Control of PDE and Inverse Problems



Control of PDE

From a mathematical point of view, a control system is a dynamical system on which we can act by means of a control. This system is described by a state variable, and the aim of introducing a control is forcing the system to have a prescribed behavior. For instance, we can try to drive the system from an initial state to a final one, either in a final time or asymptotically as the time goes to infinity. Or we can look for a control minimizing some quantity given by a functional depending on the evolution of the control system.

The properties of a control system given by a PDE depend on the underlying equation. In this way, we consider, for instance, parabolic, hyperbolic or dispersive systems, and linear or nonlinear systems.

The main topics are controllability (exact, to zero or approximately), stabilization and optimal ontrol. They are studied from both theoretical (existence of controls, characterization, etc.) and numerical point of view (discretization, convergence analysis, implementation, etc.).

Inverse problems

An inverse problem is a task where the values of some model parameters must be obtained from the observed data resulting from the model. For instance, finding some properties of the medium from knowledge of some fields interacting with the medium. Inverse problems are natural for several applications. Some examples are Geophysical Prospecting, where given a signal and an unknown obstacle, the problem is to find what does the obstacle look like; and Medical Imaging, where the objective is to find out something about the inside of a body from non-invasive measurements.

We are mainly interested in the mathematical theory of inverse problems where the underlying phenomena is modeled by a Partial Differential Equation (PDE). In this case, an inverse problem is usually stated in this way: given some information about the solutions to a PDE, find the unknown coefficients in the governing equation. Depending on the specific problem, we are concerned with the uniqueness, stabilization and reconstruction of the unknown parameter.



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