TITLE. Optimal control of switching systems in infinite dimension.

DESCRIPTION OF THE SUBJECT. Let us consider the following particular case of semilinear heat equation (see the proposed subject for “Laurea Triennale” Equazioni alle derivate parziali evolutive non lineari)

u_t - Du = b(u),

where b is a discontinuous function which may assume only two values, let us say 1 and -1, and the switching rule for passing from one value to the other depends one some suitable threshold values which must be reached by u. We can think to b as a thermostat acting on a heating device which is on if b=1 and is off if b=-1. The heating device passes from the position “on” to the position “off” (and viceversa) depending on some suitable values (thresholds) reached by the function u (the temperature: if it is too low then the device is on, if it is too high then the device is off).

Now let us suppose that the switching (i.e. the thermostat) is in some sense “intrinsic” to the model described by the equation (in another framework than temperature and heat equation, we can think for instance to a car with a two-modes automatic gear: the switching rule for passing from one mode to the other is “intrinsic” to the car, and it depends on suitable values (thresholds) reached by the velocity of the car; however, several other examples of systems with such a kind of intrinsic switching may be found in many applied disciplines).

Now, let us suppose that we are facing a semilinear switching heat equation as above described, and suppose that we want to control it in order to optimize some performance criterium (a cost). The problem may be formalized in the following way. We consider the controlled switching heat equation

u_t - Du = b(u)+f(u,a),

where the switching is intrinsic (and represented by b, which is a datum of the problem) and f represents an external heat source which is at our disposal: we can make it be on, be off, be “partially” on, and so on, depending on the value of the control a(·) (a function of time) which is at our disposal, and which is choosen by us in order to minimize a cost J given by an integral of the following type

J(u₀,a)=òℓ(u,a)dt,

where u₀ is the initial datum (the initial temperature) and is a function belonging to a suitable function space (still considering the example of automatic gear as above, the control a may represent the acceleration, which actually is at disposal of the driver through the pressure of his foot on the accelerator).

Apart from the switching feature, this is an optimal control problem for a partial differential equation which in some sense generalize the optimal control problems for systems of ordinary differential equations (see the proposed subject for “Laurea Triennale” Controllo Ottimo). Also for optimal control problems for partial differential equations, we can applied the so-called Dynamic Programming Method, in order to find a partial differential equation satisfied by the optimum function

V(u₀)=inf_a J(u₀,a).

The main difference with respect to the “ordinary differential case” is that, for the optimal control of a partial differential equation, the problem has an infinite dimension: the state variable is the function u of (x,t), that is, for every fixed t, u(·,t) is a function of x, the space-variable. Hence, the trajectory to be controlled is a trajectory inside an infinite dimensional space of functions of x (a Banach space) , whereas in the ordinary differential case, the trajectory is inside the finite dimensional space Rⁿ. This fact of course causes some new troubles in performing a suitable analysis. However, the theory and the results in the literature are good enough.

Coming back to the presence of the switching term b, for the infinite dimensional case this seems to be a new feature, not yet studied in the literature.

GOAL OF THE THESIS. Starting from some previous results concerning the Dynamic Programming approach to some switching (thermostatic) systems in finite dimension (ordinary differential equation), the idea is to try to perform a similar analysis for the infinite dimension case (switching heat equation), adapting those results with the already existing results for the optimal control of the (non-switching) heat equation.

REQUIRED NOTIONS. Differential and integral calculus for functions of several variables, Banach, Sobolev and Hilbert spaces.