American Institute of Mathematical Sciences

September  2013, 2(3): 543-556. doi: 10.3934/eect.2013.2.543

On the structural properties of an efficient feedback law

 1 Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cédex, France

Received  February 2013 Revised  May 2013 Published  July 2013

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.
Citation: Ambroise Vest. On the structural properties of an efficient feedback law. Evolution Equations & Control Theory, 2013, 2 (3) : 543-556. doi: 10.3934/eect.2013.2.543
References:
 [1] F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems,, SIAM J. Control Optim., 37 (1999), 521. doi: 10.1137/S0363012996313835. Google Scholar [2] 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. Google Scholar [3] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems,", Second edition, (2007). Google Scholar [4] 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. doi: 10.1016/j.jsv.2003.10.053. Google Scholar [5] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Springer, (2011). Google Scholar [6] J.-S. Briffaut., "Méthodes Numériques Pour le Contrôle et la Stabilisation Rapide des Grandes Strucutures Flexibles,", PhD thesis, (1999). Google Scholar [7] T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford University Press, (1998). Google Scholar [8] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain,, J. Math. Pures Appl. 58 (1979), 58 (1979), 249. Google Scholar [9] K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000). Google Scholar [10] 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. doi: 10.1007/BF01762397. Google Scholar [11] D. L. Kleinman, An easy way to stabilize a linear constant system,, IEEE Transactions on Automatic Control, 15 (1970). doi: 10.1109/TAC.1970.1099612. Google Scholar [12] V. Komornik, Rapid boundary stabilization of linear distributed systems,, SIAM J. Control Optim., 35 (1997), 1591. doi: 10.1137/S0363012996301609. Google Scholar [13] V. Komornik, Rapid boundary stabilization of Maxwell's equations,, in, (1998), 611. Google Scholar [14] V. Komornik and P. Loreti, "Fourier Series in Control Theory,", Springer-Verlag, (2005). Google Scholar [15] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163. doi: 10.1016/0022-0396(83)90073-6. Google Scholar [16] 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. doi: 10.1007/BF01448390. Google Scholar [17] 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. doi: 10.1016/0022-0396(87)90025-8. Google Scholar [18] 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, (2000). Google Scholar [19] J.-L. Lions, Exact controllability, stabilizability and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001. Google Scholar [20] D. L. Lukes, Stabilizability and optimal control,, Funkcial. Ekvac., 11 (1968), 39. Google Scholar [21] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [22] 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. Google Scholar [23] D. L. Russell, "Mathematics of Finite-Dimensional Control Systems,", Marcel Dekker Inc., (1979). Google Scholar [24] M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM J. Control, 12 (1974), 500. doi: 10.1137/0312038. Google Scholar [25] 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. doi: 10.1109/TAC.2009.2013038. Google Scholar [26] J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators,, SIAM J. Control Optim., 43 (2005), 2233. doi: 10.1137/S0363012901388452. Google Scholar [27] A. Vest, Rapid stabilization in a semigroup framework,, Preprint, (). Google Scholar [28] H. Zwart, Invertible solutions of the Lyapunov and algebraic Riccati equation,, preprint, (). Google Scholar

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References:
 [1] F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems,, SIAM J. Control Optim., 37 (1999), 521. doi: 10.1137/S0363012996313835. Google Scholar [2] 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. Google Scholar [3] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, "Representation and Control of Infinite Dimensional Systems,", Second edition, (2007). Google Scholar [4] 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. doi: 10.1016/j.jsv.2003.10.053. Google Scholar [5] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Springer, (2011). Google Scholar [6] J.-S. Briffaut., "Méthodes Numériques Pour le Contrôle et la Stabilisation Rapide des Grandes Strucutures Flexibles,", PhD thesis, (1999). Google Scholar [7] T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford University Press, (1998). Google Scholar [8] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain,, J. Math. Pures Appl. 58 (1979), 58 (1979), 249. Google Scholar [9] K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000). Google Scholar [10] 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. doi: 10.1007/BF01762397. Google Scholar [11] D. L. Kleinman, An easy way to stabilize a linear constant system,, IEEE Transactions on Automatic Control, 15 (1970). doi: 10.1109/TAC.1970.1099612. Google Scholar [12] V. Komornik, Rapid boundary stabilization of linear distributed systems,, SIAM J. Control Optim., 35 (1997), 1591. doi: 10.1137/S0363012996301609. Google Scholar [13] V. Komornik, Rapid boundary stabilization of Maxwell's equations,, in, (1998), 611. Google Scholar [14] V. Komornik and P. Loreti, "Fourier Series in Control Theory,", Springer-Verlag, (2005). Google Scholar [15] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163. doi: 10.1016/0022-0396(83)90073-6. Google Scholar [16] 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. doi: 10.1007/BF01448390. Google Scholar [17] 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. doi: 10.1016/0022-0396(87)90025-8. Google Scholar [18] 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, (2000). Google Scholar [19] J.-L. Lions, Exact controllability, stabilizability and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001. Google Scholar [20] D. L. Lukes, Stabilizability and optimal control,, Funkcial. Ekvac., 11 (1968), 39. Google Scholar [21] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [22] 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. Google Scholar [23] D. L. Russell, "Mathematics of Finite-Dimensional Control Systems,", Marcel Dekker Inc., (1979). Google Scholar [24] M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM J. Control, 12 (1974), 500. doi: 10.1137/0312038. Google Scholar [25] 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. doi: 10.1109/TAC.2009.2013038. Google Scholar [26] J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators,, SIAM J. Control Optim., 43 (2005), 2233. doi: 10.1137/S0363012901388452. Google Scholar [27] A. Vest, Rapid stabilization in a semigroup framework,, Preprint, (). Google Scholar [28] H. Zwart, Invertible solutions of the Lyapunov and algebraic Riccati equation,, preprint, (). Google Scholar
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