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On the structural properties of an efficient feedback law

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  • We investigate some structural properties of an efficient feedback law that stabilize linear time-reversible systems with an arbitrarily large decay rate. After giving a short proof of the generation of a group by the closed-loop operator, we focus on the domain of the infinitesimal generator in order to illustrate the difference between a distributed control and a boundary control, the latter being technically more complex. We also give a new proof of the exponential decay of the solutions and we provide an explanation of the higher decay rate observed in some experiments.
    Mathematics Subject Classification: Primary: 93D15; Secondary: 93C05, 47D06, 93C20.


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  • [1]

    F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM J. Control Optim., 37 (1999), 521-542.doi: 10.1137/S0363012996313835.


    A. Benabdallah and M. Lenczner, Estimation du taux de décroissance pour la solution de problèmes de stabilisation, application à la stabilisation de l'équation des ondes, RAIRO Modél. Math. Anal. Numér., 30 (1996), 607-635.


    A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems," Second edition, Birkhäuser, Boston, 2007.


    F. Bourquin, M. Joly, M. Collet and L. Ratier, An efficient feedback control algorithm for beams: Experimental investigations, Journal of Sound and Vibration, 278 (2004), 181-206.doi: 10.1016/j.jsv.2003.10.053.


    H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Springer, New York, 2011.


    J.-S. Briffaut., "Méthodes Numériques Pour le Contrôle et la Stabilisation Rapide des Grandes Strucutures Flexibles," PhD thesis, École Nationale des Ponts et Chaussées, 1999.


    T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations," Oxford University Press, New York, 1998.


    G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. 58 (1979), 249-273.


    K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, 2000.


    F. Flandoli, I. Lasiecka and R. Triggiani, Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems, Ann. Mat. Pura Appl., 153 (1988), 307-382.doi: 10.1007/BF01762397.


    D. L. Kleinman, An easy way to stabilize a linear constant system, IEEE Transactions on Automatic Control, 15 (1970), 692.doi: 10.1109/TAC.1970.1099612.


    V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim., 35 (1997), 1591-1613.doi: 10.1137/S0363012996301609.


    V. Komornik, Rapid boundary stabilization of Maxwell's equations, in "Équations aux Dérivées Partielles et Applications," Gauthier-Villars, Paris, (1998), 611-622.


    V. Komornik and P. Loreti, "Fourier Series in Control Theory," Springer-Verlag, New York, 2005.


    J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.doi: 10.1016/0022-0396(83)90073-6.


    I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_{2}(0,T;L_{2}(\Gamma ))$-Dirichlet boundary terms, Appl. Math. Optim., 10 (1983), 275-286.doi: 10.1007/BF01448390.


    I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2(0,\infty; L_2(\Gamma))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390.doi: 10.1016/0022-0396(87)90025-8.


    I. Lasiecka and R.Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories, volume II, Abstract Hyperbolic-like Systems over a Finite Time Horizon," Cambridge University Press, Cambridge, 2000.


    J.-L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.doi: 10.1137/1030001.


    D. L. Lukes, Stabilizability and optimal control, Funkcial. Ekvac., 11 (1968), 39-50.


    A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, 1983, viii+279 pp.doi: 10.1007/978-1-4612-5561-1.


    J. P. Quinn and D. L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping, Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 97-127.


    D. L. Russell, "Mathematics of Finite-Dimensional Control Systems," Marcel Dekker Inc., New York, 1979.


    M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control, 12 (1974), 500-508.doi: 10.1137/0312038.


    A. Smyshlyaev, B.-Z. Guo and M. Krstic, Arbitrary decay rate for Euler-Bernoulli beam by backstepping boundary feedback, IEEE Trans. Automat. Control, 54 (2009), 1134-1140.doi: 10.1109/TAC.2009.2013038.


    J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., 43 (2005), 2233-2244.doi: 10.1137/S0363012901388452.


    A. VestRapid stabilization in a semigroup framework, Preprint, arXiv:1301.5744.


    H. ZwartInvertible solutions of the Lyapunov and algebraic Riccati equation, preprint, http://wwwhome.math.utwente.nl/ zwarthj/.

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