Science, engineering and economics are full of situations in which one observes the evolution of a given system in time.
The systems of interest can differ a lot in nature and their description may require finitely many, as well as
infinitely many, variables. Nevertheless, the above models can be formulated in terms of evolution
equations, a mathematical structure where the dependence on time plays an essential role. Such equations have long been the object of intensive theoretical study as well as the source of an enormous number of applications.
A typical class of problems that have been addressed over the years is concerned with the well-posedness of an evolution equation with given initial and boundary conditions (the so-called direct problems). In several applied situations, however, initial conditions are hard to know exactly while measurements of the solution at different stages of its evolution might be available. Different techniques have been developed to recover, from such pieces of information, specific parameters governing the evolution such as forcing terms or diffusion coefficients. The whole body of results in this direction is usually referred to as inverse problems. A third way to approach the subject is to try to influence the evolution of a given system through some kind of external action called control. Control problems may be of very different nature: one may aim at bringing a given system to a desired configuration in finite or infinite time (positional control), or rather try to optimize a performance criterion (optimal control).
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