WKB expansion for a fractional Schrödinger equation with applications to controllability

PDF version | Download Matlab Code… In [3], we develop a WKB analysis for the propagation of the solutions to the following one-dimensional nonlocal Schrödinger equation \begin{align}\label{main_eq} \mathcal{P}_s u:= \left[i\partial_t + \ffl{s}{}\right]u = 0, &\;\; (x,t)\in\RR\times(0,+\infty), \end{align} with highly oscillatory initial datum \begin{align}\label{in_dat} u(x,0) = u_{\textrm{\small in}}(x) e^{i\frac{\xi_0}{\varepsilon} x}:=u_0(x),\;\;\; \xi_0\in\RR. \end{align} In (\ref{main_eq}), $\ffl{s}{}$ is…

Solving an optimal control problem arised in ecology with AMPL

PDF version…  |   Download Code… Introduction We are interested in optimal control problems subject to a class of diffusion-reaction systems that describes the growth and spread of an introduced population of organisms \begin{equation} \label{pde} y_t – y_{xx} = f(y), \quad x\in \mathbb{R}, \quad t \in \mathbb{R}^+, \end{equation} where \begin{equation} \label{fyatet} f(y)=a y(1-y)(\theta-y), \end{equation} is the reaction…

Control of PDEs involving non-local terms

Relevant models in Continuum Mechanics, Mathematical Physics and Biology are of non-local nature. Moreover, these models are applied for the description of several complex phenomena for which a local approach is inappropriate or limiting. In this setting, classical PDE theory fails because of non-locality. Yet many of the existing techniques can be tuned and adapted, although this is often a delicate matter…

Models involving memory terms & hybrid PDE+ODE systems (MHM)

Control theory for PDEs has been quite exhaustively developed for model problems (heat and wave equations). But other important models in applications, of hybrid nature, remain poorly understood. This is particularly the case for models involving memory terms in viscoelasticity. Our recent contributions in this area are inspired in the interpretation of memory models as…