Preliminary Sketch of Possible Fixed Point Transformations for Use in Adaptive Control

In this paper a further step towards a novel approach to adaptive nonlinear control developed at Budapest Tech in the past few years is reported. Its main advantage in comparison with the complicated Lyapunov function based techniques is that it is based on simple geometric considerations on the basis of which the control task can be formulated as a Fixed Point Problem for the solution of which a Contractive Mapping is created that generates an Iterative Cauchy Sequence for Single Input - Single Output (SISO) systems. Consequently it converges to the fixed point that is the solution of the control task. In the formerly developed approaches for monotone increasing or monotone decreasing systems the proper fixed points had only a finite basin of attraction outside of which the iteration might become divergent. The here sketched potential solutions apply a special function built up of the “response function” of the excited system under control and of a few parameters. This function has almost constant value apart from a finite region in which it has a “wrinkle” in the vicinity of the desired solution that is the “proper” fixed point of this function. By the use of an affine approximation of the response function around the solution it is shown that at one of its sides this fixed point is repulsive, while at the opposite side it is attractive. It is shown, too, that at the repulsive side another, so called “false” fixed point is present that is globally attractive, with the exception of the basin of attraction of the “proper” one. This structure is advantageous because a) no divergence can occur in the iteration, b) the convergence to the “false” value can easily be detected, and c) by using some ancillary tricks in the most of the cases the solution can be kicked from the wrong fixed point into the basin of attraction of the “proper one”. In the paper preliminary calculations are presented.N/

A general fractional porous medium equation

A general fractional porous medium equatio

The Cauchy-Dirichlet problem for singular nonlocal diffusions on bounded domains

This article has been published in a revised form in Discrete and Continuous Dynamical Systems https://doi.org/10.3934/dcds.202211. This version is free to download for private research and study only. Not for redistribution, re-sale or use in derivative worksWe study the Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form ∂tu = −Lum posed on a bounded Euclidean domain Ω ⊂ RN with smooth boundary and N ≥ 1. The linear diffusion operator L is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely um = |u|m−1u with 0 < m < 1. The prototype equation is the Fractional Fast Diffusion Equation (FFDE), when L is one of the three possible Dirichlet Fractional Laplacians on Ω. We provide a complete basic theory for solutions to (CDP): existence and uniqueness in the biggest class of data known so far, both for nonnegative and signed solutions; sharp smoothing estimates: classical Lp−L∞ smoothing effects, and new weighted estimates, which represent a novelty also in local case, i.e. ut = ∆um. We compare two strategies to prove smoothing effects: Moser iteration VS Green function method. Due to the singular nonlinearity and to presence of nonlocal diffusion operators, the question of how solutions satisfy the lateral boundary conditions is delicate and we answer it by quantitative upper boundary estimates. Finally, we show that solutions extinguish in finite time and we provide upper and lower estimates for the extinction time, together with explicit sharp extinction rates in different normsMTM2017-85757-P, PID2020-113596GB-I00, CEX2019-000904-S, H2020 MSCA programme, grant agreement 77782