Seminari di Equazioni Differenziali e Applicazioni

Lunedì 11 Aprile 2016, ore 11:30 - Aula 1C150 - Simone Cacace, Anna Chiara Lai



Lunedì 11 Aprile 2016 dalle ore 11:30 in Aula 1C150, si terrano i seguenti seminari.

Simone Cacace (La Sapienza, Università di Roma)
"A new approach to the numerical solution of ergodic problems for Hamilton-Jacobi equations"
We propose a new approach to the numerical solution of ergodic problems arising in the homogenization of Hamilton-Jacobi (HJ) equations. It is based on a Newton-like method for solving inconsistent systems of nonlinear equations, coming from the discretization of the corresponding ergodic HJ equations. We show that our method is able to solve efficiently cell problems in very general contexts, e.g., for first and second order scalar convex and nonconvex Hamiltonians, weakly coupled systems, dislocation dynamics and mean field games, also in the case of more competing populations. We present a large collection of numerical tests in dimension one and two, showing the performance of the proposed method in terms of accuracy, convergence and computational time. Joint work with Fabio Camilli (Sapienza Università di Roma).

Anna Chiara Lai (La Sapienza, Università di Roma)
"Global asymptotic controllability and cost estimates for systems with unbounded controls"
We present some results on control systems characterized by a Lagrangean with non-negative values and by the unboundedness of controls. In the first part of the talk we show sufficient conditions for global asymptotic controllability and a state-dependent upper bound for the infimum value. The result relies on the notion of Mininum Restraint Function — a special kind of Lyapunov Function earlier introduced by Motta and Rampazzo for the case of compact control ranges — and it holds under a quite mild assumption concerning the dependence of the data on inputs. The result applies, for instance, to control vector fields that are compositions of Lipschitz maps with polynomials and exponentials of the control variable. We then discuss the particular case of systems with a polynomial dependence on controls. This class of systems includes control-quadratic systems, that are suitable to model mechanical systems controlled via time-varying, frictionless, holonomic constrains. We show that algebraic and convexity properties of control-polynomial systems provide simplified versions of the main result. This is a joint work with Monica Motta and Franco Rampazzo.

Rif. Int. M. Bardi, C. Marchi, F. Rampazzo.

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