Abstract
The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish a complete theory on the HUM method and give its applications to first order hyperbolic systems. Thus, a deeper relationship between the controllability and the observability can be revealed. Moreover, at the end of the paper, a comparison will be made between these two methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chapelon, A. and Xu, C.-Z., Boundary control of a class of hyperbolic systems, Eur. J. Control, 9(6), 2003, 589–604.
Coron, J.-M., Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematics Society, Providence, 2007.
Lax, P. D., Functional Analysis, Wiley-Interscience, New York, 2002.
Li, T. T., Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series on Applied Mathematics, Vol. 3, American Institute of Mathematical Sciences & Higher Education Press, Beijing, 2010.
Li, T. T. and Rao, B. P., Local exact boundary controllability for a class of quasilinear hyperbolic systems, Chin. Ann. Math. Ser. B, 23(2), 2002, 209–218.
Li, T. T. and Rao, B. P., Exact boundary controllability for quasi-linear hyperbolic systems, SIAM J. Control Optim., 41, 2003, 1748–1755.
Li, T. T. and Rao, B. P., Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems, Chin. Ann. Math. Ser. B, 31(5), 2010, 723–742.
Li, T. T. and Rao, B. P., Synchronisation exacte d’un système couplé d’équations des ondes par des contrôles frontières de Dirichlet, C. R. Math. Acad. Sci. Paris, 350(15–16), 2012, 767–772.
Li, T. T. and Rao, B. P., Exact boundary controllability for a coupled system of wave equations with Neumann controls, Chin. Ann. Math. Ser. B, 38(2), 2017, 473–488.
Lions, J.-L., Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Vol. 1 and 2, Masson, Paris, 1988.
Lions, J.-L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30, 1988, 1–68.
Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676.
Author information
Authors and Affiliations
Corresponding authors
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 11831011, 11901082), the Natural Science Foundation of Jiangsu Province (No. BK20190323) and the Fundamental Research Funds for the Central Universities of China.
Rights and permissions
About this article
Cite this article
Lu, X., Li, T. Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System — the HUM Method. Chin. Ann. Math. Ser. B 43, 1–16 (2022). https://doi.org/10.1007/s11401-022-0300-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-022-0300-2