Averaged controllability for random evolution partial differential equations

TL;DR

This paper investigates the controllability of random evolution PDEs, demonstrating that averaging can induce controllability properties similar to heat equations, with behavior varying based on the probability distribution of parameters.

Contribution

It introduces the concept of averaged controllability for random PDEs, showing how averaging affects controllability properties in heat and Schrödinger equations.

Findings

Averages of parabolic equations exhibit null-controllability in short time.

The behavior of averaged Schrödinger equations depends on the probability density.

Averaging can lead to either conservative or parabolic-like dynamics.

Abstract

We analyze the averaged controllability properties of random evolution Partial Differential Equations. We mainly consider heat and Schr\"odinger equations with random parameters, although the problem is also formulated in an abstract frame. We show that the averages of parabolic equations lead to parabolic-like dynamics that enjoy the null-controllability properties of solutions of heat equations in an arbitrarily short time and from arbitrary measurable sets of positive measure. In the case of Schr\"odinger equations we show that, depending on the probability density governing the random parameter, the average may behave either as a conservative or a parabolic-like evolution, leading to controllability properties, in average, of very different kind.