TITLE. Controllability for semilinear parabolic equations with hysteresis.

DESCRIPTION OF THE SUBJECT. First of all the reader is referred to the subjects of two theses for “Laurea Triennale”:

Equazioni alle derivate parziali evolutive non lineari, e L'operatore matematico di isteresi di Preisach, e altri operatori di isteresi,

for some remarks on the concepts of “semilinear parabolic partial differential equation” and of “hysteresis operator and hysteresis phenomenon”.

Given a linear parabolic equation as the heat equation:

ut(x¸t)–Δu(x¸t)=0,       

given a initial value for the solution u (i.e. the value of u at the time t=0), and a finite time T>0, the problem of controllability is the following: “There exists a suitable function v of (x,t), such that, if we consider the perturbed heat equation

ut(x¸t)–Δu(x¸t)=v(x,t),

then the solution u is such that u(.,T)=0?” It is clear why this problem is called a “controllability problem”: if u is the temperature of a material occupying a certain domain, then the function v may represent a temperature-control that we may act on a (small) fixed region of the domain (i.e. we can heat or cool that region) and the question is “is it possible to choose a suitable heating/cooling strategy (i.e. the function v) in order to make the temperature u be equal to 0, all over the domain, at the fixed time T?

The controllability problem for the linear heat equation is well studied and well understood (under suitable conditions, the answer to the controllability problem is “yes, the function v exists”). The linearity of the equation is crucial for obtaining the results.

The results for the linear heat equation also extend to the semilinear heat equation

ut(x¸t)−Δu(x¸t)=β(u(x¸t))¸

that is, if we consider the heat equation with the semilinear term β(u(x¸t)), then also in this case we may find a function v, such that the solution of

ut(x¸t)−Δu(x¸t)=β(u(x¸t))+v(x,t)

satisifies u(.,T)=0. The main step for the possibility of studying the controllability of the semilinear equation is to suitably use the results for the linear equations coupling them with a suitable ``fixed point procedure``.

Now, let us suppose that the semilinear term β is a hysteresis operator, that is it depends on u with a suitable kind of memory : the value of β(u(x,t) depends on the whole past history of u. Then, the fixed point procedure, as in the simple semilinear case, does not seem to work anymore. This is certainly due to the presence of memory. This kind of problem is not yet studied in the literature.

GOAL OF THE THESIS. The goal of the thesis may be to study, in a particular and (hopefully) easy case and under a guided supervision, the controllability of the semilinear heat equation with hysteresis, and try to get some results about it. The idea is that, making some suitable hypotheses on the hysteresis operator β, it may be still possible to adapt some of the already known results for the semilinear equation.

REQUIRED NOTIONS. Integral and differential calculus for functions of several variables, Sobolev and Hilbert spaces.