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Exact Controllability of the Wave Equation on Graphs

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Abstract

In this paper we study the exact controllability problem for the wave equation on a finite metric graph with the Kirchhoff–Neumann matching conditions. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if Neumann controllers are placed at the active vertices and Dirichlet controllers are placed at the active edges. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the exact controllability utilizes both dynamical and moment method approaches. The control time for this construction is determined by the chosen orientation and path decomposition of the graph. We indicate the optimal time that guarantees the exact controllability for all systems of a described class on a given graph. While the choice of the active vertices and edges is not unique, we find the minimum number of controllers to guarantee the exact controllability as a graph invariant.

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Acknowledgements

The authors gratefully thank Julian Edward for very useful conversations about several parts of this manuscript and anonymous referee for valuable remarks. The research of Sergei Avdonin was supported in part by the National Science Foundation, Grant DMS 1909869. The research of Yuanyuan Zhao was supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1242789.

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  1. Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK, 99775, USA

    Sergei Avdonin & Yuanyuan Zhao

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia, 119991

    Sergei Avdonin

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Appendix

Appendix

We give equations in Sects. 3.4 and  3.7 for nonzero potentials. The solution to the forward problem for nonzero potentials are in parallel to the zero potentials case in Sect. 3.7, with Eq. (3.11) replaced by

$$\begin{aligned} f'_i(t) =&\sum _{j \in J^+(v_i) \setminus J^-(\Gamma )} \left\{ -h'_j(t)+\int _0^t (\omega _j)_x(0,s)h_j(t-s)\, ds \right. \\&-2h_j'(t-2l_j)-2\omega _j(2l_j,2l_j)h_j(t-2l_j)\\&+2\int _{2l_j}^t (\omega _j)_x(2l_j,s)h_j(t-s) \, ds \\&-2h_j'(t-4l_j)-2\omega _j(4l_j,4l_j)h_j(t-4l_j)\\&\left. +2\int _{4l_j}^t (\omega _j)_x(4l_j,s)h_j(t-s) \, ds - \dots \right\} \\&+ \sum _{j \in J^+(v_i) \setminus J^-(\Gamma )} \left\{ 2g'_{k(j)}(t-l_j)+2 \bar{\omega }_j(l_j,l_j)g_{k(j)}(t-l_j)\right. \\&-2\int _{l_j}^t (\bar{\omega }_j)_x(l_j,s) g_{k(j)}(t-s) \, ds \\&+2g'_{k(j)}(t-3_j)+2 \bar{\omega }_j(3_j,3_j)g_{k(j)}(t-3l_j)\\&\left. -2\int _{3l_j}^t (\bar{\omega }_j)_x(3l_j,s) g_{k(j)}(t-s) \, ds + \dots \right\} \\&+ \sum _{j \in J^+(v_i) \cap J^-(\Gamma )} \left\{ -h'_j(t)+\int _0^t (\omega _j)_x(0,s)h_j(t-s)\, ds \right. \\&+2h_j'(t-2l_j)+2\omega _j(2l_j,2l_j)h_j(t-2l_j)\\&-2\int _{2l_j}^t (\omega _j)_x(2l_j,s)h_j(t-s) \, ds \\&-2h_j'(t-4l_j)-2\omega _j(4l_j,4l_j)h_j(t-4l_j)\\&\left. +2\int _{4l_j}^t (\omega _j)_x(4l_j,s)h_j(t-s) \, ds + \dots \right\} \,, \end{aligned}$$

(3.12) replaced by

$$\begin{aligned} 0 =&\sum _{j \in J^-(v_i) \cap J^+(\Gamma ^*)} \left\{ -2 f_{k(j)}'(t-l_j) - 2 \mu _j(l_j,l_j) f_{k(j)}(t-l_j)\right. \\&+2 \int _{l_j}^t (\mu _j)_x(l_j,t) f_{k(j)}(t-s) \, ds \\&+2 f_{k(j)}'(t-3l_j) + 2 \mu _j(3l_j,3l_j) f_{k(j)}(t-3l_j)\\&\left. -2 \int _{3l_j}^t (\mu _j)_x(3l_j,t) f_{k(j)}(t-s) \, ds - \dots \right\} \\&+ \sum _{j \in J^-(v_i) \cap J^+(\Gamma ^*)} \left\{ -g'_i(t)+\int _0^t (\bar{\omega }_j)_x(0,s)g_i(t-s)\, ds \right. \\&+2g_i'(t-2l_j)+2\bar{\omega }_j(2l_j,2l_j)g_i(t-2l_j)-2\int _{2l_j}^t (\bar{\omega }_j)_x(2l_j,s)g_i(t-s) \, ds \\&\left. -2g_i'(t-4l_j)-2\bar{\omega }_j(4l_j,4l_j)g_i(t-4l_j)+2\int _{4l_j}^t (\bar{\omega }_j)_x(4l_j,s)g_i(t-s) \, ds - \dots \right\} \\&+\!\!\sum _{j \in J^-(v_i) \setminus J^+(\Gamma ^*)} \left\{ 2 h_j'(t-l_j) +\!2\omega _j(l_j,l_j)h_j(t-l_j)-2\int _{l_j}^t (\omega _j)_x(l_j,s)h_j(t-s) \, ds \right. \\&\left. +2 h_j'(t-3l_j) +2\omega _j(3l_j,3l_j)h_j(t-3l_j)-2\int _{3l_j}^t (\omega _j)_x(3l_j,s)h_j(t-s) \, ds +\dots \right\} \\&+ \sum _{j \in J^-(v_i) \setminus J^+(\Gamma ^*)} \left\{ -g'_i(t)+\int _0^t (\bar{\omega }_j)_x(0,s)g_i(t-s)\, ds \right. \\&-2g_i'(t-2l_j)-2\bar{\omega }_j(2l_j,2l_j)g_i(t-2l_j)+2\int _{2l_j}^t (\bar{\omega }_j)_x(2l_j,s)g_i(t-s) \, ds \\&\left. -2g_i'(t-4l_j)-2\bar{\omega }_j(4l_j,4l_j)g_i(t-4l_j)+2\int _{4l_j}^t (\bar{\omega }_j)_x(4l_j,s)g_i(t-s) \, ds - \dots \right\} \\&+ \sum _{j \in J^+(v_i) \setminus J^-(\Gamma )} \left\{ -h'_j(t)+\int _0^t (\omega _j)_x(0,s)h_j(t-s)\, ds \right. \\&-2h_j'(t-2l_j)-2\omega _j(2l_j,2l_j)h_j(t-2l_j)+2\int _{2l_j}^t (\omega _j)_x(2l_j,s)h_j(t-s) \, ds \\&\left. -2h_j'(t-4l_j)-2\omega _j(4l_j,4l_j)h_j(t-4l_j)+2\int _{4l_j}^t (\omega _j)_x(4l_j,s)h_j(t-s) \, ds - \dots \right\} \\&+ \sum _{j \in J^+(v_i) \cap J^-(\Gamma )} \left\{ -h'_j(t)+\int _0^t (\omega _j)_x(0,s)h_j(t-s)\, ds \right. \\&+2h_j'(t-2l_j)+2\omega _j(2l_j,2l_j)h_j(t-2l_j)-2\int _{2l_j}^t (\omega _j)_x(2l_j,s)h_j(t-s) \, ds \\&\left. -2h_j'(t-4l_j)-2\omega _j(4l_j,4l_j)h_j(t-4l_j)+2\int _{4l_j}^t (\omega _j)_x(4l_j,s)h_j(t-s) \, ds + \dots \right\} \\&+ \sum _{j \in J^+(v_i) \setminus J^-(\Gamma )} \left\{ 2g'_{k(j)}(t-l_j)+2 \bar{\omega }_j(l_j,l_j)g_{k(j)}(t-l_j)\right. \\&-2\int _{l_j}^t (\bar{\omega }_j)_x(l_j,s) g_{k(j)}(t-s) \, ds +2g'_{k(j)}(t-3_j)+2 \bar{\omega }_j(3_j,3_j)g_{k(j)}(t-3l_j)\\&\left. -2\int _{3l_j}^t (\bar{\omega }_j)_x(3l_j,s) g_{k(j)}(t-s) \, ds + \dots \right\} \,. \end{aligned}$$

The solution to the shape/velocity control problems for nonzero potentials are in parallel to the zero potentials case in Sect. 3.4, with Eq. (3.19) replaced by

$$\begin{aligned}&2 h_{m(i)}'(t-l_{m(i)}) +2\omega _{m(i)}(l_{m(i)}, l_{m(i)}) h_{m(i)}(t-l_{m(i)}) \\&\qquad -2\int _{l_{m(i)}}^t (\omega _{m(i)})_x(l_{m(i)},s) h_{m(i)}(t-s) \, ds \\&\quad = \sum _{j \in J^-(v_1)} \left\{ - g_1' (t) +\int _0^t (\bar{\omega }_j)_(0,s) g_1(t-s)\, ds \right\} \,, \end{aligned}$$

(3.20) replaced by

$$\begin{aligned}&h_{m(i)}(T-x)+\int _x^T \omega _{m(i)}(x,s) h_{m(i)} (T-s) \, ds \\&\quad - h_{m(i)}(T-2{m(i)}+x) -\int _{2l_{m(i)}-x}^T \omega _{m(i)}(2l_{m(i)}-x,s) h_{m(i)} (T-s) \, ds \\&\quad +g_1(T-l_{m(i)}+x) +\int _{l_{m(i)}-x}^T \bar{\omega }_{m(i)}({l_{m(i)}-x},s) g_1 (T-s) \, ds =\phi _{m(i)}(x)\,, \end{aligned}$$

(3.21) replaced by

$$\begin{aligned}&h_{j}(T-x)+\int _x^T \omega _{j}(x,s) h_{j} (T-s) \, ds +g_1(T-l_{j}+x) \\&\quad +\int _{l_j-x}^T \bar{\omega }_{j}(l_j-x,s) g_1 (T-s) \, ds =\phi _{j}(x)\,, \end{aligned}$$

(3.29) replaced by

$$\begin{aligned} 0 =&\left\{ -2 f_{k(m(i))}'(t-l_{m(i)}) - 2 \mu _{m(i)}(l_{m(i)},l_{m(i)}) f_{k(m(i))}(t-l_{m(i)}) \right. \\&+2 \int _{l_{m(i)}}^t (\mu _{m(i)})_x(l_{m(i)},t) f_{k(m(i))}(t-s) \, ds \\&+2 f_{k(m(i))}'(t-3l_{m(i)}) + 2 \mu _{m(i)}(3l_{m(i)},3l_{m(i)}) f_{k(m(i))}(t-3l_{m(i)}) \\&\left. -2 \int _{3l_{m(i)}}^t (\mu _{m(i)})_x(3l_{m(i)},t) f_{k(m(i))}(t-s) \, ds - \dots \right\} \\&+ \left\{ -g'_i(t)+\int _0^t (\bar{\omega }_{m(i)})_x(0,s)g_i(t-s)\, ds \right. \\&+2g_i'(t-2l_{m(i)})+2\bar{\omega }_{m(i)}(2l_{m(i)},2l_{m(i)})g_i(t-2l_{m(i)}) \\&-2\int _{2l_{m(i)}}^t (\bar{\omega }_{m(i)})_x(2l_{m(i)},s)g_i(t-s) \, ds \\&-2g_i'(t-4l_{m(i)})-2\bar{\omega }_{m(i)}(4l_{m(i)},4l_{m(i)})g_i(t-4l_{m(i)}) \\&\left. +2\int _{4l_{m(i)}}^t (\bar{\omega }_{m(i)})_x(4l_{m(i)},s)g_i(t-s) \, ds - \dots \right\} \\&+ \sum _{j \in J^+(v_i) \setminus J^-(\Gamma )} \left\{ -h'_j(t)+\int _0^t (\omega _j)_x(0,s)h_j(t-s)\, ds \right. \\&-2h_j'(t-2l_j)-2\omega _j(2l_j,2l_j)h_j(t-2l_j)+2\int _{2l_j}^t (\omega _j)_x(2l_j,s)h_j(t-s) \, ds \\&\left. -2h_j'(t-4l_j)-2\omega _j(4l_j,4l_j)h_j(t-4l_j)+2\int _{4l_j}^t (\omega _j)_x(4l_j,s)h_j(t-s) \, ds - \dots \right\} \\&+ \sum _{j \in J^+(v_i) \cap J^-(\Gamma )} \left\{ -h'_j(t)+\int _0^t (\omega _j)_x(0,s)h_j(t-s)\, ds \right. \\&+2h_j'(t-2l_j)+2\omega _j(2l_j,2l_j)h_j(t-2l_j)-2\int _{2l_j}^t (\omega _j)_x(2l_j,s)h_j(t-s) \, ds \\&\left. -2h_j'(t-4l_j)-2\omega _j(4l_j,4l_j)h_j(t-4l_j)+2\int _{4l_j}^t (\omega _j)_x(4l_j,s)h_j(t-s) \, ds + \dots \right\} \\&+ \sum _{j \in J^+(v_i) \setminus J^-(\Gamma )} \left\{ 2g'_{k(j)}(t-l_j)+2 \bar{\omega }_j(l_j,l_j)g_{k(j)}(t-l_j)\right. \\&-2\int _{l_j}^t (\bar{\omega }_j)_x(l_j,s) g_{k(j)}(t-s) \, ds \\&\left. +2g'_{k(j)}(t-3_j)+2 \bar{\omega }_j(3_j,3_j)g_{k(j)}(t-3l_j)\right. \\&\left. -2\int _{3l_j}^t (\bar{\omega }_j)_x(3l_j,s) g_{k(j)}(t-s) \, ds + \dots \right\} \,, \end{aligned}$$

(3.30) replaced by

$$\begin{aligned} 0 =&\left\{ 2 h_{m(i)}'(t-l_{m(i)}) +2\omega _{m(i)}(l_{m(i)},l_{m(i)})h_{m(i)}(t-l_{m(i)}) \right. \\&-2\int _{l_{m(i)}}^t (\omega _{m(i)})_x(l_{m(i)},s)h_{m(i)}(t-s) \, ds \\&+2 h_{m(i)}'(t-3l_{m(i)}) +2\omega _{m(i)}(3l_{m(i)},3l_{m(i)})h_{m(i)}(t-3l_{m(i)}) \\&\left. -2\int _{3l_{m(i)}}^t (\omega _{m(i)})_x(3l_{m(i)},s)h_{m(i)}(t-s) \, ds +\dots \right\} \\&+ \left\{ -g'_i(t)+\int _0^t (\bar{\omega }_{m(i)})_x(0,s)g_i(t-s)\, ds \right. \\&-2g_i'(t-2l_{m(i)})-2\bar{\omega }_{m(i)}(2l_{m(i)},2l_{m(i)})g_i(t-2l_{m(i)}) \\&+2\int _{2l_{m(i)}}^t (\bar{\omega }_{m(i)})_x(2l_{m(i)},s)g_i(t-s) \, ds \\&-2g_i'(t-4l_{m(i)})-2\bar{\omega }_{m(i)}(4l_{m(i)},4l_{m(i)})g_i(t-4l_{m(i)}) \\&\left. +2\int _{4l_{m(i)}}^t (\bar{\omega }_{m(i)})_x(4l_{m(i)},s)g_i(t-s) \, ds - \dots \right\} \\&+ \sum _{j \in J^+(v_i) \setminus J^-(\Gamma )} \left\{ -h'_j(t)+\int _0^t (\omega _j)_x(0,s)h_j(t-s)\, ds \right. \\&-2h_j'(t-2l_j)-2\omega _j(2l_j,2l_j)h_j(t-2l_j)+2\int _{2l_j}^t (\omega _j)_x(2l_j,s)h_j(t-s) \, ds \\&\left. -2h_j'(t-4l_j)-2\omega _j(4l_j,4l_j)h_j(t-4l_j)+2\int _{4l_j}^t (\omega _j)_x(4l_j,s)h_j(t-s) \, ds - \dots \right\} \\&+ \sum _{j \in J^+(v_i) \cap J^-(\Gamma )} \left\{ -h'_j(t)+\int _0^t (\omega _j)_x(0,s)h_j(t-s)\, ds \right. \\&+2h_j'(t-2l_j)+2\omega _j(2l_j,2l_j)h_j(t-2l_j)-2\int _{2l_j}^t (\omega _j)_x(2l_j,s)h_j(t-s) \, ds \\&\left. -2h_j'(t-4l_j)-2\omega _j(4l_j,4l_j)h_j(t-4l_j)+2\int _{4l_j}^t (\omega _j)_x(4l_j,s)h_j(t-s) \, ds + \dots \right\} \\&+ \sum _{j \in J^+(v_i) \setminus J^-(\Gamma )} \left\{ 2g'_{k(j)}(t-l_j)+2 \bar{\omega }_j(l_j,l_j)g_{k(j)}(t-l_j)\right. \\&-2\int _{l_j}^t (\bar{\omega }_j)_x(l_j,s) g_{k(j)}(t-s) \, ds +2g'_{k(j)}(t-3_j)+2 \bar{\omega }_j(3_j,3_j)g_{k(j)}(t-3l_j)\\&\left. -2\int _{3l_j}^t (\bar{\omega }_j)_x(3l_j,s) g_{k(j)}(t-s) \, ds + \dots \right\} \,. \end{aligned}$$

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Avdonin, S., Zhao, Y. Exact Controllability of the Wave Equation on Graphs. Appl Math Optim 85, 1 (2022). https://doi.org/10.1007/s00245-022-09869-w

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