PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

On controllability of second order dynamical systems – a survey

Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The paper presents a survey of recent results in the area of controllability of second order dynamical systems. Controllability problem for finite and infinite dimensional, linear, semilinear, deterministic and stochastic dynamical systems (with delays and undelayed) is taken into consideration. Different types of controllability are discussed.
Twórcy
autor
  • Silesian University of Technology, 16 Akademicka St., 44-100 Gliwice, Poland
autor
  • Silesian University of Technology, 16 Akademicka St., 44-100 Gliwice, Poland
autor
  • Silesian University of Technology, 16 Akademicka St., 44-100 Gliwice, Poland
Bibliografia
  • [1] R. E. Kalman, "Contributions to the theory of optimal control", Boletin de la Sociedad Matematica Mexicana 5, 102-119 (1960).
  • [2] R. E. Kalman, "On general theory of control systems", in: Proceedings of 1th IFAC Congress, 481-493 (1960).
  • [3] R. E. Kalman, “Canonical structure of linear dynamical systems”, Proceedings of the National Academy of Sciences of the United States of America 48, 596–600 (1962).
  • [4] R.E. Kalman, “Mathematical description of linear dynamical systems”, Journal of SIAM Series A Control 1 (2), 152–192 (1963).
  • [5] R.E. Kalman, P.L. Falb, and M.A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969.
  • [6] C.T. Chen, Introduction to Linear Systems Theory, Holt, Rinehart and Winston Inc., New York, 1970.
  • [7] J. Klamka, Controllability of Dynamical Systems, Kluwer Academic, Dordrecht, 1991.
  • [8] B. De Schutter, “Minimal state-space realization in linear system theory: an overview”, Journal of Computational and Applied Mathematics 121 (1–2), 331–354 (2000).
  • [9] B.C. Moore, “Principal component analysis in linear systems: controllability, observability and model reduction”, IEEE Transactions on Automatic Control 26 (1), 17–31 (1981).
  • [10] L.M. Silverman, “Realization of linear dynamical systems”, IEEE Transactions on Automatic Control 16 (6), 54–567 (1971).
  • [11] E.G. Gilbert, “Controllability and observability in multivariable control systems”, Journal of SIAM Series A Control 1 (2), 128–151 (1963).
  • [12] D.G. Luenberger, “Canonical forms for linear multivariable systems”, IEEE Transactions on Automatic Control 12 (3), 290–290 (1967).
  • [13] M.K. Solak, “Transformations between canonical forms for multivariable linear constant systems”, International Journal of Control 40 (1), 141–148 (1984).
  • [14] J. Klamka, “Minimum energy control of discrete systems with delays in control”, International Journal of Control 26 (5), 737–744 (1977).
  • [15] J. Klamka, “Minimum energy control of 2-D systems in Hilbert spaces”, Systems Science 9 (1–2), 33–42 (1983).
  • [16] J. Klamka, “Stochastic controllability and minimum energy control of systems with multiple delays in control”, Applied Mathematics and Computation 206 (2), 704–715 (2008).
  • [17] T. Kaczorek and J. Klamka, “Minimum energy control of 2-D linear systems with variable coefficients”, International Journal of Control 44 (3), 645–650 (1986).
  • [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
  • [19] R. Triggiani, “On the lack of exact controllability for mild solutions in Banach spaces”, Journal of Mathematical Analysis and Applications 50 (2), 438–446 (1975).
  • [20] Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Infinite Delay, Springer, Berlin, 1991.
  • [21] J.K. Hale and J. Kato, “Phase space for retarded equations with infinite delay”, Funkcialaj Ekvacioj 21, 11–41 (1978).
  • [22] G. Arthi and K. Balachandran, “Controllability of second-order impulsive evolution systems with infinite delay”, Nonlinear Analysis: Hybrid Systems 11 (1), 139–153 (2014).
  • [23] L. Caccetta and V.G. Rumchev, “A survey of reachability and controllability for positive linear systems”, Annals of Operations Research 98 (1), 101–122 (2000).
  • [24] C.J. Wang, “Controllability and observability of linear time varying singular systems”, IEEE Transactions on Automatic Control 44 (10), 1901–1905 (1999).
  • [25] L. Pandolfi, “Controllability and stabilization for linear systems of algebraic and differential equations”, Journal of Optimization Theory & Applications 30 (4), 601–620 (1980).
  • [26] L. Armentano, “The pencil (sE-A) and controllability-observability for generalized linear systems: a geometric approach”, SIAM Journal on Control Optimization 24 (4), 616–638 (1986).
  • [27] L. Banaszuk and K.M. Przyłuski, “On perturbations of controllable implicit linear systems”, IMA Journal of Mathematical Control and Information 16 (1), 91–102 (1999).
  • [28] D.J. Bender and A.J. Laub, “Controllability and observability at infinity of multivariable linear second-order models”, IEEE Transactions on Automatic Control 30 (12), 1234–1237 (1985).
  • [29] T. Berger, and T. Reis, Surveys in Differential-Algebraic Equations I, Springer-Verlag, Berlin, 2013.
  • [30] R.E. O’Brien, “Perturbation of controllable systems”, SIAM Journal on Control and Optimization 17 (2), 462–491 (1975).
  • [31] T. Kobayashi, “Simplified conditions of controllability and observability for a model of flexible structures”, Archives of Control Sciences 4, 251–259 (1995).
  • [32] J. Klamka and J.Wyrwał, “Controllability of second-order infinite-dimensional systems”, Systems and Control Letters 57 (5), 386– 391 (2008).
  • [33] A. Bashirov and M. Jneid, “On partial complete controllability of semilinear systems”, Abstract and Applied Analysis 2013, 1–8 (2013).
  • [34] H. Leiva, N. Merentes, and J. Sanchez, “A characterization of semilinear dense range operators and applications”, Abstract and Applied Analysis 2013, 1–11 (2013).
  • [35] H. Leiva, N. Merentes, and J. Sanchez, “Approximate controllability of semilinear reaction diffusion equations”, Mathematical Control and Related Fields 2 (2), 171–182 (2012).
  • [36] R. Sakthivel, R. Ganesh, Y. Ren, and S. Anthoni, “Approximate controllability of nonlinear fractional dynamical systems”, Communications in Nonlinear Science and Numerical Simulation 18 (12), 3498–3508 (2013).
  • [37] J. Klamka, “Constrained controllability of semilinear systems”, Nonlinear Analysis 47 (5), 2939–2949 (2001).
  • [38] J. Klamka, “Constrained controllability of semilinear systems with multiple delays in control”, Bull. Pol. Ac.: Tech. 52 (1), 25–30 (2004).
  • [39] J. Klamka, “Constrained controllability of semilinear systems with delays”, Nonlinear Dynamics 56 (1), 169–177 (2009).
  • [40] J. Klamka, “Constrained controllability of second order dynamical systems with delay”, Proceedings of the 14th International Conference on Methods and Models in Automation and Robotics, 15–20 (2009).
  • [41] J.Y. Park, K. Balachandran, and G. Arthi, “Controllability of impulsive neutral integrodifferential systems with infinite delay in Banach spaces”, Nonlinear Analysis: Hybrid Systems 3 (3), 184–194 (2009).
  • [42] J. Klamka, “Constrained controllability of semilinear systems with delayed controls”, Bull. Pol. Ac.: Tech. 56 (4), 333–337 (2008).
  • [43] D.N. Chalishajar and H. Chalishajar, “Trajectory controllability of second order nonlinear integro-differential system: An analytical and a numerical estimation”, Differential Equations and Dynamical Systems 23 (4), 467–481 (2015).
  • [44] N.I. Mahmudov and S.J. Zorlu, “Controllability of nonlinear stochastic systems”, International Journal of Control 76 (2), 95–104 (2003).
  • [45] N.I. Mahmudov and A. Denker, “On controllability of linear stochastic systems”, International Journal of Control 73 (2), 144–151 (2000).
  • [46] J. Klamka, “Stochastic controllability of linear systems with delay in control”, Bull. Pol. Ac.: Tech. 55 (1), 23–29 (2007).
  • [47] J. Klamka, “Stochastic controllability of systems with variable delay in control”, Bull. Pol. Ac.: Tech. 56 (3), 279–284 (2008).
  • [48] J. Klamka, “Stochastic controllability of linear systems with state delays”, International Journal of Applied Mathematics and Computer Science 17 (1), 5–13 (2007).
  • [49] J. Klamka, “Controllability of dynamical systems. A survey”, Bull. Pol. Ac.: Tech. 61 (2), 335–342 (2013).
  • [50] S. Das, D. Pandey, and N. Sukavanam, “Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay”, Acta Mathematica Scientia 36 (5), 1509–1523 (2016).
  • [51] U. Arora and N. Sukavanam, “Approximate controllability of second order semilinear stochastic system with variable delay in control and with nonlocal conditions”, Rendiconti del Circolo Matematico di Palermo 65 (2), 307–322 (2016).
  • [52] P. Muthukumar and C. Rajivganthi, “Approximate controllability of second-order neutral stochastic differential equations with infinite delay and poisson jumps”, Journal of Systems Science & Complexity 28 (5), 1033–1048 (2015).
  • [53] G. Chen and X. Dong, “On feedback control of chaotic continuous time systems”, IEEE Transactions on Circuits Systems I: Fundamental Theory and Applicatons 40 (9), 591–601 (1993).
  • [54] H. Nijmeijer and H. Berghuis, “On Lyapunov control of the duffing equation”, IEEE Transactions on Circuits Systems I: Fundamental Theory and Applicatons 42 (8), 473–477 (1995).
  • [55] G. Duffing, Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz, F. Vieweg & Sohn, Braunschweig, 1918.
  • [56] N. Van Loi and V. Obukhovskii, “On controllability of duffing equation”, Applied Mathematics and Computation 219 (29), 10468– 10474 (2013).
  • [57] A. Ammar-Kohdia, M. Benabdallah, L. Gonzáles-Burgos, and L. de Teresa, “Recent results on the controllability of coupled parabolic problems: a survey”, Mathematical Control and Related Fields 1 (3), 267–306 (2011).
  • [58] R. Dáger, “Insensitizing controls for the 1-d wave equation”, SIAM J. Control Optim. 45 (5), 1758–1768 (2006).
  • [59] L. Tebou, “Locally distributed desensitizing controls for the wave equation”, C. R. Math. Acad. Sci. Paris 346 (7), 407–412 (2008).
  • [60] F. Alabau-Boussouira, “A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems”, SIAM J. Control Optim. 42 (3), 871–906 (2003).
  • [61] F. Alabau-Boussouira and M. Leautaud, “Indirect controllability of locally coupled systems under geometric conditions”, C. R. Math. Acad. Sci. Paris 349 (7), 395–400 (2011).
  • [62] D. Russell, “Controllability and stabilizability theory for linear partial differential equations”, SIAM Review 20 (4), 639–739 (1978).
  • [63] S. Avdonin and S. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, 1995.
  • [64] S. Hansen and E. Zuazua, “Exact controllability and stabilization of a vibrating spring with an interior point mass”, SIAM J. Control Optim. 33 (5), 1357–1391 (1995).
  • [65] S. Avdonin, A. Choque Rivero, and L. de Teresa, “Exact boundary controllability of coupled hyperbolic equations”, International Journal of Applied Mathematics and Computer Science 23 (4), 701–710 (2013).
  • [66] A. Gallo and A. Piccirillo, “About new analogies of Gronwall-Bellman- Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems”, Nonlinear Analysis 67 (5), 1550–1559 (2007).
  • [67] V. Lakshmikantham, D.D. Bainov, and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
  • [68] Y. Rogovchenko, “Impulsive evolution systems: main results and new trends”, Dynamics of Continuous, Discrete and Impulsive Systems 3 (1), 57–88 (1997).
  • [69] A. Samoilenko and N. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
  • [70] S. Zavalishchin and A. Sesekin, Dynamic Impulse Systems. Theory and Applications, Kluwer Academic Publishers Group, Dordrecht, 1997.
  • [71] M. Choisy, J.F. Guegan, and P. Rohani, “Dynamics of infectious diseases and pulse vaccination: tearing apart the embedded resonance effect”, Physica D 223 (1), 26–35 (2006).
  • [72] W. Wang, H. Wang, and Z. Li, “The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy”, Chaos Solitions and Fractals 32 (5), 1772–1785 (2007).
  • [73] W. Zhang and M. Fan, “Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays”, Mathematical and Computer Modelling 39 (4–5), 479– 493 (2004).
  • [74] J.K. Hale and J. Kato, “Phase space for retarded equations with infinite delay”, Funkcialaj Ekvacioj 21, 11–41 (1978).
  • [75] F. Kappel and W. Schappacher, “Some considerations to the fundamental theory of infinite delay equations”, Journal of Differential Equations 37 (2), 141–183 (1980).
  • [76] J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
  • [77] G. Arthi and J. Park, “On controllability of second-order impulsive neutral integrodifferential systems with infinite delay”, IMA Journal of Mathematical Control and Information 32 (3), 639–657 (2015).
  • [78] L. Chen and G. Li, “Approximate controllability of impulsive differential equations with nonlocal conditions”, International Journal of Nonlinear Science 10 (4), 438–446 (2010).
  • [79] H. Larez, H. Leiva, J. Rebaza, and A. Rios, “Approximate controllability of semilinear impulsive strongly damped wave equation”, Journal of Applied Analysis 21 (1), 45–58 (2015).
  • [80] S. Chen and R. Triggiani, “Proof of extensions of two conjectures on structural damping for elastic systems”, Pacific Journal of Mathematics 136 (1), 15–55 (1989).
  • [81] J. Bana՚s and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.
  • [82] W. Mitkowski, Stabilization of Dynamical Systems, WNT, Warsaw, 1991.
  • [83] W. Mitkowski, W. Bauer, and M. Zagórowska, “Discrete-time feedback stabilization”, Proceedings of the 20th National Conference on Automation of Discrete Processes, (2016).
  • [84] J.P. LaSale, The Stability and Control of Discrete Processes, Springer-Verlag, New York, 1986.
  • [85] V.N. Phat, Constrained Control Problems of Discrete Processes, World Scientific, Singapore, 1996.
  • [86] R. Zawiski, “On controllability and measures of noncompactness”, Proceedings of the 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2014 1637, 1241– 1246 (2014).
Uwagi
PL
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-13825c54-52fa-4017-9ece-c2937d688bb6
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.