Francesca Carlotta Chittaro: Hamiltonian approach to sufficient optimality conditions
The celebrated Pontryagin Maximum Principle (PMP) provides a (first order) necessary condition for the optimality of trajectories of optimal control problems. In most cases, however, a trajectory satisfying PMP is not optimal. For these reasons, additional optimality conditions are required.
In this context, Hamiltonian methods are quite effective in establishing sufficient optimality conditions. In this talk, after a brief review of the main ideas of the general method, we will focus on optimal control problems associated with control-affine dynamics and costs of the form
\[\int_0^T |u(t)\varphi(x(t))|dt.\]
These kind of cost are very common in problems modeling neurobiology, mechanics and fuel-consumption.
This is a joint work with L. Poggiolini (DIMAI).
Matteo Franca: Stability and separation properties of Ground States for reaction-diffusion equations
In this talk we consider the long time behavior of positive solutions of the following Cauchy problem:
$$ \left\{ \begin{array}{l}
u_t=\Delta u + u^{q}\\
u(0,x)= \phi(x)\,,
\end{array} \right. $$
where $u :\mathbb{R} \times \mathbb{R}^{n} \to \mathbb{R}$, $q> \frac{n}{n-2}$ and of its generalization to spatial dependent potentials.
This equation can be regarded as a model for an exothermic reaction which may produce an explosion ($u$ is the temperature).
Hence we have two main expected behaviors: if “$\phi$ is large” $u$ blows up in finite time, while if “$\phi$ is small”
$u$ converges to $0$ for $t$ large. Our aim is to explore the threshold between these two behaviors.
In this context a key role is played by radial stationary solutions, i.e. Ground States, both regular and singular, and in particular by their separation properties.
In fact, roughly speaking, they determine the threshold between blowing up and fading solutions, and if suitable ordering properties are satisfied they gain some stability.
If there will be time we will discuss some recent results concerning separation properties of the stationary problem where the Laplace operator is replaced by its $p$-Laplace generalization.
Nicola Visciglia: On the Nonlinear Schroedinger Equation with multiplicative white noise
We prove global existence, uniqueness and convergence almost surely of solutions to a family of properly regularized and renormalized approximating equations to the Nonlinear Schroedinger Equation. One of the difficulties is connected with the low regularity of the white noise potential, that does not allow the use of the standard definition of product between functions.
We show how this problem can be settled by using a suitable renormalization argument, once we exploit a transformation first introduced by Hairer in the context of the heat equation. Next we show how the use of suitable energies allow to extend the solutions for every time almost surely w.r.t. to the probabilistic parameter defining the white noise potential.
In particular we extend a previous result by A. Debussche and H. Weber available in the case of a cubic Nonlinearity.
This is a joint work with N. Tzvetkov.
