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.
Sophisticated mathematical models and algorithms are at the heart of mathematical finance and high-frequency trading. Traders can be regarded as competing rational agents, which is precisely the setting of MFGs. Understanding MFGs will give insights on the stability of financial markets, explain price bubbles, and help evaluate the appropriate levels of regulation.
MFGs also examine socio-economic issues such as economic growth, inequality, adoption of renewable technologies, and segregation. These matters are essential for the balanced growth of any society. MFGs provide a theoretical framework for their analysis and numerical tools for the simulation of different scenarios.
Experimental data show with clear evidence that for some viscoelastic materials, such as polymers, suspensions and emulsions, one has to take into account the effects of the so-called "memory": the present state depends also on the previous ones. The evolutive models of materials with memory lead to particular wave equations with integro-differential terms, which represent the memory of the system and take into consideration the characteristics of the material. By the use of fractional calculus , the linear theory of viscoelasticity may be enlarged to innovative materials