Stable observation of additive superpositions of Partial Differential Equations

Abstract

We address the problem of observing the solutions of a given Partial Differential Equations (PDE), submitted to unknown perturbations. The problem under consideration is motivated by the issue of averaged control and observation. In the latter, one aims at determining the initial datum of a dynamics, submitted to unknown perturbations, by means of measurements done on the average of solutions with respect to the unknown parameters.We do it in two different cases. First, we assume that a given dynamics is perturbed in an additive manner by a state governed by a PDE, but not necessarily of the same type. In particular we consider the additive superposition of multi-physics solutions of wave and heat type, for instance.Assuming the PDEs under consideration have constant coefficients, we prove that, the by now well-known observability properties of PDE (mainly heat and wave equations) from open subsets of the domain where the dynamics evolves, are stable under these additive perturbations.The proof is rather systematic and easy to apply. It consists in composing all but one (the observable one) PDE operators so to reduce the problem to the consideration of a single equation and applying its known observability properties. The same proof leads to very interesting open problems for systems involving PDEs with variable coefficients where commutators lead to non-local lower order perturbations.We then consider a particular case of a PDE depending on a parameter in a continuous manner. We show that the average is governed by the superposition of two distinct solutions, corresponding to the extremal values of the parameter, and experiencing an extra regularizing effect. This allows showing that the averaged observability property holds, for the continuous average, but to the prize of loosing a finite number of derivatives. This opens a wide range of interesting problems to be explored.

Internal Observability for Coupled Systems of Linear Partial Differential Equations

“…• Article [49] analyzes coupled systems of two heat equations, or the coupling of different dynamics, e.g. heat and wave equations, and it inspires the first part of the present article.…”

“…• Can we obtain the same characterizations if we couple different dynamics (for example systems of mixed heat and wave equations), as in [49]?…”

Averaged controllability of parameter dependent conservative semigroups

“…with boundary or internal controls of different nature. However, the proof, which is rather straightforward, is based on a smallness argument and, hence, it does not cover the results in [24] for the averaged controllability of two wave equations with internal control, or the ones in [40] for the additive superposition of wave and heat equations.…”

“…In the PDE context, the problem of averaged control was considered in [24] for two different wave equations by means of a common interior control, using H-measure techniques. Other situations were also considered in [40], when, for instance, the solution of a given PDE is perturbed additively by the solution of another one. Furthermore, in [30], the authors considered one-parameter families of Schrödinger and heat equations in the multi-dimensional case, with controls distributed in some interior sub-domain, showing that, depending on the averaging measure, one can obtain either the controllability results corresponding to time-reversible or parabolic-like equations.…”

“…In order to handle this case and prove the needed averaged Ingham inequalities we use a different argument. Instead of arguing through a perturbation principle, we shall rather use a method inspired by [10] and [40] whose key tool is to use the fact the solutions of the model under consideration, for given values of the parameter, are annihilated by a given linear bounded operator commuting with all other equations. This is the case for the 1d wave and Schrödinger equations with Dirichlet boundary conditions, for which the solutions are time-periodic.…”