affected by a stochastic volatility ev... more We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenisation and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity.

Indiana University Mathematics Journal, 2003

Page 1. Propagation of Maxima and Strong Maximum Principle for Viscosity Solutions of Degenerate ... more Page 1. Propagation of Maxima and Strong Maximum Principle for Viscosity Solutions of Degenerate Elliptic Equations. II: Concave Operators MARTINO BARDI & FRANCESCA DA LIO ABSTRACT. ... 52, No. 3 (2003) Page 2. 608 MARTINO BARDI & FRANCESCA DA LIO ...

Bollettino Dell Unione Matematica Italiana, Feb 11, 2008

We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type e... more We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group.

... II, entitled" An Introduction to Gambling Theory and Its Applications to Stochastic Game... more ... II, entitled" An Introduction to Gambling Theory and Its Applications to Stochastic Games," A. Maitra ... such as Discrete Red and Black, Get-ting to a Set, and Avoiding Bankruptcy. ... to the countable state space case, avoiding Bewley-Kohlberg machinery for undiscounted games. ...

We study general 2nd order fully nonlinear degenerate elliptic equations on an arbitrary closed s... more We study general 2nd order fully nonlinear degenerate elliptic equations on an arbitrary closed set with generalized Dirichlet boundary conditions in the viscosity sense. We prove some properties of the maximal subsolution and the minimal supersolution of the Dirichlet type problem. Under a sort of compatibility condition on the boundary data we show that the maximal subsolution is the natural generalized solution of the boundary value problem, even if it is not necessarily continuous, and we give approximation theorems, in particular by penalization. We apply these results to the Hamilton-Jacobi-Bellman-Isaacs equations of time-optimal stochastic control and pursuit-evasion games whose value functions, which are discontinuous in general, are characterized as the unique solution in the previous sense of the appropriate Dirichlet type problem.

Siam Journal on Mathematical Analysis, Aug 1, 2006

Simple explicit estimates are presented for the viscosity solution of the Cauchy problem for the ... more Simple explicit estimates are presented for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation where either the Hamiltonian or the initial data are the sum of a convex and a concave function. The estimates become equalities whenever a "minmax" equals a "maxmin" and thus a representation formula for the solution is obtained, generalizing the classical Hopf formulas as well as some formulas of Kružkov [Functional Anal. Appl., 2 (1969), pp. 128-136].

We consider stochastic control systems affected by a fast mean reverting volatility Y(t)Y(t)Y(t) driven... more We consider stochastic control systems affected by a fast mean reverting volatility Y(t)Y(t)Y(t) driven by a pure jump L\'evy process. Motivated by a large literature on financial models, we assume that Y(t)Y(t)Y(t) evolves at a faster time scale fractvarepsilon\frac{t}{\varepsilon}fractvarepsilon than the assets, and we study the asymptotics as varepsilonto0\varepsilon\to 0varepsilonto0. This is a singular perturbation problem that we study mostly by PDE methods within the theory of viscosity solutions.