2016, 9(4): 1069-1094. doi: 10.3934/dcdss.2016042

Null controllable sets and reachable sets for nonautonomous linear control systems

1. 

Dipartimento di Matematica e Informatica "Ulisse Dine", Università di Firenze, Via di Santa Marta 3, 50139 Firenze

2. 

Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid

3. 

Departamento de Matemática Aplicada, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid

Received  July 2015 Revised  October 2015 Published  August 2016

Under the assumption of lack of uniform controllability for a family of time-dependent linear control systems, we study the dimension, topological structure and other dynamical properties of the sets of null controllable points and of the sets of reachable points. In particular, when the space of null controllable vectors has constant dimension for all the systems of the family, we find a closed invariant subbundle where the uniform null controllability holds. Finally, we associate a family of linear Hamiltonian systems to the control family and assume that it has an exponential dichotomy in order to relate the space of null controllable vectors to one of the Lagrange planes of the continuous hyperbolic splitting.
Citation: Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042
References:
[1]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory,, Springer-Verlag, (1982). doi: 10.1007/978-1-4615-6927-5.

[2]

R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969).

[3]

R. Fabbri, R. Johnson and C. Núñez, Rotation number for non-autonomous linear Hamiltonian systems I: Basic properties,, Z. Angew. Math. Phys., 54 (2003), 484. doi: 10.1007/s00033-003-1068-1.

[4]

R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non-autonomous control processes,, Discrete Contin. Dynam. Systems, 9 (2003), 677. doi: 10.3934/dcds.2003.9.677.

[5]

R. Fabbri, R. Johnson, S. Novo and C. Núñez, On linear-quadratic dissipative control processes with time-varying coefficients,, Discrete Contin. Dynam. Systems, 33 (2013), 193. doi: 10.3934/dcds.2013.33.193.

[6]

R. Johnson and M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems,, Mem. Amer. Math. Soc., 136 (1998). doi: 10.1090/memo/0646.

[7]

R. Johnson, S. Novo, C. Núñez and R. Obaya, Uniform weak disconjugacy and principal solutions for linear Hamiltonian systems,, in: Recent Advances in Delay Differential and Difference Equations, 94 (2014), 131. doi: 10.1007/978-3-319-08251-6.

[8]

R. Johnson, S. Novo, C. Núñez and R. Obaya, Nonautonomous linear-quadratic dissipative control processes without uniform null controllability,, J. Dynam. Differential Equations, (2015), 1. doi: 10.1007/s10884-015-9495-1.

[9]

R. Johnson and C. Núñez, Remarks on linear-quadratic dissipative control systems,, Discr. Cont. Dyn. Sys. B, 20 (2015), 889. doi: 10.3934/dcdsb.2015.20.889.

[10]

R. Johnson, R. Obaya, S. Novo, C. Núñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control,, Developments in Mathematics 36, 36 (2016). doi: 10.1007/978-3-319-29025-6.

[11]

W. Kratz, A limit theorem for monotone matrix functions,, Linear Algebra Appl., 194 (1993), 205. doi: 10.1016/0024-3795(93)90122-5.

[12]

W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory,, Mathematical Topics 6, 6 (1995).

[13]

W. Kratz, Definiteness of quadratic functionals,, Analysis (Munich), 23 (2003), 163. doi: 10.1524/anly.2003.23.2.163.

[14]

Y. Matsushima, Differentiable Manifolds,, Marcel Dekker, (1972).

[15]

A. S. Mishchenko, V. E. Shatalov and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator,, Springer-Verlag, (1990). doi: 10.1007/978-3-642-61259-6.

[16]

S. Novo, C. Núñez and R. Obaya, Ergodic properties and rotation number for linear Hamiltonian systems,, J. Differential Equations, 148 (1998), 148. doi: 10.1006/jdeq.1998.3469.

[17]

W. T. Reid, Sturmian Theory for Ordinary Differential Equations,, Applied Mathematical Sciences 31, 31 (1980). doi: 10.1007/978-1-4612-6110-0.

[18]

W. T. Reid, Principal solutions of nonoscillatory linear differential systems,, J. Math. Anal. Appl., 9 (1964), 397. doi: 10.1016/0022-247X(64)90026-5.

[19]

P. Šepitka and R. Šimon Hilscher, Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems,, J. Dynam. Differential Equations, 26 (2014), 57. doi: 10.1007/s10884-013-9342-1.

[20]

P. Šepitka and R. Šimon Hilscher, Principal Solutions at Infinity of Given Ranks for Nonoscillatory Linear Hamiltonian Systems,, J. Dynam. Differential Equations, 27 (2015), 137.

[21]

R. Šimon Hilscher, Sturmian theory for linear Hamiltonian systems without controllability,, Math. Nachr., 248 (2011), 831. doi: 10.1002/mana.201000071.

[22]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. doi: 10.1016/0022-0396(78)90057-8.

[23]

M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems,, International J. Difference Equ., 2 (2007), 221.

[24]

V. A. Yakubovich, A linear-quadratic optimization problem and the frequency theorem for periodic systems. I,, Siberian Math. J., 27 (1986), 614. doi: 10.1007/bf00969175.

[25]

V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems. II,, Siberian Math. J., 31 (1990), 1027. doi: 10.1007/BF00970068.

[26]

V. A. Yakubovich, A. L. Fradkov, D. J. Hill and A. V. Proskurnikov, Dissipativity of $T$-periodic linear systems,, IEEE Trans. Automat. Control, 52 (2007), 1039.

show all references

References:
[1]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory,, Springer-Verlag, (1982). doi: 10.1007/978-1-4615-6927-5.

[2]

R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969).

[3]

R. Fabbri, R. Johnson and C. Núñez, Rotation number for non-autonomous linear Hamiltonian systems I: Basic properties,, Z. Angew. Math. Phys., 54 (2003), 484. doi: 10.1007/s00033-003-1068-1.

[4]

R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non-autonomous control processes,, Discrete Contin. Dynam. Systems, 9 (2003), 677. doi: 10.3934/dcds.2003.9.677.

[5]

R. Fabbri, R. Johnson, S. Novo and C. Núñez, On linear-quadratic dissipative control processes with time-varying coefficients,, Discrete Contin. Dynam. Systems, 33 (2013), 193. doi: 10.3934/dcds.2013.33.193.

[6]

R. Johnson and M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems,, Mem. Amer. Math. Soc., 136 (1998). doi: 10.1090/memo/0646.

[7]

R. Johnson, S. Novo, C. Núñez and R. Obaya, Uniform weak disconjugacy and principal solutions for linear Hamiltonian systems,, in: Recent Advances in Delay Differential and Difference Equations, 94 (2014), 131. doi: 10.1007/978-3-319-08251-6.

[8]

R. Johnson, S. Novo, C. Núñez and R. Obaya, Nonautonomous linear-quadratic dissipative control processes without uniform null controllability,, J. Dynam. Differential Equations, (2015), 1. doi: 10.1007/s10884-015-9495-1.

[9]

R. Johnson and C. Núñez, Remarks on linear-quadratic dissipative control systems,, Discr. Cont. Dyn. Sys. B, 20 (2015), 889. doi: 10.3934/dcdsb.2015.20.889.

[10]

R. Johnson, R. Obaya, S. Novo, C. Núñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control,, Developments in Mathematics 36, 36 (2016). doi: 10.1007/978-3-319-29025-6.

[11]

W. Kratz, A limit theorem for monotone matrix functions,, Linear Algebra Appl., 194 (1993), 205. doi: 10.1016/0024-3795(93)90122-5.

[12]

W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory,, Mathematical Topics 6, 6 (1995).

[13]

W. Kratz, Definiteness of quadratic functionals,, Analysis (Munich), 23 (2003), 163. doi: 10.1524/anly.2003.23.2.163.

[14]

Y. Matsushima, Differentiable Manifolds,, Marcel Dekker, (1972).

[15]

A. S. Mishchenko, V. E. Shatalov and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator,, Springer-Verlag, (1990). doi: 10.1007/978-3-642-61259-6.

[16]

S. Novo, C. Núñez and R. Obaya, Ergodic properties and rotation number for linear Hamiltonian systems,, J. Differential Equations, 148 (1998), 148. doi: 10.1006/jdeq.1998.3469.

[17]

W. T. Reid, Sturmian Theory for Ordinary Differential Equations,, Applied Mathematical Sciences 31, 31 (1980). doi: 10.1007/978-1-4612-6110-0.

[18]

W. T. Reid, Principal solutions of nonoscillatory linear differential systems,, J. Math. Anal. Appl., 9 (1964), 397. doi: 10.1016/0022-247X(64)90026-5.

[19]

P. Šepitka and R. Šimon Hilscher, Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems,, J. Dynam. Differential Equations, 26 (2014), 57. doi: 10.1007/s10884-013-9342-1.

[20]

P. Šepitka and R. Šimon Hilscher, Principal Solutions at Infinity of Given Ranks for Nonoscillatory Linear Hamiltonian Systems,, J. Dynam. Differential Equations, 27 (2015), 137.

[21]

R. Šimon Hilscher, Sturmian theory for linear Hamiltonian systems without controllability,, Math. Nachr., 248 (2011), 831. doi: 10.1002/mana.201000071.

[22]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems,, J. Differential Equations, 27 (1978), 320. doi: 10.1016/0022-0396(78)90057-8.

[23]

M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems,, International J. Difference Equ., 2 (2007), 221.

[24]

V. A. Yakubovich, A linear-quadratic optimization problem and the frequency theorem for periodic systems. I,, Siberian Math. J., 27 (1986), 614. doi: 10.1007/bf00969175.

[25]

V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems. II,, Siberian Math. J., 31 (1990), 1027. doi: 10.1007/BF00970068.

[26]

V. A. Yakubovich, A. L. Fradkov, D. J. Hill and A. V. Proskurnikov, Dissipativity of $T$-periodic linear systems,, IEEE Trans. Automat. Control, 52 (2007), 1039.

[1]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684

[2]

Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-605. doi: 10.3934/dcds.2019024

[3]

Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393

[4]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001

[5]

Victor Ayala, Adriano Da Silva, Luiz A. B. San Martin. Control systems on flag manifolds and their chain control sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2301-2313. doi: 10.3934/dcds.2017101

[6]

María Barbero-Liñán, Miguel C. Muñoz-Lecanda. Strict abnormal extremals in nonholonomic and kinematic control systems. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 1-17. doi: 10.3934/dcdss.2010.3.1

[7]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623

[8]

Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519

[9]

Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361

[10]

Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043

[11]

D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401

[12]

Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147

[13]

Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275

[14]

Magdi S. Mahmoud, Mohammed M. Hussain. Control design of linear systems with saturating actuators: A survey. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 413-435. doi: 10.3934/naco.2012.2.413

[15]

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

[16]

Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889

[17]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

[18]

Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207

[19]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361

[20]

Mark A. Pinsky, Alexandr A. Zevin. Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 243-250. doi: 10.3934/dcds.2005.12.243

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

[Back to Top]