# Control of PDE and Inverse Problems

**Members:** Nicolás Carreño, Patricio Guzmán, Rodrigo Lecaros, Alberto Mercado

Given a mathematical model, in this research area we are concerned with problems related to obtaining information by indirect or partial measurements of a given variable, or the feasibility of imposing a particular behavior of the system by modifying some parameters. Usually, the model describes some phenomenon in sciences/engineering, and is given by a system of partial differential equations (PDE’s).

We can wonder if we can retrieve properties of a medium by knowing how the system responds to some perturbations or by measuring some particular solution, which is the typical form of an inverse problem. On the other hand, we can think about choosing some source term in a given equation in order to drive the state from a given initial condition to the rest, which is a controllability question and is one of the main topics of control theory.

In our group, the research focuses on control and inverse problems for mathematical models coming from a variety of situations in nature, such as water waves, diffusion and vibration phenomena, inviscid fluids, cell biology, underwater vehicles, to name some. These models are described by linear and nonlinear parabolic, hyperbolic, and dispersive equations, including coupled systems. Our research lies in the intersection of functional and complex analysis, theory of PDE’s, calculus of variations and other fields, using mathematical tools as spectral analysis, Carleman estimates, Lyapunov techniques, and integral transformations, among others.