Control of PDEs involving non-local terms 1 2 Enrique Zuazua * DeustoTech - Bilbao, Basque Country, Spain * Universidad Aut´onoma de Madrid, Spain * Laboratoire Jacques Louis Lions, UPMC, Paris Motivation Table of Contents 1 Motivation 2 Viscoelasticity The model PDE aspects Moving control 3 Models involving memory terms Heat with memory Waves with memory 4 Fractional wave and Schro¨dinger equations The model Controlability 5 Equations involving space-like lower order terms 6 Fractional time derivatives 7 Perspectives Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 2 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54 Motivation Why nonlocal? Strictly speaking relevant models in Continuum Mechanics, Math Physics and Biology are of nonlocal nature: Boltzmann equations in gas dynamics; Navier-Stokes equations in Fluid Mechanics; Keller-Segel model for Chemotaxis. Here, however, we shall deal with other non-local effects, due to anomalous dispersion and diffusion and memory terms. This leads to PDE involving nonlocal terms in the form of integrals either in space or time or both. In that setting, of course, classical PDE theory fails because of non-locality. Yet many of the existing techniques can be tuned and adapted, although this is often a delicate matter because modern PDE analysis is based on the use of localisation arguments (test and cut-off functions) that do not apply in a straightforward manner in the nonlocal context. Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 3 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54 Motivation Goal Try to develop a systematic analysis of the control theoretical consequences of the possible presence of non-local terms in the model. We do it for the following model cases: Viscoelasticity Models involving memory terms Fractional Laplacian Lower order space-like nonlocal terms. Fractional time derivatives Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 4 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54 Viscoelasticity Table of Contents 1 Motivation 2 Viscoelasticity The model PDE aspects Moving control 3 Models involving memory terms Heat with memory Waves with memory 4 Fractional wave and Schro¨dinger equations The model Controlability 5 Equations involving space-like lower order terms 6 Fractional time derivatives 7 Perspectives Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 5 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54 Viscoelasticity The model Viscoelasticity A wave equation with both viscous Kelvin-Voigt damping: ytt − ∆y −∆yt = 1ωh, x ∈ Ω, t ∈ (0, T), (1) y = 0, x ∈ ∂Ω, t ∈ (0, T), (2) y(x, 0) = y0(x), yt(x, 0) = y1(x) x ∈ Ω. (3) N Here, Ω is a smooth, bounded open set in R and h = h(x, t) is a control located in a open subset ω of Ω. We address the controllability problem: Given (y0, y1), to find a control h such that the associated solution to (1)-(3) satisfies y(T) = yt(T) = 0. Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 6 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54 Viscoelasticity The model Viscoelasticity arises in areas such as biomechanics, power industry or heavy construction: Synthetic polymers; Wood; Human tissue, cartilage; Metals at high temperature; Concrete, bitumen; ... Viscoelastic materials are those for which the behavior combines liquid-like 3 and solid-like characteristics. 3 See H. T. Banks, S. Hu and Z. R. Kenz, A Brief Review of Elasticity and Viscoelasticity for Solids, Adv. Appl. Math. Mech., 3 (1), (2011), 1-51. Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 7 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54 Viscoelasticity PDE aspects How do you classify this PDE? ytt − ∆y −∆yt = 0 Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 8 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54 Viscoelasticity PDE aspects The first key for control: Space-like propagation For the action of controls/actuators, localised in a subset ω of the domain where the PDE evolves Ω, to be effective everywhere one requires the PDE to enjoy the property of propagation in the space-like direction. In particular, one needs of Unique Continuation results ensuring that, if y solves ytt − ∆y −∆yt = 0 and y = 0 in ω × (0, T) then y ≡ 0. Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 9 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54 Viscoelasticity PDE aspects A geometric obstruction Standard results on unique continuation do not apply. The principal part of the operator is ∂t∆. Then characteristic hyperplanes are of the form t = t0 and x · e = 1. And the zero sets do not propagate by standard unique continuation arguments. This phenomenon was previously observed by S. Micu in the context of the 4 5 Benjamin-Bona-Mahoni equation 3 In that frame the underlying operator is ∂t − ∂ xxt but its principal part is the same 3 ∂ . xxt 4 S. Micu, SIAM J. Control Optim., 39(2001), 1677–1696. 5 X. Zhang and E. Z. Matematische Annalen, 325 (2003), 543-582. Micrololocal Analysis, Resonances, and Control Theory in PDE, October 2017 10 E. Zuazua (DeustoTech-UAM-UPMC) Control & Nonlocality / 54