Analisi e Controllo di Modelli Evolutivi con Fenomeni Non Locali
(Analysis and Control of Evolutionary Models with Nonlocal Phenomena)
The Site of an Italian GNAMPA-INdAM 2025 Project on Mathematical Analysis
This Research Project brings together experts from different research topics but linked by a common thread: study of differential equations and inclusions/variational inequalities, of an evolutionary type, with non-local phenomena. By "non-local phenomena" here we mean functional relationships which involve the whole values that a function takes on its domain or on some part of it. Two important examples of these phenomena are: i) "rate independence", i.e. memory-type relationships between time-dependent functions that do not depend on the time derivative of the functions but only on the values that they have reached in their history; ii) “downstream traffic”, speed changes in vehicular traffic that depend on the density of vehicles “downstream” of the driver.
More specific examples of the models we intend to study are:
Conservation laws and/or first order hyperbolic systems with flow dependent on the past history of the solution (temporal non-locality: memory, rate-independent hysteresis, delay), or dependent on values that the solution takes on other areas of space ( spatial non-locality: vehicular traffic with dependence on "downstream" density, opening and closing of valves for the regulation of fluid flows and related threshold effects).
Problems of controllability and optimal control, even infinite dimensional, in which non-locality is represented by phenomena such as those in point 1), or by the fact that the state of the problem itself is a set that moves in the phase space ( moving sets) and therefore represented by a function in an infinite dimensional environment.
Sweeping process. It is a model that represents well, perhaps in the best way, memory and rate-independence effects in a vector environment and consists of a particular evolutionary law of the moving sets type.