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Exponential stabilization of conservation systems with interior disturbance. (English) Zbl 1330.93210
Summary: In this paper, we consider the stabilization problem of the conservation systems with interior disturbance. Employing the idea of sliding-mode control, we design a nonlinear distributed feedback controller. We prove the solvability of the resulted closed-loop system by the maximal monotone operator theory. Further, we prove the exponential stability of the closed-loop system. In particular, we prove that the requirement of a classical solution in the Lyapunov approach is unnecessary.

93D21 Adaptive or robust stabilization
93B12 Variable structure systems
93B52 Feedback control
47H05 Monotone operators and generalizations
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[1] Azocar, A.; Gimenez, J.; Nikodem, K.; Sanchez, J. L., On strongly midconvex functions, Opuscula Math., 31, 1, 15-26, (2011) · Zbl 1234.26035
[2] Barbu, V., Nonlinear differential equations of monotone types in Banach spaces, (2010), Springer New York · Zbl 1197.35002
[3] Chen, B. S.; Lee, T. S.; Feng, J. H., A nonlinear \(H_\infty\) control design in robotic systems under parameter perturbations and external disturbances, Internat. J. Control, 59, 439-461, (1994) · Zbl 0807.93018
[4] Chen, G.; Delfour, M. C.; Krall, A. M.; Payre, G., Modeling, stabilization and control of serially connected beam, SIAM J. Control Optim., 25, 526-546, (1987) · Zbl 0621.93053
[5] Cheng, M. B.; Radisavljevic, V.; Su, W. C., Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties, Automatica, 47, 381-387, (2001) · Zbl 1213.35239
[6] Christofides, P. D., Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes, Systems & Control: Foundations Applications Series, (2001), Birkhäuser Boston, MA · Zbl 1018.93001
[7] Cox, S.; Zuazua, E., The rate at which energy decays in a damped string, Comm. Partial Differential Equations, 19, 213-243, (1994) · Zbl 0818.35072
[8] Dieci, L.; Lopez, L., Sliding motion in Filippov differential systems: theoretical results and a computational approach, SIAM J. Numer. Anal., 47, 2023-2051, (2009) · Zbl 1197.34009
[9] Drakunov, S.; Barbieri, E.; Silver, D. A., Sliding mode control of a heat equation with application to arc welding, (International Conference on Control Applications, (1996)), 668-672
[10] Fridman, E.; Orlov, Y., An LMI approach to boundary control of semilinear parabolic and hyperbolic systems, Automatica, 45, 2060-2066, (2009) · Zbl 1175.93107
[11] Ge, S. S.; Zhang, S.; He, W., Vibration control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance, Internat. J. Control, 84, 947-960, (2011) · Zbl 1245.93065
[12] Guo, B. Z.; Jin, F. F., The active disturbance rejection and sliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbance, Automatica, 49, 2911-2918, (2013) · Zbl 1364.93637
[13] Guo, B. Z.; Kang, W., Lyapunov approach to the boundary stabilization of a beam equation with boundary disturbance, Internat. J. Control, 87, 925-939, (2014) · Zbl 1291.93070
[14] Han, Z. J.; Xu, G. Q., Dynamical behavior of a hybrid system of nonhomogeneous Timoshenko beam with partial non-collocated inputs, J. Dyn. Control Syst., 17, 77-121, (2011) · Zbl 1211.93054
[15] Han, Z. J.; Xu, G. Q., Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement, Asian J. Control, 14, 95-108, (2012) · Zbl 1282.93207
[16] Komornik, V.; Zuazua, E., A direct method for the boundary stability of the wave equation, J. Math. Pures Appl., 69, 33-54, (1990) · Zbl 0636.93064
[17] LaSalle, J. P., The stability and control of discrete processes, (1986), Springer-Verlag Berlin · Zbl 0606.93001
[18] Levaggi, L., Infinite dimensional systems sliding motions, Eur. J. Control, 8, 508-518, (2002) · Zbl 1293.93160
[19] Liu, J. J.; Wang, J. M., Active disturbance rejection control and sliding mode control of one-dimensional unstable heat equation with boundary uncertainties, IMA J. Math. Control Inform., 32, 1, 97-117, (2015) · Zbl 1308.93057
[20] Morgül, Ö., Stabilization and disturbance rejection for the beam equation, IEEE Trans. Automat. Control, 46, 1913-1918, (2001) · Zbl 1009.93040
[21] Nakao, M., Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann., 305, 403-417, (1996) · Zbl 0856.35084
[22] Nikodem, K.; Pales, Z. S., Characterization of inner product spaces by strongly convex functions, Banach J. Math. Anal., 5, 1, 83-87, (2011) · Zbl 1215.46016
[23] Orlov, Y., Discontinuous unit feedback control of uncertain infinite-dimensional systems, IEEE Trans. Automat. Control, 45, 834-843, (2000) · Zbl 0973.93018
[24] Orlov, Y., Discontinuous systems - Lyapunov analysis and robust synthesis under uncertainty conditions, (2009), Springer-Verlag London · Zbl 1180.37004
[25] Orlov, Y.; Utkin, V. I., Sliding mode control in indefinite-dimensional systems, Automatica, 23, 753-757, (1987) · Zbl 0661.93036
[26] Orlov, Y.; Liu, Y.; Christofides, P. D., Robust stabilization of infinite-dimensional systems using sliding-mode output feedback control, Internat. J. Control, 77, 1115-1136, (2004) · Zbl 1062.93012
[27] Pisano, A.; Orlov, Y.; Usai, E., Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques, SIAM J. Control Optim., 49, 363-382, (2011) · Zbl 1217.93136
[28] Polyak, B. T., Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7, 72-75, (1966) · Zbl 0171.09501
[29] Shang, Y. F.; Xu, G. Q., Dynamic feedback control and exponential stabilization of a compound system, J. Math. Anal. Appl., 422, 2, 858-879, (2015) · Zbl 1297.93131
[30] Shang, Y. F.; Xu, G. Q.; Chen, Y. L., Stability analysis of Euler-Bernoulli beam with input delay in the boundary control, Asian J. Control, 14, 186-196, (2012) · Zbl 1282.93199
[31] Utkin, V. I., Sliding modes in control and optimization, (1992), Springer-Verlag Berlin · Zbl 0748.93044
[32] Wang, H.; Xu, G. Q., Exponential stabilization of 1-d wave equation with input delay, WSEAS Trans. Math., 12, 10, 1001-1013, (2013)
[33] Wang, J. M.; Liu, J. J.; Ren, B.; Chen, J., Sliding mode control to stabilization of cascaded heat PDE-ODE systems subject to boundary control matched disturbance, Automatica, 52, 23-34, (2015) · Zbl 1309.93124
[34] Xu, G. Q.; Guo, B. Z., Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM J. Control Optim., 42, 966-984, (2003) · Zbl 1066.93028
[35] Xu, G. Q.; Yung, S. P.; Li, L. K., Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12, 4, 770-785, (2006) · Zbl 1105.35016
[36] Zhou, J.; Wen, C.; Zhang, Y., Adaptive backstepping control of a class of uncertain nonlinear systems with unknown backlash-like hysteresis, IEEE Trans. Automat. Control, 49, 1751-1759, (2004) · Zbl 1365.93251
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