On Convergence for Piecewise Affine Models of Gene Regulatory Networks via a Lyapunov Approach
On Convergence for Piecewise Affine Models of Gene Regulatory Networks via a Lyapunov Approach
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DOI：
10.1109/tac.2019.2945214
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发表时间：
2020-08
影响因子：
6.8
通讯作者：
Mirko Pasquini;D. Angeli
中科院分区：
文献类型：
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作者：
Mirko Pasquini;D. Angeli
本文利用分段二次李雅普诺夫函数分析了基因调控网络分段仿射模型的收敛性和稳定性。在第一部分中，我们提出了基因调控网络的分段仿射模型，然后证明了存在一个最小的，吸引的，正不变的盒子，我们描述了一个线性矩阵不等式的框架来构造一个系统的李雅普诺夫函数。在第二部分的工作中，系统的解决方案的表征后，考虑也孤立芝诺行为的轨迹，一个单调性的性质，因此发现李雅普诺夫函数正式证明，连同结果的轨迹的收敛性，在某种意义上的措施。最后，我们提出了两个例子，显示的适用性和效用的主要理论结果前面解释。
In this paper, we analyse convergence and stability properties for piecewise affine models of gene regulatory networks, by means of piecewise quadratic Lyapunov functions. In the first part, we present the piecewise affine model of a gene regulatory network then, after proving a result on the existence of a minimal, attractive, and positively invariant box, we describe a linear matrix inequalitie framework to construct a Lyapunov function for the system. In the second part of the work, after a characterization of the system solutions when considering also isolated Zeno behavior of the trajectories, a monotonicity property of the so found Lyapunov function is formally proved, together with a result on the convergence of the trajectories, in the sense of measure. Finally, we present two examples, showing the applicability and utility of the main theoretical results previously explained.