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Spectral Conditions for Stability and Stabilization of Positive Equilibria

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Spectral Conditions for Stability and Stabilization of Positive Equilibria Spectral Conditions for Stability and Stabilization of Positive Equilibria for a Class of Nonlinear Cooperative Systems Precious Ugo Abara, Francesco Ticozzi and Claudio Altafini The self-archived postprint version of this journal article is available at Linköping University Institutional Repository (DiVA): http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-145240 N.B.: When citing this work, cite the original publication. Abara, P. U., Ticozzi, F., Altafini, C., (2018), Spectral Conditions for Stability and Stabilization of Positive Equilibria for a Class of Nonlinear Cooperative Systems, IEEE Transactions on Automatic Control, 63(2), 402-417. https://doi.org/10.1109/TAC.2017.2713241 Original publication available at: https://doi.org/10.1109/TAC.2017.2713241 Copyright: Institute of Electrical and Electronics Engineers (IEEE) http://www.ieee.org/index.html ©2018 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. 1 Spectral conditions for stability and stabilization of positive equilibria for a class of nonlinear cooperative systems Precious Ugo Abara, Francesco Ticozzi and Claudio Altafini Abstract—Nonlinear cooperative systems associated to vector While the assumption of cooperativity is standard in the fields that are concave or subhomogeneous describe well inter- context of nonlinear positive systems [31], [1], [7], [27], connected dynamics that are of key interest for communication, as it guarantees invariance in the positive orthant, concavity biological, economical and neural network applications. For this class of positive systems, we provide conditions that guarantee corresponds to vector fields that “decline” when the state existence, uniqueness and stability of strictly positive equilibria. grows, and in such a way it favours the boundedness of the These conditions can be formulated directly in terms of the trajectories. Concavity appears naturally as a common feature spectral radius of the Jacobian of the system. If control inputs are in all examples mentioned above: subhomogeneity of order available, then it is shown how to use state feedback to stabilize 1 is a proxy for concavity, sigmoidal functions are concave an equilibrium point in the interior of the positive orthant. when restricted to the positive semiaxis, and so are Michaelis- Index terms – Nonlinear Cooperative Systems; Positive equi- Menten functions. Combining the monotonic behavior of librium points; Concave Systems; Subhomogeneous systems; cooperative systems (with its lack of limit cycles and the Stability and Stabilization. relation of order it implies on the trajectories [31]) with the boundedness induced by concavity helps in achieving unique- I. INTRODUCTION ness and asymptotic stability of the equilibrium point. In fact, Positive nonlinear systems are widely used as models of the asymptotic behavior of cooperative concave systems has dynamical systems in which the state variables represent been known for more than thirty years [32]. In particular, it is intrinsically non-negative quantities, such as concentrations, known that they admit a “limit set trichotomy” [23], i.e., three masses, populations, probabilities, etc. [13], [22], [31]. They possible types of asymptotic behaviors: i) convergence to the are used for instance to model biochemical reactions [33], gene origin; ii) convergence to a unique positive fixed point; iii) regulatory networks [24], population dynamics [15], epidemic divergence to 1, see [32], [30], [23], [31]. In [32], conditions processes [26], [21], compartmental systems [18], economic for distinguishing between the first two cases are given, based systems [22], hydrological networks, power control in wireless on the spectral radius of the Jacobian. Alternatively, nonlinear networks [11], [36], [10], certain types of neural networks versions of the Perron-Frobenius theorem for positive cones [17], [14] and many other systems. can be used to show “ray convergence” in a special metric, The aim of this paper is to investigate the existence, variously called part metric or Hilbert projective metric [6], uniqueness, stability and stabilizability properties of positive [23]. The limit set trichotomy can all be placed along this ray. equilibria of certain classes of nonlinear positive systems In this paper we follow the approach of [32] and look at recurrent in applications. The main assumptions we make on spectral conditions on the Jacobian of the system in order to our systems is that they are i) cooperative and ii) concave (or describe the convergence to the two stable situations of the subhomogeneous). For example, in wireless networks, most limit set trichotomy. In particular, we obtain novel sufficient power control algorithms assume that the (nonlinear) “inter- conditions that guarantee the existence of strictly positive ference functions” are scalable [36] (i.e., subhomogeneous [5], equilibria for monotone concave systems, but also for the more [9]). In a completely different field, the (nonlinear) “activation general class of monotone sub-homogeneous systems [5] and function” of a Hopfield-type neural network [17], [14] is often for that of monotone contractive systems [10]. monotone and sigmoidal [14], [37], which means that it lacks It is worth remarking that in all the applications mentioned inflection points once it is restricted to positive values. In gene above the equilibrium point is normally required to be positive. regulatory network theory, the cooperative case appears as In fact, the origin is typically not very interesting as an a special case (all activatory links), and asymptotic stability equilibrium: for example, a power control algorithm that is achieved making use of saturated monotonicities such as converges to zero power is meaningless, and similarly for the Michaelis-Menten functional forms [24]. other applications. A common trick to move the equilibrium point from the origin to the interior of the positive orthant is to Work supported in part by a grant from the Swedish Research Council add a non-vanishing positive input, usually a constant, to the (grant n. 2015-04390 to C.A.). Preliminary versions of this manuscript were presented at the 54th and 55th Conf. on Decision and Control, see [34], [35] system dynamics (a current in a neural network, a noise power P. Ugo Abara was with the Dept. of Information Engineering, via Gradenigo in a wireless interference function, a constant mRNA synthesis 6B, University of Padova, 35131, Padova, Italy. He is now with the Technical rate in a gene network) [7], [27]. This trick is standard University of Munich, Germany. F. Ticozzi is with the Dept. of Information Engineering, via Gradenigo 6B, (and necessary) in linear positive systems [8], but not strictly University of Padova, 35131, Padova, Italy and the Physics and Astronomy necessary in the nonlinear case. As a matter of fact, in some Dept., Dartmouth College, 6127 Wilder, Hanover, NH (USA). cases the extra constant term seems more artificially motivated C.Altafini is with the Division of Automatic Control, Dept. of Electrical Engineering, Linkoping¨ University, SE-58183, Linkoping,¨ Sweden. email: by the need of guaranteeing positivity of the equilibrium, [email protected] rather than emerging from the problem setting. The conditions 2 we provide in this paper do not require the use of additive Sect. V and most of Sect. VI are novel material. external inputs to shift the stable equilibrium from the origin. For concave systems, the properties of uniqueness and II. PRELIMINARY MATERIAL global attractivity of a positive equilibrium correspond to a bound on the spectral radius of the Jacobian at the origin, A. Notation and linear algebra n plus an additional condition that has to hold inside the positive Throughout this paper R+ denotes the positive orthant of n n n n n orthant. When the concave nonlinearities are also bounded, the R , int(R+) its interior, and bd(R+) = R+ n int(R+) its n spectral radius of the Jacobian at the origin alone decides all boundary. If x1; x2 2 R , x1 6 x2 means x1;i 6 x2;i 8 i = the global dynamical features of the system [32]. 1; : : : ; n (x1;i = i-th component of x1), while x1 < x2 means n×n Our conditions extend those provided in [32] is several x1;i < x2;i 8 i = 1; : : : ; n, A matrix A = [aij] 2 R is directions. For instance, we show in the paper that when said nonnegative (in the following indicated A > 0) if aij > 0 feedback design makes sense, then simple linear diagonal 8 i; j, and Metzler if aij > 0 8 i 6= j. A is irreducible if @ a feedback can be used to choose any type of behavior in the permutation matrix Π that renders it block diagonal: trichotomy of possible asymptotic characters. A A As another improvement with respect to [32], we show ΠT AΠ = 11 12 that our results hold essentially unchanged when we replace 0 A22 concave functionals with the broader class of subhomogeneous for nontrivial square matrices A11, A22. The spectrum of A functionals. This is another class for which stability has been is denoted Λ(A) = fλ1(A); : : : ; λn(A)g, where λi(A), i = studied mostly at the origin [5], [9], or in presence of constant 1; : : : ; n, are the eigenvalues of A. The spectral radius of A, additive terms [11], [36]. The existence of simple spectral ρ(A), is the smallest positive real number such that ρ(A) > characterizations for uniqueness and stability of equilibria jλi(A)j, 8i = 1; : : : ; n. The spectral abscissa of A is µ(A) = is not limited to concave/subhomogeneous vector fields: the maxfRe(λi(A); i = 1; : : : ; ng. c-contractive vector fields of [10] (which lead to a similar The following standard properties of nonnegative matrices single global attractor) also admit an easy characterization in will be needed later on. terms of the spectral radius of the Jacobian.
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