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December  2019, 9(4): 643-671. doi: 10.3934/mcrf.2019045

## Strong stabilization of (almost) impedance passive systems by static output feedback

 1 Dept. of Mathematics, University of Groningen, 9700 AV Groningen, The Netherlands 2 School of Electrical Eng., Tel Aviv University, Ramat Aviv 69978, Israel

*The second author is the coordinator of the ETN network ConFlex, funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 765579

Received  July 2018 Revised  August 2019 Published  November 2019

The plant to be stabilized is a system node $\Sigma$ with generating triple $(A,B,C)$ and transfer function ${\bf G}$, where $A$ generates a contraction semigroup on the Hilbert space $X$. The control and observation operators $B$ and $C$ may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator $E$ such that, if we replace ${\bf G}(s)$ by ${\bf G}(s)+E$, the new system $\Sigma_E$ becomes impedance passive. An easier case is when ${\bf G}$ is already impedance passive and a special case is when $\Sigma$ has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback $u = - {\kappa} y+v$, where $u$ is the input of the plant and ${\kappa}>0$, stabilizes $\Sigma$, strongly or even exponentially. Here, $y$ is the output of $\Sigma$ and $v$ is the new input. Our main result is that if for some $E\in {\mathcal L}(U)$, $\Sigma_E$ is impedance passive, and $\Sigma$ is approximately observable or approximately controllable in infinite time, then for sufficiently small ${\kappa}$ the closed-loop system is weakly stable. If, moreover, $\sigma(A)\cap i {\mathbb R}$ is countable, then the closed-loop semigroup and its dual are both strongly stable.

Citation: Ruth F. Curtain, George Weiss. Strong stabilization of (almost) impedance passive systems by static output feedback. Mathematical Control & Related Fields, 2019, 9 (4) : 643-671. doi: 10.3934/mcrf.2019045
##### References:
 [1] K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback, Mathematics of Control, Signals and Systems, 15 (2002), 229-255.  doi: 10.1007/s004980200009.  Google Scholar [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. of the American Mathematical Society, 306 (1988), 837–852. doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar [3] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander., Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar [4] T. Bailey and J. E. Hubbard jr., Distributed piezoelectric polymer active vibration control of a cantilever beam, AIAA Journal on Guidance, Control and Dynamics, 8 (1985), 605-611.  doi: 10.2514/3.20029.  Google Scholar [5] A. V. Balakrishnan, Compensator design for stability enhancement with collocated controllers, IEEE Trans. Autom. Control, 36 (1991), 994-1007.  doi: 10.1109/9.83531.  Google Scholar [6] A. V. Balakrishnan, Shape control of plates with piezo actuators and collocated position / rate sensors, Applied Math. and Comput., 63 (1994), 213-234.  doi: 10.1016/0096-3003(94)90196-1.  Google Scholar [7] C. J. K. Batty and V. Q. Phong, Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc., 322 (1990), 805-818.  doi: 10.1090/S0002-9947-1990-1022866-5.  Google Scholar [8] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM Journal on Control and Optim., 16 (1978), 373-379.  doi: 10.1137/0316023.  Google Scholar [9] J. Bontsema, Dynamic Stabilization of Large Flexible Space Structures, Ph.D.Thesis, Rijksuniversiteit Groningen, The Netherlands, 1989. Google Scholar [10] R. F. Curtain and G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, W. Schappacher, eds.), 41–59, Birkhäuser-Verlag, Basel, 1989.  Google Scholar [11] R. F. Curtain and G. Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control and Optim., 45 (2006), 273-297.  doi: 10.1137/040610489.  Google Scholar [12] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar [13] R. F. Curtain and B. Jacob, Spectral properties of pseudo-resolvents under structured perturbations, Mathematics of Control, Signals and Systems, 21 (2008), 21-50.  doi: 10.1007/s00498-008-0035-y.  Google Scholar [14] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980.   Google Scholar [15] K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math. vol. 194, Springer-Verlag, New York, 2000.  Google Scholar [16] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact feedback, SIAM J. Control and Optim., 18 (1980), 311-316.  doi: 10.1137/0318022.  Google Scholar [17] Y. Le Gorrec, H. J. Zwart and B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators., SIAM J. of Control and Optim., 44 (2005), 1864-1892.  doi: 10.1137/040611677.  Google Scholar [18] B.-Z. Guo and Z.-H. Luo, Controllability and stability of a second-order hyperbolic system with colocated sensor/actuator, Systems & Control Letters, 46 (2002), 45-65.  doi: 10.1016/S0167-6911(01)00201-8.  Google Scholar [19] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugaliae Mathematica, 46 (1989), 245-258.   Google Scholar [20] I. Lasiecka and R. Triggiani, $L_2(\Sigma)$-regular ity of the boundary to boundary operator $B^*L$ for hyperbolic and Petrowski PDE's, Abstract and Applied Analysis, 2003 (2003), 1061-1139.  doi: 10.1155/S1085337503305032.  Google Scholar [21] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II, Abstract Hyperbolic-type Systems over a Finite Time Horizon, Encyclopedia of Mathematics and its Applications, 75, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511574801.002.  Google Scholar [22] K. Liu, Local distributed control and damping for the conservative systems, SIAM J. Control and Optim., 35 (1997), 1574-1590.  doi: 10.1137/S0363012995284928.  Google Scholar [23] M. S. Livšsic, Operators, Oscillations, Waves (Open Systems), volume 34 of Translations of Mathematical Monographs. American Mathematical Society, Providence, Rhode Island, 1973.  Google Scholar [24] Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.  doi: 10.4064/sm-88-1-37-42.  Google Scholar [25] Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag, London, 1999. doi: 10.1007/978-1-4471-0419-3.  Google Scholar [26] J. Malinen, O. J. Staffans and G. Weiss, When is a linear system conservative?, Quarterly of Applied Math., 64 (2006), 61-91.  doi: 10.1090/S0033-569X-06-00994-7.  Google Scholar [27] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970 (transl. of the French edition of 1967).  Google Scholar [28] J. C. Oostveen, Strongly Stabilizable Infinite-Dimensional Systems, Frontiers in Applied Mathematics, SIAM, Philadelphia, 2000. Google Scholar [29] M. R. Opmeer, Infinite-dimensional linear systems: A distributional approach, Proc. London Math. Society, 91 (2005), 738-760.  doi: 10.1112/S0024611505015315.  Google Scholar [30] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford University Press, Oxford, 1985.   Google Scholar [31] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.  Google Scholar [32] D. L. Russell, Linear stabilization of the linear oscillator in Hilbert space, J. Math. and Applications, 25 (1969), 663-675.  doi: 10.1016/0022-247X(69)90264-9.  Google Scholar [33] D. Salamon, Infinite dimensional systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar [34] A. van der Schaft, $L_2$-Gain and Passivity Techniques in Nonlinear Control, second (enlarged) edition, Springer-Verlag, London, 2000. doi: 10.1007/978-1-4471-0507-7.  Google Scholar [35] M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control and Optim., 12 (1974), 500-508.  doi: 10.1137/0312038.  Google Scholar [36] M. Slemrod, Stabilization of boundary control systems, J. of Differential Equations, 22 (1976), 402-415.  doi: 10.1016/0022-0396(76)90036-X.  Google Scholar [37] M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. of Control, Signals and Systems, 2 (1989), 265-285.  doi: 10.1007/BF02551387.  Google Scholar [38] O. J. Staffans, Quadratic optimal control of stable well-posed linear systems, Trans. American Math. Society, 349 (1997), 3679-3715.  doi: 10.1090/S0002-9947-97-01863-1.  Google Scholar [39] O. J. Staffans, Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems, Math. Control, Signals and Systems, 15 (2002), 291-315.  doi: 10.1007/s004980200012.  Google Scholar [40] O. J. Staffans, Stabilization by collocated feedback, Directions in Mathematical Systems Theory and Optimization, A. Rantzer and C.I. Byrnes, eds, LNCIS, vol. 286, Springer-Verlag, Berlin, 2003, 275–292. doi: 10.1007/3-540-36106-5_21.  Google Scholar [41] O. J. Staffans, Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view), In Mathematical Systems Theory in Biology, Communication, Computation, and Finance, volume 134 of IMA Volumes in Mathematics and its Applications, pages 375–413. Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21696-6_14.  Google Scholar [42] O. J. Staffans, Well-Posed Linear Systems, Cambridge University Press, Cambridge, UK, 2005.  doi: 10.1017/CBO9780511543197.  Google Scholar [43] O. J. Staffans and G. Weiss, Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup, Trans. American Math. Society, 354 (2002), 3229-3262.  doi: 10.1090/S0002-9947-02-02976-8.  Google Scholar [44] V. L. Syrmos, C. T. Abdallah, P. Dorato and K. Grigoriadis, Static output feedback - A survey, Automatica, 33 (1997), 125-137.  doi: 10.1016/S0005-1098(96)00141-0.  Google Scholar [45] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbations, Proc. Amer. Math. Soc., 105 (1989), 375-383.  doi: 10.1090/S0002-9939-1989-0953013-0.  Google Scholar [46] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, J. Math. Anal. Appl., 137 (1989), 438-461.  doi: 10.1016/0022-247X(89)90255-2.  Google Scholar [47] M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. Part II: Controllability and stability, SIAM J. Control and Optim., 42 (2003), 907-935.  doi: 10.1137/S0363012901399295.  Google Scholar [48] J. A. Villegas, H. Zwart, Y. Le Gorrec, B. Maschke and A. 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show all references

##### References:
 [1] K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback, Mathematics of Control, Signals and Systems, 15 (2002), 229-255.  doi: 10.1007/s004980200009.  Google Scholar [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. of the American Mathematical Society, 306 (1988), 837–852. doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar [3] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander., Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar [4] T. Bailey and J. E. Hubbard jr., Distributed piezoelectric polymer active vibration control of a cantilever beam, AIAA Journal on Guidance, Control and Dynamics, 8 (1985), 605-611.  doi: 10.2514/3.20029.  Google Scholar [5] A. V. Balakrishnan, Compensator design for stability enhancement with collocated controllers, IEEE Trans. Autom. Control, 36 (1991), 994-1007.  doi: 10.1109/9.83531.  Google Scholar [6] A. V. Balakrishnan, Shape control of plates with piezo actuators and collocated position / rate sensors, Applied Math. and Comput., 63 (1994), 213-234.  doi: 10.1016/0096-3003(94)90196-1.  Google Scholar [7] C. J. K. Batty and V. Q. Phong, Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc., 322 (1990), 805-818.  doi: 10.1090/S0002-9947-1990-1022866-5.  Google Scholar [8] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM Journal on Control and Optim., 16 (1978), 373-379.  doi: 10.1137/0316023.  Google Scholar [9] J. Bontsema, Dynamic Stabilization of Large Flexible Space Structures, Ph.D.Thesis, Rijksuniversiteit Groningen, The Netherlands, 1989. Google Scholar [10] R. F. Curtain and G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch, W. Schappacher, eds.), 41–59, Birkhäuser-Verlag, Basel, 1989.  Google Scholar [11] R. F. Curtain and G. Weiss, Exponential stabilization of well-posed systems by colocated feedback, SIAM J. Control and Optim., 45 (2006), 273-297.  doi: 10.1137/040610489.  Google Scholar [12] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar [13] R. F. Curtain and B. Jacob, Spectral properties of pseudo-resolvents under structured perturbations, Mathematics of Control, Signals and Systems, 21 (2008), 21-50.  doi: 10.1007/s00498-008-0035-y.  Google Scholar [14] E. B. Davies, One-Parameter Semigroups, Academic Press, London, 1980.   Google Scholar [15] K. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math. vol. 194, Springer-Verlag, New York, 2000.  Google Scholar [16] J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact feedback, SIAM J. Control and Optim., 18 (1980), 311-316.  doi: 10.1137/0318022.  Google Scholar [17] Y. Le Gorrec, H. J. Zwart and B. Maschke, Dirac structures and boundary control systems associated with skew-symmetric differential operators., SIAM J. of Control and Optim., 44 (2005), 1864-1892.  doi: 10.1137/040611677.  Google Scholar [18] B.-Z. Guo and Z.-H. Luo, Controllability and stability of a second-order hyperbolic system with colocated sensor/actuator, Systems & Control Letters, 46 (2002), 45-65.  doi: 10.1016/S0167-6911(01)00201-8.  Google Scholar [19] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugaliae Mathematica, 46 (1989), 245-258.   Google Scholar [20] I. Lasiecka and R. Triggiani, $L_2(\Sigma)$-regular ity of the boundary to boundary operator $B^*L$ for hyperbolic and Petrowski PDE's, Abstract and Applied Analysis, 2003 (2003), 1061-1139.  doi: 10.1155/S1085337503305032.  Google Scholar [21] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II, Abstract Hyperbolic-type Systems over a Finite Time Horizon, Encyclopedia of Mathematics and its Applications, 75, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511574801.002.  Google Scholar [22] K. Liu, Local distributed control and damping for the conservative systems, SIAM J. Control and Optim., 35 (1997), 1574-1590.  doi: 10.1137/S0363012995284928.  Google Scholar [23] M. S. Livšsic, Operators, Oscillations, Waves (Open Systems), volume 34 of Translations of Mathematical Monographs. American Mathematical Society, Providence, Rhode Island, 1973.  Google Scholar [24] Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.  doi: 10.4064/sm-88-1-37-42.  Google Scholar [25] Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag, London, 1999. doi: 10.1007/978-1-4471-0419-3.  Google Scholar [26] J. Malinen, O. J. Staffans and G. Weiss, When is a linear system conservative?, Quarterly of Applied Math., 64 (2006), 61-91.  doi: 10.1090/S0033-569X-06-00994-7.  Google Scholar [27] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970 (transl. of the French edition of 1967).  Google Scholar [28] J. C. Oostveen, Strongly Stabilizable Infinite-Dimensional Systems, Frontiers in Applied Mathematics, SIAM, Philadelphia, 2000. Google Scholar [29] M. R. Opmeer, Infinite-dimensional linear systems: A distributional approach, Proc. London Math. Society, 91 (2005), 738-760.  doi: 10.1112/S0024611505015315.  Google Scholar [30] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford University Press, Oxford, 1985.   Google Scholar [31] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.  Google Scholar [32] D. L. Russell, Linear stabilization of the linear oscillator in Hilbert space, J. Math. and Applications, 25 (1969), 663-675.  doi: 10.1016/0022-247X(69)90264-9.  Google Scholar [33] D. Salamon, Infinite dimensional systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar [34] A. van der Schaft, $L_2$-Gain and Passivity Techniques in Nonlinear Control, second (enlarged) edition, Springer-Verlag, London, 2000. doi: 10.1007/978-1-4471-0507-7.  Google Scholar [35] M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control and Optim., 12 (1974), 500-508.  doi: 10.1137/0312038.  Google Scholar [36] M. Slemrod, Stabilization of boundary control systems, J. of Differential Equations, 22 (1976), 402-415.  doi: 10.1016/0022-0396(76)90036-X.  Google Scholar [37] M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. of Control, Signals and Systems, 2 (1989), 265-285.  doi: 10.1007/BF02551387.  Google Scholar [38] O. J. Staffans, Quadratic optimal control of stable well-posed linear systems, Trans. American Math. Society, 349 (1997), 3679-3715.  doi: 10.1090/S0002-9947-97-01863-1.  Google Scholar [39] O. J. Staffans, Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems, Math. Control, Signals and Systems, 15 (2002), 291-315.  doi: 10.1007/s004980200012.  Google Scholar [40] O. J. Staffans, Stabilization by collocated feedback, Directions in Mathematical Systems Theory and Optimization, A. Rantzer and C.I. Byrnes, eds, LNCIS, vol. 286, Springer-Verlag, Berlin, 2003, 275–292. doi: 10.1007/3-540-36106-5_21.  Google Scholar [41] O. J. Staffans, Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view), In Mathematical Systems Theory in Biology, Communication, Computation, and Finance, volume 134 of IMA Volumes in Mathematics and its Applications, pages 375–413. Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21696-6_14.  Google Scholar [42] O. J. Staffans, Well-Posed Linear Systems, Cambridge University Press, Cambridge, UK, 2005.  doi: 10.1017/CBO9780511543197.  Google Scholar [43] O. J. Staffans and G. Weiss, Transfer functions of regular linear systems, Part II: The system operator and the Lax-Phillips semigroup, Trans. American Math. Society, 354 (2002), 3229-3262.  doi: 10.1090/S0002-9947-02-02976-8.  Google Scholar [44] V. L. Syrmos, C. T. Abdallah, P. Dorato and K. Grigoriadis, Static output feedback - A survey, Automatica, 33 (1997), 125-137.  doi: 10.1016/S0005-1098(96)00141-0.  Google Scholar [45] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbations, Proc. Amer. Math. Soc., 105 (1989), 375-383.  doi: 10.1090/S0002-9939-1989-0953013-0.  Google Scholar [46] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, J. Math. Anal. Appl., 137 (1989), 438-461.  doi: 10.1016/0022-247X(89)90255-2.  Google Scholar [47] M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air. Part II: Controllability and stability, SIAM J. Control and Optim., 42 (2003), 907-935.  doi: 10.1137/S0363012901399295.  Google Scholar [48] J. A. Villegas, H. Zwart, Y. Le Gorrec, B. Maschke and A. 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The open-loop system node $\Sigma$ with static output feedback. If ${\kappa}>0$ is sufficiently small, then this feedback results in a closed-loop system $\Sigma^ {\kappa}$ that is well-posed and system stable. Under suitable additional assumptions, the operator semigroup of $\Sigma^ {\kappa}$ and its dual are strongly stable
The scattering passive system $\Sigma^s$ with input $u^s$ and output $y^s$, obtained from the impedance passive system node $\Sigma^p$ via the diagonal transformation, as in Proposition 4.1
The feedback system $\Sigma^ {\kappa}$ from Figure 1, with input $v$ and output $y$, as obtained from the scattering passive system $\Sigma^s$ with input $u^s$ and output $y^s$. Note that there is no feedback loop involved in this transformation
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