A unified analytical framework for a class of optimal control problems on networked systems

We consider a class of optimal control problems on networks that generically permits a reduction to a universal set of reference problems without differential constraints that may be solved analytically. The derivation shows that input homogeneity across the network results in universally constant optimal control inputs. These predictions are validated using numerical analysis of problems of synchronization of coupled phase oscillators and spreading dynamics on time-varying networks.

[1]  A. Goltsev,et al.  Critical behavior of the relaxation rate, the susceptibility, and a pair correlation function in the Kuramoto model on scale-free networks. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Jari Saramäki,et al.  Small But Slow World: How Network Topology and Burstiness Slow Down Spreading , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  George J. Pappas,et al.  Analysis and Control of Epidemics: A Survey of Spreading Processes on Complex Networks , 2015, IEEE Control Systems.

[4]  Harry Dankowicz,et al.  Optimization with equality and inequality constraints using parameter continuation , 2020, Appl. Math. Comput..

[5]  Vladimir Protopopescu,et al.  Controlling synchrony in a network of Kuramoto oscillators with time-varying coupling , 2015 .

[6]  Yoshiki Kuramoto,et al.  Self-entrainment of a population of coupled non-linear oscillators , 1975 .

[7]  Lu-Xing Yang,et al.  The optimal dynamic immunization under a controlled heterogeneous node-based SIRS model , 2016 .

[8]  E. Doedel,et al.  Optimization in Bifurcation Problems using a Continuation Method , 1987 .

[9]  J. Crawford,et al.  Amplitude expansions for instabilities in populations of globally-coupled oscillators , 1993, patt-sol/9310005.

[10]  Maurizio Porfiri,et al.  Continuous-Time Discrete-Distribution Theory for Activity-Driven Networks. , 2016, Physical review letters.

[11]  Frank Schilder,et al.  Recipes for Continuation , 2013, Computational science and engineering.

[12]  Mark E. J. Newman,et al.  Power-Law Distributions in Empirical Data , 2007, SIAM Rev..

[13]  János Kertész,et al.  Spreading dynamics on networks: the role of burstiness, topology and non-stationarity , 2014, 1404.2468.

[14]  Thomas K. D. M. Peron,et al.  The Kuramoto model in complex networks , 2015, 1511.07139.

[15]  Florian Dörfler,et al.  Synchronization in complex networks of phase oscillators: A survey , 2014, Autom..

[16]  R. Pastor-Satorras,et al.  Activity driven modeling of time varying networks , 2012, Scientific Reports.

[17]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[18]  Harry Dankowicz,et al.  Staged Construction of Adjoints for Constrained Optimization of Integro-Differential Boundary-Value Problems , 2018, SIAM J. Appl. Dyn. Syst..

[19]  George J. Pappas,et al.  Optimal Resource Allocation for Control of Networked Epidemic Models , 2017, IEEE Transactions on Control of Network Systems.

[20]  Vincent D. Blondel,et al.  Bursts of Vertex Activation and Epidemics in Evolving Networks , 2013, PLoS Comput. Biol..

[21]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[22]  E. Ott,et al.  Low dimensional behavior of large systems of globally coupled oscillators. , 2008, Chaos.

[23]  Hyunsuk Hong,et al.  Conformists and contrarians in a Kuramoto model with identical natural frequencies. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  S. Strogatz,et al.  Stability of incoherence in a population of coupled oscillators , 1991 .

[25]  Harry Dankowicz,et al.  Lifetime-preserving reference models for characterizing spreading dynamics on temporal networks , 2017, Scientific Reports.