Differential dissipativity analysis of reaction–diffusion systems☆
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 denote the Hilbert space of square integrable functions mapping to with the conventional inner product and norm denoted by . When clear from the context, we will drop the subindex. For vectors in , the inner product is denoted as and the associated norm as . The set denotes the set of complex numbers with non-negative real part, whereas denotes the set of non-negative real numbers. A symmetric, positive (semi-) definite matrix is denoted as () , whereas, represents the identity matrix of dimension .
Section snippets
Reaction–diffusion systems
The family of distributed systems under consideration has the form where denotes the state of the system at position and time . The nonlinear function represents a static nonlinearity and its properties are stated below. Spatial diffusion is modeled via the matrix which is symmetric and positive definite, and the Laplace operator with domain specified below, whereas the matrices and are constant
Differential analysis of reaction diffusion systems
Differential analysis consists in analyzing the properties of infinitesimal variations around an arbitrary solution of (1b)–(2) as is made in [19], [20] for the case of finite-dimensional systems.
Namely, let denote the solution of the reaction–diffusion system (1)–(2) at position and time with initial condition . Let and be two given initial conditions and let be a smooth curve such that , and
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 -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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