Abstract : This minisymposium highlights recent applications of control-theoretic methods in the study of problems arising in mathematical physics and engineering. Special attention is paid to the interplay between deterministic controllability of dynamical systems and qualitative properties of their counterparts driven by stochastic noise. Both, the regulation of dynamical systems via control forces, and the qualitative investigation of randomness in mathematical models, are strongly motivated by real-world applications. The talks will reflect on current challenges concerning finite-dimensional and infinite-dimensional stochastic systems, deterministic control theory, and the synergetic effects that arise when bringing both topics closer together.
Abstract : In this talk, we will review some recent results where controllability is used to derive mixing for PDEs perturbed by bounded noise.
[05602] Multi-bubble blow-ups and multi-solitons to focusing (stochastic) nonlinear Schrödinger equations
Format : Talk at Waseda University
Author(s) :
Deng Zhang (Shanghai Jiao Tong University)
Abstract : In this talk we mainly review the recent results on multi-bubble blow-ups and multi-solitons to the focusing (stochastic) nonlinear Schrödinger equations. In the mass-critical case, the construction and conditional uniqueness of multi-bubble Bourgain-Wang type blow-up solutions will be presented, which provide new examples for the mass quantization conjecture. In the deterministic case without noise, this also provides new examples of non-pure multi-solitons (including dispersive part) for the soliton resolution conjecture. Furthermore, the refined uniqueness of pure multi-bubble blow-ups and multi-solitons in the very low asymptotic regime is obtained. At last, in both the mass critical and subcritical cases, the direct construction of stochastic multi-solitons will also be presented, particularly, in the absence of the classical pseudo-conformal symmetry.
[04445] Controllability results for a class of bilinear degenerate wave equations
Format : Talk at Waseda University
Author(s) :
Piermarco Cannarsa (University of Rome Tor Vergata)
Patrick Martinez (Université de Toulouse)
Cristina Urbani (University of Rome Tor Vergata & Accademia Nazionale dei Lincei)
Abstract : I will present a result of exact controllability along the ground state for a degenerate wave equation by means of a bilinear control. We prove that there exists a threshold time $T_0$ such that: for $T>T_0$ (and $T=T_0$ and strong degeneracy) a classical controllability result can be achieved; for $T
[02831] Small-time approximate controllability for nonlinear Schrödinger equations via bilinear controls
Format : Talk at Waseda University
Author(s) :
Alessandro Duca (Centre Inria Nancy - Grand Est)
Abstract : Consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension in presence of an external potential field whose time-dependent amplitude plays the role of control. We ensure the approximate controllability between eigenstates in arbitrarily small time with respect to the $L^2-$norm. We use specific saturation properties to develop a multiplicative version of the geometric approach introduced for additive controls by Agrachev and Sarychev.
[03022] Small-time control of bilinear PDEs via infinite-dimensional Lie brackets
Format : Online Talk on Zoom
Author(s) :
Eugenio Pozzoli (Università di Bari)
Abstract : We consider PDEs with multiplicative control terms, such as Schrödinger, heat and wave equations. Following a bilinear control strategy recently introduced by Duca and Nersesyan, that is a small-time asymptotic of conjugated dynamics, we investigate some approximate controllability properties of these systems, which hold in arbitrarily small times. We moreover comment on the relation between the controllability properties of the systems and the Lie brackets (a.k.a. commutators) of the operators that generate the dynamics.
