The Ramsey model is one of the most popular neoclassical growth models in economics. The basic time-dependent model has been extended by a spatial component in recent years, i.e. capital accumulation is modeled as a process not only in time but in space as well. We consider a Ramsey economy where the value of capital depends not only on the respective location but is influenced by the surrounding areas as well. Here, we do not only take spatially local effects modeled by a diffusion operator into account but we also include a nonlocal diffusion term in integral form leading to control PIDEs.
We analyze the controllability properties under positivity constraints of the heat equation involving the fractional Laplacian on the interval (-1,1). We prove the existence of a minimal (strictly positive) time T_{min} such that the fractional heat dynamics can be controlled from any non-negative initial datum to a positive steady state, when s>1/2. Moreover, we prove that in this minimal time constrained, controllability can also be achieved through a control that belongs to a certain space of Radon measures. We also give some numerical results.
We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques.