Ricerc@Sapienza

In the recent times, several classical parabolic equations, such as the Heat and the Porous Medium equation, have been revisited by replacing the standard time derivative with a fractional one. Time-fractional derivatives are given by convolution integrals of the time derivative with power-law kernels and arise in several phenomena in connection with anomalous diffusion, a class of nonmarkovian stochastic processes which model phenomena with jumps, waiting times and long tail distributions. Moreover, time-fractional derivatives are typical for memory effects in complex systems, for example in visco-elasticity and the study of smart materials.
In this project, we investigate optimal control of time-fractional partial differential equations, with the aim of extending to this framework some tools developed in the classical setting. Note that, from a mathematical point of view, the presence of nonlocal terms with respect to the time variable poses several technical difficulties. For example, the standard integration by part formula is not true for time-fractional derivatives and this require the introduction of new (Lebesgue and Sobolev) functional spaces where the problem has to be posed.