American Institute of Mathematical Sciences

Advanced
December  2014, 7(6): 1321-1334. doi: 10.3934/dcdss.2014.7.1321

Control via decoupling of a class of second order linear hybrid systems

David L. Russell 1,

1. 

Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, VA 24060

Received  April 2013 Revised  November 2013 Published  June 2014

We study a terminal state control (reachability) problem for certain elastic systems of ``hybrid" type consisting of a single space dimension distributed parameter part coupled, at one endpoint of the relevant spatial, $x$, interval, to a lumped mass component. Two such systems are studied in detail. The first is a vibrating string system fixed at $x = 0$ and attached to a point mass at the right hand endpoint $x = L$. The second example concerns an Euler - Bernoulli beam system ``clamped" at $x = 0$ and attached, at $x = L$, to a mass with both translational and rotational inertia. In both cases the controls act on the mass, which is modeled by a finite dimensional system of differential equations. Analysis of the reachability problem is facilitated by a preliminary ``feedback type" transformation of the control variable which decouples the point mass from the distributed system. In both examples a concluding analysis is required to show that the component of the control generated by feedback lies in the same space as the originally applied control.
Keywords: PDE control, hybrid systems, decoupling, stabilization..
Mathematics Subject Classification: Primary: 74C05, 74K10, 74K20, 90C25, 93D1.
Citation: David L. Russell. Control via decoupling of a class of second order linear hybrid systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1321-1334. doi: 10.3934/dcdss.2014.7.1321
References:
[1]

Int. J. Math. & Stat., 7 (2010), 46-53.  Google Scholar

[2]

Cambridge University Press, Cambridge, New York, Melbourne, 1995.  Google Scholar

[3]

IEEE Trans. Aerosp. & Electr. Syst., 28 (1992), 80-91. doi: 10.1109/7.135434.  Google Scholar

[4]

C. R. Acad. Sci. Paris Sér. I, Math., 326 (1998), 453-458. doi: 10.1016/S0764-4442(97)89791-1.  Google Scholar

[5]

Angew. Math. & Phys., 12 (1961), 369-392. doi: 10.1007/BF01600687.  Google Scholar

[6]

SIAM J. Control. & Opt., 36 (1998), 1962-1986. doi: 10.1137/S0363012996302366.  Google Scholar

[7]

J. Math. Anal. & and Appl., 361 (2010), 392-400. doi: 10.1016/j.jmaa.2009.06.059.  Google Scholar

[8]

SIAM J. Control. & Opt., 39 (2001), 1736-1747. doi: 10.1137/S0363012999354880.  Google Scholar

[9]

CRC Press, Boca Raton, 2002. Google Scholar

[10]

Mathem. Zeitschrift, 41 (1936), 367-379. doi: 10.1007/BF01180426.  Google Scholar

[11]

Ann. Mat. Pura & Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.  Google Scholar

[12]

Arch. Rat. Mech. & Anal., 103 (1988), 193-236. doi: 10.1007/BF00251758.  Google Scholar

[13]

IEEE Trans. Automat. Control, 39 (1994), 2140-2145. doi: 10.1109/9.328811.  Google Scholar

[14]

SIAM J. Control. & Opt., 33 (1995), 440-454. doi: 10.1137/S0363012992239879.  Google Scholar

[15]

J. Math. Anal. & Appl., 18 (1967), 542-560. doi: 10.1016/0022-247X(67)90045-5.  Google Scholar

[16]

Int. J. Tomogr. & Stat., 10 (2008), 125-140.  Google Scholar

show all references

References:
[1]

Int. J. Math. & Stat., 7 (2010), 46-53.  Google Scholar

[2]

Cambridge University Press, Cambridge, New York, Melbourne, 1995.  Google Scholar

[3]

IEEE Trans. Aerosp. & Electr. Syst., 28 (1992), 80-91. doi: 10.1109/7.135434.  Google Scholar

[4]

C. R. Acad. Sci. Paris Sér. I, Math., 326 (1998), 453-458. doi: 10.1016/S0764-4442(97)89791-1.  Google Scholar

[5]

Angew. Math. & Phys., 12 (1961), 369-392. doi: 10.1007/BF01600687.  Google Scholar

[6]

SIAM J. Control. & Opt., 36 (1998), 1962-1986. doi: 10.1137/S0363012996302366.  Google Scholar

[7]

J. Math. Anal. & and Appl., 361 (2010), 392-400. doi: 10.1016/j.jmaa.2009.06.059.  Google Scholar

[8]

SIAM J. Control. & Opt., 39 (2001), 1736-1747. doi: 10.1137/S0363012999354880.  Google Scholar

[9]

CRC Press, Boca Raton, 2002. Google Scholar

[10]

Mathem. Zeitschrift, 41 (1936), 367-379. doi: 10.1007/BF01180426.  Google Scholar

[11]

Ann. Mat. Pura & Appl., 152 (1988), 281-330. doi: 10.1007/BF01766154.  Google Scholar

[12]

Arch. Rat. Mech. & Anal., 103 (1988), 193-236. doi: 10.1007/BF00251758.  Google Scholar

[13]

IEEE Trans. Automat. Control, 39 (1994), 2140-2145. doi: 10.1109/9.328811.  Google Scholar

[14]

SIAM J. Control. & Opt., 33 (1995), 440-454. doi: 10.1137/S0363012992239879.  Google Scholar

[15]

J. Math. Anal. & Appl., 18 (1967), 542-560. doi: 10.1016/0022-247X(67)90045-5.  Google Scholar

[16]

Int. J. Tomogr. & Stat., 10 (2008), 125-140.  Google Scholar

[1]

Sebastian Hage-Packhäuser, Michael Dellnitz. Stabilization via symmetry switching in hybrid dynamical systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 239-263. doi: 10.3934/dcdsb.2011.16.239
[2]

Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021055
[3]

Sorin Micu, Jaime H. Ortega, Lionel Rosier, Bing-Yu Zhang. Control and stabilization of a family of Boussinesq systems. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 273-313. doi: 10.3934/dcds.2009.24.273
[4]

Fatiha Alabau-Boussouira, Piermarco Cannarsa, Roberto Guglielmi. Indirect stabilization of weakly coupled systems with hybrid boundary conditions. Mathematical Control & Related Fields, 2011, 1 (4) : 413-436. doi: 10.3934/mcrf.2011.1.413
[5]

Kokum R. De Silva, Tuoc V. Phan, Suzanne Lenhart. Advection control in parabolic PDE systems for competitive populations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1049-1072. doi: 10.3934/dcdsb.2017052
[6]

Kathrin Flasskamp, Sebastian Hage-Packhäuser, Sina Ober-Blöbaum. Symmetry exploiting control of hybrid mechanical systems. Journal of Computational Dynamics, 2015, 2 (1) : 25-50. doi: 10.3934/jcd.2015.2.25
[7]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[8]

Lian Zhang, Mingchao Cai, Mo Mu. Decoupling PDE computation with intrinsic or inertial Robin interface condition. Electronic Research Archive, 2021, 29 (2) : 2007-2028. doi: 10.3934/era.2020102
[9]

Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063
[10]

Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004
[11]

Yanli Han, Yan Gao. Determining the viability for hybrid control systems on a region with piecewise smooth boundary. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 1-9. doi: 10.3934/naco.2015.5.1
[12]

Kobamelo Mashaba, Jianxing Li, Honglei Xu, Xinhua Jiang. Optimal control of hybrid manufacturing systems by log-exponential smoothing aggregation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1711-1719. doi: 10.3934/dcdss.2020100
[13]

Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934/mbe.2007.4.633
[14]

David L. Russell. Modeling and control of hybrid beam systems with rotating tip component. Evolution Equations & Control Theory, 2014, 3 (2) : 305-329. doi: 10.3934/eect.2014.3.305
[15]

A. Alessandri, F. Bedouhene, D. Bouhadjra, A. Zemouche, P. Bagnerini. Observer-based control for a class of hybrid linear and nonlinear systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1213-1231. doi: 10.3934/dcdss.2020376
[16]

Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034
[17]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020
[18]

K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 237-247. doi: 10.3934/naco.2019050
[19]

Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303
[20]

Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control & Related Fields, 2020, 10 (4) : 715-734. doi: 10.3934/mcrf.2020017

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences