Spatial uniformity in diffusively-coupled systems using weighted L 2 norm contractions S. Yusef Shafi *,‡ Zahra Aminzare †,‡ Murat Arcak * , Eduardo D. Sontag † Abstract— We present conditions that guarantee spatial uni- formity in diffusively-coupled systems. Diffusive coupling is a ubiquitous form of local interaction, arising in diverse areas including multiagent coordination and pattern formation in biochemical networks. The conditions we derive make use of the Jacobian matrix and Neumann eigenvalues of elliptic operators, and generalize and unify existing theory about asymptotic convergence of trajectories of reaction-diffusion partial differential equations as well as compartmental ordinary differential equations. We present numerical tests making use of linear matrix inequalities that may be used to certify these con- ditions. We discuss an example pertaining to electromechanical oscillators. The paper’s main contributions are unified verifiable relaxed conditions that guarantee synchrony. I. I NTRODUCTION Diffusively coupled models are crucial to understanding the dynamical behavior of a range of engineering and bio- logical systems. Understanding the conditions under which a distributed system exhibits spatial uniformity is a central question in many application fields concerned with pattern formation, ranging from biology (morphogenesis develop- mental biology, species competition and cooperation in ecol- ogy, epidemiology) [1], [2], [3] and enzymatic reactions in chemical engineering [4] to spatio-temporal dynamics in semiconductors [5]. This paper studies reaction-diffusion partial differential equations (PDEs) of the form ∂u ∂t (ω,t)= F (u(ω,t),t)+ Lu(ω,t), (1) where L denotes a diffusion operator, and compartmental systems of ordinary differential equations (ODEs): ˙ u(t)= ˜ F (u(t)) −L (d) u(t), (2) where ˜ F (u)= ( F (u 1 ) T , ··· ,F (u N ) T ) T and L (d) denotes diffusive coupling over a graph. We prove a two-part result that addresses the question of how the stability of solutions of the PDE or compartmental system of ODEs relates to stability of solutions of the underlying ordinary differential equation (ODE) dx dt (t)= F (x(t),t). The first part of our result shows that when solutions of the ODE have a certain contraction property, namely * Department of Electrical Engineering and Computer Sciences, Uni- versity of California, Berkeley, CA, USA. Email: yusef@eecs.berkeley.edu, arcak@eecs.berkeley.edu. Work supported in part by grants NSF ECCS- 1101876 and AFOSR FA9550-11-1-0244. †Department of Mathematics, Rutgers University, Piscataway, NJ, USA. Emails: aminzare@math.rutgers.edu, sontag@math.rutgers.edu. Work sup- ported in part by grants NIH 1R01GM086881 and 1R01GM100473, and AFOSR FA9550-11-1-0247. ‡The first two authors contributed equally. μ 2,Q (J F (u, t)) < 0 uniformly on u and t, where μ 2,Q is a logarithmic norm (matrix measure) associated to a Q- weighted L 2 norm, the associated PDE, subject to no-flux (Neumann) boundary conditions, and compartmental system of ODEs, enjoy a similar property. This result complements a similar result shown in [6] which, while allowing norms L p with p not necessarily equal to 2, had the restriction that it only applied to diagonal matrices Q and L was the standard Laplacian. Logarithmic norm or “contraction” approaches arose in the dynamical systems literature [7], [8], [9], and were extended and much further developed in work by Slotine e.g. [10]; see also [11] for historical comments. The second, and complementary, part of our result shows that when μ 2,Q (J f (u, t) − Λ 2 ) < 0, where Λ 2 is a nonneg- ative diagonal matrix whose entries are the second smallest Neumann eigenvalues of the diffusion operators in (1), or respectively the second smallest eigenvalues of the diffusive coupling matrix in (2), the solutions become spatially ho- mogeneous as t →∞. This result generalizes the previous work [12] to allow for spatially-varying diffusion, and makes a contraction principle implicitly used in [12] explicit. We next derive convex linear matrix inequality [13] tests as in [12] that can be used to certify the conditions. Our discussion concludes with an example of synchronization in coupled ring oscillators, which have been studied in the context of cross-coupled circuits [14] and gene regulatory networks [15]. II. PRELIMINARIES For any invertible matrix Q, and any 1 ≤ p ≤∞, and continuous u :Ω → R n , we denote the weighted L p,Q norm, ‖u‖ p,Q = ‖Qu‖ p , where (Qu)(ω)= Qu(ω) and ‖·‖ p indicates the norm in L p (Ω, R n ). Definition 1: Let (X, ‖·‖ X ) be a finite dimensional normed vector space over R or C. The space L(X, X) of linear transformations M : X → X is also a normed vector space with the induced operator norm ‖M ‖ X→X = sup ‖x‖ X =1 ‖Mx‖ X . The logarithmic norm μ X (·) induced by ‖·‖ X is defined as the directional derivative of the matrix norm, that is, μ X (M )= lim h→0 + 1 h (‖I + hM ‖ X→X − 1) , where I is the identity operator on X. The following lemma relates the logarithmic norm of a matrix to its satisfaction of a certain linear matrix inequality, and will be useful in proving our main results about spatial 2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013 978-1-4799-0176-0/$31.00 ©2013 AACC 5639