Stability and transient behavior of composite nonlinear systems

Araki, Mikiya; Kondo, B.

doi:10.1109/tac.1972.1100042

Cited by 115 publications

(21 citation statements)

References 9 publications

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“…Since the matrix (A N p À b N )I À B 11 has nonpositive off-diagonal entries, by [8,Theorem 3] it is an M-matrix so every entry of its inverse matrix is nonnegative. Hence, if we take v N ¼ 1 then it follows from (15) that…”

Section: Lemma 23mentioning

confidence: 99%

Fixed point global attractors and repellors in competitive Lotka–Volterra systems

Hou

Baigent

2011

Dynamical Systems

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Zeeman and Zeeman [E.C. Zeeman and M.L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc. 355 (2003), pp. 713-734] show that if a strongly competitive Lotka-Volterra system (i) has a unique interior fixed point p and (ii) the carrying simplex AE lies below (above) the strongly balanced tangent plane to AE at p then the system has no periodic orbits and p is a global attractor (repellor) relative to AE. Condition (ii) is then translated into the definiteness of a certain quadratic function on the tangent plane, which is equivalent to the definiteness of an (N À 1) Â (N À 1) real symmetric matrix that can be computed. Here we adapt these methods to show that the above conclusions are still true without the assumption (i). Hence, our results apply to globally attracting or repelling fixed points on the boundary, as well as in the interior, of R N þ . Moreover, the algebraic condition for global attraction also implies global asymptotic stability of the fixed point. We also show that the global attraction holds not just relative to AE, but also relative to the interior of the first quadrant.

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Section: Lemma 23mentioning

confidence: 99%

Fixed point global attractors and repellors in competitive Lotka–Volterra systems

Hou

Baigent

2011

Dynamical Systems

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“…If the matrix A is a Metzler matrix, the off-diagonal elements of −A are nonpositive. On the basis of the results in [59], we obtain that there always exists a positive constant vector ρ ∈ R n with ρ > 0 such that (−A) T ρ > 0, which further can be written by ρ T A < 0. Since the system (4) is positive, the state variable x(t) ≥ 0.…”

Section: Problem Statement and Preliminariesmentioning

confidence: 93%

Stabilization Control of Continuous-Time Fractional Positive Systems Based on Disturbance Observer

Shao

2016

IEEE Access

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This paper studies the problem of stabilization control for a class of continuous-time fractional systems subject to external constant disturbances based on a disturbance observer. To tackle unknown disturbances, a fractional disturbance observer (FDO) is introduced for the fractional system. A stabilization controller incorporating the disturbance observer is then designed for the fractional system. Under the developed control scheme and linear program (LP) method, the sufficient conditions are proposed to guarantee that the fractional systems with closed-loop positivity are asymptotically stable. Numerical simulation results further demonstrate the effectiveness of the results.

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“…If Ω ∈ M + n (R) (i.e. has positive diagonal entries), the row-or column-diagonal dominance conditions suffice to ensure that there exists a positive diagonal matrix Q such that QΩ is positive definite-see Araki and Kondo (1972), Ikeda and Siljak (1980), and Theorem 2 below.…”

Section: Remark 2 Clearlyω Is An M-matrix If and Only Ifωmentioning

confidence: 99%

“…We adapt an argument originally due to Araki and Kondo (1972). Let c and d be positive vectors such thatΩd > 0 andΩ T c > 0 (for brevity, the argument x is temporarily suppressed).…”

Section: A2 Proof Of Theoremmentioning

confidence: 99%

Conditions for Global Dynamic Stability of a Class of Resource-Bounded Model Ecosystems

Seymour¹,

Knight²,

Fung

2010

Bull. Math. Biol.

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This paper studies a class of dynamical systems that model multi-species ecosystems. These systems are 'resource bounded' in the sense that species compete to utilize an underlying limiting resource or substrate. This boundedness means that the relevant state space can be reduced to a simplex, with coordinates representing the proportions of substrate utilized by the various species. If the vector field is inward pointing on the boundary of the simplex, the state space is forward invariant under the system flow, a requirement that can be interpreted as the presence of non-zero exogenous recruitment. We consider conditions under which these model systems have a unique interior equilibrium that is globally asymptotically stable. The systems we consider generalize classical multispecies Lotka-Volterra systems, the behaviour of which is characterized by properties of the community (or interaction) matrix. However, the more general systems considered here are not characterized by a single matrix, but rather a family of matrices. We develop a set of 'explicit conditions' on the basis of a notion of 'uniform diagonal dominance' for such a family of matrices, that allows us to extract a set of sufficient conditions for global asymptotic stability based on properties of a single, derived matrix. Examples of these explicit conditions are discussed.

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Stability and transient behavior of composite nonlinear systems

Cited by 115 publications

References 9 publications

Fixed point global attractors and repellors in competitive Lotka–Volterra systems

Fixed point global attractors and repellors in competitive Lotka–Volterra systems

Stabilization Control of Continuous-Time Fractional Positive Systems Based on Disturbance Observer

Conditions for Global Dynamic Stability of a Class of Resource-Bounded Model Ecosystems

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