Mixed integer predictive control and shortest path reformulation

Mixed integer predictive control deals with optimizing integer and real control variables over a receding horizon. The mixed integer nature of controls might be a cause of intractability for instances of larger dimensions. To tackle this little issue, we propose a decomposition method which turns the original $n$-dimensional problem into $n$ indipendent scalar problems of lot sizing form. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon. This last reformulation step mirrors a standard procedure in mixed integer programming. The approximation introduced by the decomposition can be lowered if we operate in accordance with the predictive control technique: i) optimize controls over the horizon ii) apply the first control iii) provide measurement updates of other states and repeat the procedure

BV-regularity for the Malliavin Derivative of the Maximum of the Wiener Process

We prove that, on the classical Wiener space, the random variable $\sup_{0\le t \le T} W_t$ admits a measure as second Malliavin derivative, whose total variation measure is finite and singular w.r.t.\ the Wiener measure

Loop group actions on categories and Whittaker invariants

We develop some aspects of the theory of $D$-modules on ind-schemes of pro-finite type. These notions are used to define $D$-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a non-degenerate character $\chi: N(\!(t)\!) \to \mathbb{G}_a$ and a category $\mathcal{C}$ acted upon by $N(\!(t)\!)$, we define the category $\mathcal{C}^{N(\!(t)\!), \chi}$ of $(N(\!(t)\!), \chi)$-invariant objects, along with the coinvariant category $\mathcal{C}_{N(\!(t)\!), \chi}$. These are the Whittaker categories of $\mathcal{C}$, which are in general not equivalent. However, there is always a family of functors $\Theta_k: \mathcal{C}_{N(\!(t)\!), \chi} \to \mathcal{C}^{N(\!(t)\!), \chi}$, parametrized by $k \in \mathbb{Z}$. We conjecture that each $\Theta_k$ is an equivalence, provided that the $N(\!(t)\!)$-action on $\mathcal{C}$ extends to a $G(\!(t)\!)$-action. Using the Fourier-Deligne transform (adapted to Tate vector spaces), we prove this conjecture for $G= GL_n$ and show that the Whittaker categories can be obtained by taking invariants of $\mathcal{C}$ with respect to a very explicit pro-unipotent group subscheme (not ind-scheme) of $G(\!(t)\!)$

Oscillating shells: A model for a variable cosmic object

A model for a possible variable cosmic object is presented. The model consists of a massive shell surrounding a compact object. The gravitational and self-gravitational forces tend to collapse the shell, but the internal tangential stresses oppose the collapse. The combined action of the two types of forces is studied and several cases are presented. In particular, we investigate the spherically symmetric case in which the shell oscillates radially around a central compact object

Zero noise limits using local times

We consider a well-known family of SDEs with irregular drifts and the correspondent zero noise limits. Using (mollified) local times, we show which trajectories are selected. The approach is completely probabilistic and relies on elementary stochastic calculus only