Infinitesimal interconnection variation in nonlinear networked systems

We propose a novel infinitesimal variation for a nonlinear networked system's behavior when its interconnection topology changes discontinuously. We introduce a variational derivative of system output with respect to the connectivity, and derive an analytic formula for the derivative using an adjoint formulation. We provide bounds relating the discontinuous change in system behavior to the proposed continuous infinitesimal variation. The variational derivative can be used as the sensitivity of the system output to the interconnection topology. The separability of the variational derivative allows us to develop a tractable algorithm for an interconnection pursuit problem applicable to optimization in biochemical reaction networks.

[1]  P. Holmes,et al.  Frontiers in Neural Circuits Neural Circuits , 2022 .

[2]  Kellen Petersen August.  Real Analysis , 2009 .

[3]  G. Stephanopoulos,et al.  Identifying gene targets for the metabolic engineering of lycopene biosynthesis in Escherichia coli. , 2005, Metabolic engineering.

[4]  D. di Bernardo,et al.  How to infer gene networks from expression profiles , 2007, Molecular systems biology.

[5]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[6]  George J. Pappas,et al.  Distributed connectivity control of mobile networks , 2007, 2007 46th IEEE Conference on Decision and Control.

[7]  W. Lim,et al.  Defining Network Topologies that Can Achieve Biochemical Adaptation , 2009, Cell.

[8]  Cheng-Lin Liu,et al.  Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations , 2009, Autom..

[9]  M. Elowitz,et al.  Regulatory activity revealed by dynamic correlations in gene expression noise , 2008, Nature Genetics.

[10]  Andrew M. Gleason,et al.  Calculus (2nd ed.) , 1998 .

[11]  Munther A. Dahleh,et al.  Robust Distributed Routing in Dynamical Networks—Part I: Locally Responsive Policies and Weak Resilience , 2013, IEEE Transactions on Automatic Control.

[12]  Eduardo D. Sontag,et al.  Synchronization of Interconnected Systems With Applications to Biochemical Networks: An Input-Output Approach , 2009, IEEE Transactions on Automatic Control.

[13]  P. Kokotovic,et al.  Direct and adjoint sensitivity equations for parameter optimization , 1967, IEEE Transactions on Automatic Control.

[14]  Naomi Ehrich Leonard,et al.  Collective Motion, Sensor Networks, and Ocean Sampling , 2007, Proceedings of the IEEE.

[15]  Sonia Martínez,et al.  Coverage control for mobile sensing networks , 2002, IEEE Transactions on Robotics and Automation.

[16]  Eduardo Sontag,et al.  A passivity-based stability criterion for a class of biochemical reaction networks. , 2008, Mathematical biosciences and engineering : MBE.

[17]  J. Hadamard.  Sur les problemes aux derive espartielles et leur signification physique , 1902 .

[18]  Mehran Mesbahi,et al.  Network Identification via Node Knockout , 2010, IEEE Transactions on Automatic Control.

[19]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[20]  John N. Tsitsiklis,et al.  Continuous-Time Average-Preserving Opinion Dynamics with Opinion-Dependent Communications , 2009, SIAM J. Control. Optim..

[21]  A. V. Meier.  Electric power systems : a conceptual introduction , 2006 .