生化网络的结构多面体稳定性等价于相应的广义Petri网的有限性
Structural polyhedral stability of a biochemical network is equivalent to finiteness of the associated generalised Petri net
摘要：
我们认为生化系统与广义类的Petri网可能是负的令牌号码。我们表明，存在一个结构多面体李雅普诺夫函数的生化系统是等价于相关的Petri网演化的有界性，或者等价地，从每个初始条件可达到的状态的数量的有限性。对于不允许多面体李雅普诺夫函数的网络，我们研究是否有可能通过在一些钉扎节点上施加强负反馈来加强多面体结构稳定性：在Petri网中，这相当于将钉扎节点变成黑洞，清除任何正或负的传入令牌。如果这样的节点被选择，使变换后的Petri网有界离散轨迹，那么存在一个稳定的钉扎控制：生化网络成为李雅普诺夫稳定，如果足够强的局部负反馈施加到钉扎节点。这些结果使我们能够从结构上确定关键节点进行局部控制，以确保整个网络的稳定性。
英文摘要：
We consider biochemical systems associated with a generalised class of Petri nets with possibly negative token numbers. We show that the existence of a structural polyhedral Lyapunov function for the biochemical system is equivalent to the boundedness of the associated Petri net evolution or, equivalently, to the finiteness of the number of states reachable from each initial condition. For networks that do not admit a polyhedral Lyapunov function, we investigate whether it is possible to enforce polyhedral structural stability by applying a strong negative feedback on some pinned nodes: in terms of the Petri net, this is equivalent to turning pinned nodes into black holes that clear any positive or negative incoming token. If such nodes are chosen so that the transformed Petri net has bounded discrete trajectories, then there exists a stabilising pinning control: the biochemical network becomes Lyapunov stable if a sufficiently strong local negative feedback is applied to the pinned nodes. These results allow us to structurally identify the critical nodes to be locally controlled so as to ensure the stability of the whole network.
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