Controllability of partial differential equations on graphs

2025/05/27 by Avdonin, S. A., Mikhaylov, V. S. · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2505.20690

Abstract

We study the boundary control problems for the wave, heat, and Schrödinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. The exact controllability in L₂-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. The null controllability for the heat equation and exact controllability for the Schrödinger equation in arbitrary time interval are obtained.

Citations

• Controllability cost of conservative systems: resolvent condition and transmutation
• The control transmutation method and the cost of fast controls
• On the eigenvalue estimates for the weighted Laplacian on metric graphs
• Kirchhoff's rule for quantum wires

Cited by

• On the inverse problem of the two-velocity tree-like graph
• On an inverse problem for tree-like networks of elastic strings

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