Rodrigo Lecaros

Control of 2D scalar conservation laws in the presence of shocks

We analyze a model optimal control problem for a 2D scalar conservation law: The so-called inverse design problem, the goal being to identify the initial datum leading to a given final time configuration. The presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We develop an alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. A numerical implementation is proposed using splitting and finite differences. The descent method we propose is of alternating nature and combines variations taking account of the shock location and those that take care of the smooth components of the solution. The efficiency of the method is illustrated by numerical experiments.

Tracking Control of 1D Scalar Conservation Laws in the Presence of Shocks

We analyze a model tracking problem for a 1D scalar conservation law. It consists in optimizing the initial datum so to minimize a weighted distance to a given target during a given finite time horizon.

Determination of the Calcium channel distribution in the olfactory system

Abstract In this paper we study a linear inverse problem with a biological interpretation, which is modeled by a Fredholm integral equation of the first kind. When the kernel in the Fredholm equation is represented by step functions, we obtain identifiability, stability and reconstruction results. Furthermore, we provide a numerical reconstruction algorithm for the kernel, whose main feature is that a non-regular mesh has to be used to ensure the invertibility of the matrix representing the numerical discretization of the system. Finally, a second identifiability result for a polynomial approximation of degree less than nine of the kernel is also established.

CONTROL OF UNDERWATER VEHICLES IN INVISCID FLUIDS

In this paper, we investigate the controllability of an underwater vehicle immersed in an infinite volume of an inviscid fluid whose flow is assumed to be irrotational. Taking as control input the flow of the fluid through a part of the boundary of the rigid body, we obtain a finite-dimensional system similar to Kirchhoff laws in which the control input appears through both linear terms (with time derivative) and bilinear terms. Applying Coron’s return method, we establish some local controllability results for the position and velocities of the underwater vehicle. Examples with six, four, or only three controls inputs are given for a vehicle with an ellipsoidal shape.

Multicolor image segmentation using Ambrosio-Tortorelli approximation

Our research aims at image segmentation using the variational framework of Mumford and Shah, following an approximation proposed by Ambrosio and Tortorelli. This technique circumvents the use of parametric contours and implicit level-set techniques, where its solution may be regarded as a soft segmentation, with a number the levels or colors being 2N. On the other hand, the implementation was based on an finite difference discretization, where two - and four - color cases are described with their corresponding numerical results.