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…

Control of the semi-discrete 1D heat equation under nonnegative control constraint

PDF version…  |   Download Code… 1 Introduction In the post IpOpt and AMPL use to solve time optimal control problems, we explain how to use IpOpt and AMPL in order to solve control problems with control constraints and possibly some state constraints. In the present post, we are going to present a numerical development in order…

Kolmogorov equation

PDF version…  |   Download Code… 1 Introduction We are interested in the numerical discretization of the Kolmogorov equation [12] [latex] \begin{equation}\label{kolmo} \left\{ \begin{array}{lll} \partial_t f – \mu \partial_{xx} f – v(x) \partial_y f =0, & (x,y)\in\R^2, t>0,\\ f(x,y,0) =f_0(x,y), & (x,y)\in\R^2 \end{array} \right. \end{equation} [/latex] where $\mu>0$ is a diffusive function and $v$ a potential function.…

IpOpt and AMPL use to solve time optimal control problems

PDF version…  |   Download Code… Featured Video Evolution of the controls and of the state for $y^0=1$, $y^1=5$, $M=20$ and the discretization parameters $N_x=30$, $N_t=450$ in the minimal computed time $T\simeq\mathtt{0.2093}$. Your browser does not support the video tag. Introduction In this short tutorial we explain how to use IpOpt in order to solve time optimal…