Differential dissipativity analysis of reaction–diffusion systems

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Abstract

This note shows how classical tools from linear control theory can be leveraged to provide a global analysis of nonlinear reaction–diffusion models. The approach is differential in nature. It proceeds from classical tools of contraction analysis and recent extensions to differential dissipativity.

Introduction

Reaction–diffusion equations are broadly used for modeling the spatio-temporal evolution of processes appearing in many fields of science such as propagation of electrical activity on cells in cellular biology [1]; reactions between substances on active media in chemistry [2]; transport phenomena in semiconductor devices in electronics [3]; and combustion processes and heat propagation in physics [4], to name a few. They have attracted recent interest in control, most notably [5] and [6], because the close link between reaction–diffusion systems and synchronization models under diffusive coupling: the linear diffusion term in reaction–diffusion partial differential equations is the continuum limit of the diffusive (or incrementally passive) interconnection network of agents sharing the same reaction dynamics. In that sense, the results in [5] and [6] are infinite dimensional generalizations of classical finite dimensional results pertaining to synchronization [7], [8], [9], [10].

Our contribution in the present note is to further emphasize the potential of classical tools from linear control theory in the analysis and design of reaction–diffusion systems. Our observation is twofold. First, we model reaction diffusion systems as the interconnection of a linear spatially and time-invariant (LTSI) model with a static nonlinearity. This natural decomposition calls for a dissipativity analysis of the interconnection, with complementary input–output dissipation inequalities imposed on the LTSI model and on the static nonlinearity, respectively. Second, we study this interconnection differentially along arbitrary solutions, thereby studying a nonlinear model through a family on linearized systems.

The proposed approach is largely inspired from [5] and [6], which analyze spatial homogeneity via contraction theory. The purely differential approach in the present paper is thought to offer further potential especially in situations where the attractor is difficult to characterize explicitly. In this note, we illustrate the benefits of a differential approach in two distinct ways: (i) we use the classical KYP lemma to complement existing state–space analysis results with a frequency-domain analysis; and (ii) we use recent results of differential dissipativity theory [11] to characterize the attractor of two classical reaction–diffusion models: Nagumo model of bistability [12], and FitzHugh–Nagumo model of oscillation [13].

Let Ln2(Ω) denote the Hilbert space of square integrable functions mapping ΩR to Rn with the conventional inner product x,yLn2(Ω)=Ωx(θ)y(θ)dθ and norm denoted by Ln2(Ω). When clear from the context, we will drop the subindex. For vectors ξ,ψ in Rn, the inner product is denoted as ξψ and the associated norm as ||. The set +{a+jb|a0} denotes the set of complex numbers with non-negative real part, whereas R+ denotes the set of non-negative real numbers. A symmetric, positive (semi-) definite matrix Π is denoted as (Π0) Π0, whereas, In represents the identity matrix of dimension n.

Section snippets

Reaction–diffusion systems

The family of distributed systems under consideration has the form xt(θ,t)=DΔx(θ,t)+Ax(θ,t)Bφ(Cx(θ,t))where x(θ,t)Rn denotes the state of the system at position θΩR and time t0. The nonlinear function φ:RmRm represents a static nonlinearity and its properties are stated below. Spatial diffusion is modeled via the matrix DRn×n which is symmetric and positive definite, and the Laplace operator Δ:Dom(Δ)Ln2(Ω)Ln2(Ω) with domain specified below, whereas the matrices A,B and C are constant

Differential analysis of reaction diffusion systems

Differential analysis consists in analyzing the properties of infinitesimal variations δx(θ,t) around an arbitrary solution x(θ,t) of (1b)–(2) as is made in [19], [20] for the case of finite-dimensional systems.

Namely, let ϕ(θ,t,x0) denote the solution of the reaction–diffusion system (1)–(2) at position θ and time t with initial condition x(θ,0)=x0(θ). Let x1(θ,0) and x2(θ,0) be two given initial conditions and let γ:S1×[0,1]Rn be a smooth curve such that γ(,0)=x1(,0), γ(,1)=x2(,0) and γ(0

Analysis in the frequency domain

The linear system (1b)–(2) is both space and time invariant (LTSI): solutions shifted in time and in space satisfy the same equation [25].

Spatial and temporal invariance properties of linear systems allow for insightful frequency domain analysis. In this section, we briefly illustrate the frequency-domain interpretation of the results of the previous sections.

Conclusions

We illustrated the potential of differential dissipativity for the analysis of nonlinear reaction–diffusion systems. The differential dynamics naturally decompose into two components, the differential inhomogeneous dynamics and the differential homogeneous dynamics. We illustrated sufficient conditions for spatial homogeneity, that is, contraction of the differential inhomogeneous dynamics, and for p-differential dissipativity of the differential homogeneous dynamics. Future work will explore

CRediT authorship contribution statement

Félix A. Miranda-Villatoro: Conceptualization, Methodology, Formal analysis, Software, Validation, Writing-original draft. Rodolphe Sepulchre: Supervision, Conceptualization, Methodology, Writing - review & editing, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645.

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