Zero Forcing, Linear and Quantum Controllability for Systems Evolving on Networks

Daniel Burgarth*, Domenico D'Alessandro, Leslie Hogben, Simone Severini, Michael Young

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Our main result says that controllability in the quantum sense, expressed by the Lie algebra rank condition, and controllability in the sense of linear systems, expressed by the controllability matrix rank condition, are equivalent conditions. We also investigate how the graph theoretic concept of a zero forcing set impacts the controllability property; if a set of vertices is a zero forcing set, the associated dynamical system is controllable. These results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena.

Original languageEnglish
Pages (from-to)2349-2354
Number of pages6
JournalIEEE Transactions on Automatic Control
Issue number9
Publication statusPublished - Sept 2013


  • Control
  • graph
  • Lie algebra
  • quantum system
  • walk matrix
  • zero forcing


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