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Controllability of partial differential equations on graphs

S. A. Avdonin, V. S. Mikhaylov

TL;DR

This work establishes boundary controllability for wave, heat, and Schrödinger equations on finite tree graphs (quantum graphs) with Dirichlet boundary control at exterior vertices. Using spectral data, the method of moments, and propagation of singularities, it proves exact controllability for the wave equation in time equal to the graph's optical diameter $d(\Omega)$, null controllability for the heat equation in arbitrary time, and exact controllability for Schrödinger flow in arbitrary time, including scenarios with partial boundary control. The analysis hinges on constructing suitable exponential families from Dirichlet spectral data, showing they form $\mathcal{L}$-bases, and relating wave controllability to parabolic and Schrödinger controllability via operator-theoretic transfers. The results advance understanding of control on networked PDEs, with implications for boundary control design on quantum graphs and related inverse problems.

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 $L_2$-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.

Controllability of partial differential equations on graphs

TL;DR

This work establishes boundary controllability for wave, heat, and Schrödinger equations on finite tree graphs (quantum graphs) with Dirichlet boundary control at exterior vertices. Using spectral data, the method of moments, and propagation of singularities, it proves exact controllability for the wave equation in time equal to the graph's optical diameter , null controllability for the heat equation in arbitrary time, and exact controllability for Schrödinger flow in arbitrary time, including scenarios with partial boundary control. The analysis hinges on constructing suitable exponential families from Dirichlet spectral data, showing they form -bases, and relating wave controllability to parabolic and Schrödinger controllability via operator-theoretic transfers. The results advance understanding of control on networked PDEs, with implications for boundary control design on quantum graphs and related inverse problems.

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 -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.

Paper Structure

This paper contains 10 sections, 16 theorems, 52 equations.

Table of Contents

  1. Introduction.
  2. Statement of the problems and main results.
  3. Dirichlet spectral problem.
  4. Initial boundary value problems. Control from the whole boundary.
  5. Initial boundary value problems. Control from a part of the boundary.
  6. Auxiliary results.
  7. Proof of Theorem $\ref{['main']}$.
  8. Proof of Theorem $\ref{['par_main']}$.
  9. Proof of Theorem $\ref{['Schr_main']}$.
  10. Acknowledgments

Key Result

Theorem 1

For arbitrary state $\{a,b\}\in \mathcal{H}\times \mathcal{H}_{-1}$, there exists such a control function $f(\gamma,t)\in \mathcal{F}^T_\Gamma$ with $T=d(\Omega)$, that solution of the initial boundary value problem (wv_eqn)--(cond_3) satisfies the equalities $u^f(\cdot,T)=a$, $u^f_t(\cdot,T)=b$.

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Proposition 1
  • Proposition 2
  • ...and 13 more