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Sidewise Profile Control of 1-D Waves

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Abstract

We analyze the sidewise controllability for the variable coefficients one-dimensional wave equation. The control is acting on one extreme of the string with the aim that the solution tracks a given path or profile at the other free end. This sidewise profile control problem is also often referred to as nodal profile or tracking control. The problem is reformulated as a dual observability property for the corresponding adjoint system, which is proved by means of sidewise energy propagation arguments in a sufficiently large time, in the class of BV-coefficients. We also present a number of open problems and perspectives for further research.

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References

  1. Bárcena-Petisco, J.A., Zuazua, E.: Tracking controllability of the heat equation (in preparation)

  2. Cannarsa, P., Komornik, V., Loreti, P.: Well posedness and control of semilinear wave equations with iterated logarithms. ESAIM Control Optim. Calc. Vib. 4, 37–56 (1999)

    Article  MathSciNet  Google Scholar 

  3. Castro, C., Zuazua, E.: Concentration and lack of observability of waves in highly heterogeneous media. Arch. Ration. Mech. Anal. 164(1), 39–72 (2002)

    Article  MathSciNet  Google Scholar 

  4. Cazenave, T., Haraux, A.: Propriétés oscillatoires des solutions de certaines équations des ondes semi-linéaires. C. R. Acad. Sci. Paris Ser. I Math. 298, 449–452 (1983)

    MATH  Google Scholar 

  5. Dáger, R., Zuazua, E.: Wave propagation, observation and control in \(1-d\) flexible multi-structures. In: Mathématiques et Applications, vol. 50. Springer, Verlag (2006)

  6. Fanelli, F., Zuazua, E.: Weak observability estimates for 1-D wave equations with rough coefficients. Ann. Inst. Henri Poincaré Anal. Non-linéaire 32, 245–277 (2015)

    Article  MathSciNet  Google Scholar 

  7. Fernández-Cara, E., Zuazua, E.: On the null controllability of the one-dimensional heat equation with BV coefficients. Comput. Appl. Math. 21(1), 167–190 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Gugat, M.: A turnpike result for convex hyperbolic optimal boundary control problems. Pure Appl. Funct. Anal. 4(4), 849–866 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Gugat, M., Herty, M., Schleper, V.: Flow control in gas networks: Exact controllability to a given demand. Math. Meth. Appl. Sci. 34(7), 745–757 (2011)

    Article  MathSciNet  Google Scholar 

  10. Gugat, M., Trélat, E., Zuazua, E.: Optimal Neumann control for the 1D wave equation: finite horizon, infinite horizon, boundary tracking terms and the turnpike property. Syst. Control Lett. 90, 61–70 (2016)

    Article  MathSciNet  Google Scholar 

  11. Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modeling, analysis and control of dynamic elastic multi-link structures. In: Systems & Control: Foundations & Applications. Springer, Berlin (2012)

  12. Lions, J.L.: Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30(1), 1–68 (1988)

    Article  MathSciNet  Google Scholar 

  13. Lions, J.L.: Contrôlabilité Exacte, Perturbations et Stabilisation de Systémes Distribués Tome 1. Masson, Contrôlabilité Exacte. Rech. Math. Appl. (1988)

  14. Li, T.: Exact boundary controllability of nodal profile for quasilinear hyperbolic systems. Math. Methods Appl. Sci. 33(17), 2101–2106 (2010)

    Article  MathSciNet  Google Scholar 

  15. Li, T., Wang, K., Gu, Q.: Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems. Springer Briefs in Mathematics, Springer, Berlin (2016)

    Book  Google Scholar 

  16. Martin, P., Rosier, L., Rouchon, P.: Null controllability of the heat equation using flatness. Automatica 50(12), 3067–3076 (2014)

    Article  MathSciNet  Google Scholar 

  17. Micu, S., Zuazua, E.: An introduction to the controllability of partial differential equations. In: Sari, T. (ed.) Quelques Questions de Théorie du Contrôle, pp. 67–150. Collection Travaux en Cours, Herman (2005)

    Google Scholar 

  18. Symes, W.: On the relation between coefficients and boundary values for solutions of Webster’s Horn equation. SIAM J. Math. Anal. 17(6), 1400–1420 (1986)

    Article  MathSciNet  Google Scholar 

  19. Wang, K.: Exact boundary controllability of nodal profile for 1-D quasilinear wave equations. Front. Math. China 6(3), 545–555 (2011)

    Article  MathSciNet  Google Scholar 

  20. Wang, K., Gu, Q.: Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree-like network of strings. Math. Methods Appl. Sci. 37(8), 1206–1218 (2014)

    Article  MathSciNet  Google Scholar 

  21. Zhang, X., Zheng, C., Zuazua, E.: Time discrete wave equations: boundary observability and control. Discrete Contin. Dyn. Syst. 23, 571–604 (2009)

    Article  MathSciNet  Google Scholar 

  22. Zhuang, K., Leugering, G., Li, T.: Exact boundary controllability of nodal profile for Saint–Venant system on a network with loops. J. Math. Pures Appl. 129, 34–60 (2019)

    Article  MathSciNet  Google Scholar 

  23. Zuazua, E.: Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. Henri Poincaré Anal. Non-linéaire 10(1), 109–129 (1993)

    Article  MathSciNet  Google Scholar 

  24. Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47(2), 197–243 (2005)

    Article  MathSciNet  Google Scholar 

  25. Zuazua, E.: Controllability of partial differential equations and its semi-discrete approximation. Discrete Contin Dyn. Syst. 8(2), 469–513 (2002)

    Article  MathSciNet  Google Scholar 

  26. Zuazua, E.: Series and Sidewise Profile Control of 1-d Waves (preprint) (2021)

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Acknowledgements

Authors thank Martin Gugat (FAU) for fruitful discussions. This work was done while the first author was visiting FAU-Erlangen during a sabbatical year. The second author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon), the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 765579-ConFlex.D.P., the Alexander von Humboldt-Professorship program, the Transregio 154, Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks, of the German DFG, project C08 and by Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain).

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Authors and Affiliations

  1. Department of Mathematics, Atatürk University, 25240, Erzurum, Turkey

    Yeşim Saraç

  2. Chair of Dynamics, Control and Numerics, Alexander von Humboldt-Professorship, Department of Data Science, Friedrich-Alexander University, Erlangen-Nürnberg, 91058, Erlangen, Germany

    Enrique Zuazua

  3. Chair of Computational Mathematics, Fundación Deusto, University of Deusto, 48007, Bilbao, Basque Country, Spain

    Enrique Zuazua

  4. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Enrique Zuazua

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  1. Yeşim Saraç
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  2. Enrique Zuazua
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Correspondence to Yeşim Saraç.

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Communicated by Boris S. Mordukhovich.

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Saraç, Y., Zuazua, E. Sidewise Profile Control of 1-D Waves. J Optim Theory Appl 193, 931–949 (2022). https://doi.org/10.1007/s10957-021-01986-w

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