We study a variant of the classical safe landing optimal control problem in aerospace, introduced by Miele in the Sixties, where the target was to land a spacecraft on the moon by minimizing the consumption of fuel. Assuming that the spacecraft has a failure and that the thrust (representing the control) can act in both vertical directions, the new target becomes to land safely by minimizing time, no matter of what the consumption is. In dependence of the initial data (height, velocity, and fuel), we prove that the optimal control can be of four different kinds, all being piecewise constant. Our analysis covers all possible situations, including the nonexistence of a safe landing strategy due to the lack of fuel or for heights/velocities for which also a total braking is insufficient to stop the spacecraft.<br><br> <b> N.B.</b>: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006

In this talk I will review some recent results obtained in collaboration with E. Runa and A. Kerschbaum on the one-dimensionality of the minimizers of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while minimizers are not. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension. In the discrete setting such results had been previously obtained for a smaller set of functionals with a different approach by Giuliani and Seiringer. <br> <b> N.B.: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006 </b> <br> The seminar will be online at the following <a href="https://teams.microsoft.com/l/meetup-join/19%3ad6108b52495242c887fce0232a1d851f%40thread.tacv2/1605026950198?context=%7b%22Tid%22%3a%2224c5be2a-d764-40c5-9975-82d08ae47d0e%22%2c%22Oid%22%3a%229bfb10cf-6b03-47dc-906c-d23eb368824c%22%7d"> link</a>

In this talk I will present a family of one dimensional systems with random additive noise such that, as the noise size increases, the Lyapunov exponent of the stationary measure transitions from positive to negative. This phenomena is known in literature as Noise Induced Order, and was first observed in a model of the Belosouv-Zhabotinsky reaction and its existence was proven only recently by Galatolo-Monge-Nisoli. In the talk I will show how this phenomena is strictly connected with non-uniform hyperbolicity and the coexistence of regions of expansion and contraction in phase space; the result is attained through a result on the continuity of the Lyapunov exponent of the stationary measure with respect to the size of the noise. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map admits invariant probability measures with positive metric entropy. The proof relies on variational techniques based on Aubry-Mather theory. Joint work with Claudio Bonanno. <br> <br> <strong> Note: </strong> <em> </strong> <em> The zoom link to the seminar will be posted on the <a href="https://www.dinamici.org/dai-seminar/"> DinAmicI website </a> and on <a href="https://mathseminars.org/seminar/DinAmicI"> Mathseminars.org</a>. Moreover, it will be also streamed live via the youtube <a href="https://www.youtube.com/channel/UCyNNg155G3iLS7l-qZjboyg"> DinAmicI channel</a>. </em> </em>

In this paper, by making use of a new variational principle, we prove existence of nontrivial solutions for two different types of semilinear problems with Neumann boundary conditions in unbounded domains. Namely, we study elliptic equations and Hamiltonian systems on the unbounded domain $Omega=R^{m} imes B_r$ where $B_r$ is a ball centered at the origin with radius $r$ in $mathbb{R}^{n}$. Our proofs consist of several new and novel ideas that can be used in broader contexts. This is a joint work with Abbas Moameni that was accepted for publication in Nonlinearity. <b>N.B.</b>: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006.

We consider the problem of following quasi-periodic tori in perturbations of some Hamiltonian systems which involve friction and external forcing. In a first goal, we use different numerical methods (Pade approximants, Newton continuation till boundary) to obtain numerically the domain of convergence. We also study the properties of the asymptotic series of the solution. In a second goal, we study rigorously the (divergent) series of formal expansions of the torus obtained using Lindstedt method. We show that, for some systems in the literature, the series is Gevrey. We hope that the method can be of independent interest: we develop KAM estimates for the divergent series. In contrast with the regular KAM method, we lose control of all the domains, so that there is no convergence, but we can generate enough control to show that the series is Gevrey. This is joint work with R. Calleja and R. de la Llave. This activity is made in collaboration with the Departments of Mathematics of the Universities of Milano, Padova, Pisa and Roma Tor Vergata (Excellence Department project MATH@TOV).

The reduced density matrix of a spatial subsystem can be written as the exponential of the modular Hamiltonian, hence this operator contains a lot of information about the entanglement of the corresponding spatial bipartition. First we consider the massless Dirac field on the half-line, imposing the most general boundary conditions that ensure the global energy conservation. This leads to two inequivalent phases where either the vector or the axial symmetry is preserved. In these two phases, we discuss the analytic expressions for the modular Hamiltonians of an interval on the half-line when the system is in its ground state, for the corresponding modular flows of the Dirac field and for the corresponding modular correlators. The method allows to obtain analytic expressions also for the modular Hamiltonians, the modular flows and the modular correlators for two disjoint equal intervals at the same distance from a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission.