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

Null controllable sets and reachable sets for nonautonomous linear control systems

Roberta Fabbri 1, , Sylvia Novo 2, , Carmen Núñez 2, and  Rafael Obaya 3,

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.
Keywords: reachable sets, abnormal systems, proper focal points., null controllable sets, rotation number, linear Hamiltonian systems, Nonautonomous linear control systems.
Mathematics Subject Classification: 37B55, 34H05, 37N35, 34D09, 49J1.
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, Berlin, Heidelberg, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[2]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar

[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), 484-502. doi: 10.1007/s00033-003-1068-1.  Google Scholar

[4]

R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non-autonomous control processes, Discrete Contin. Dynam. Systems, Ser. A, 9 (2003), 677-704. doi: 10.3934/dcds.2003.9.677.  Google Scholar

[5]

R. Fabbri, R. Johnson, S. Novo and C. Núñez, On linear-quadratic dissipative control processes with time-varying coefficients, Discrete Contin. Dynam. Systems, Ser. A , 33 (2013), 193-210. doi: 10.3934/dcds.2013.33.193.  Google Scholar

[6]

R. Johnson and M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems, Mem. Amer. Math. Soc., 136 (1998), viii+48 pp. doi: 10.1090/memo/0646.  Google Scholar

[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, Springer Proceedings in Mathematics & Statistics, Springer International Publishing Switzerland, 94 (2014), 131-159. doi: 10.1007/978-3-319-08251-6.  Google Scholar

[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), 1-29, http://link.springer.com/article/10.1007/s10884-015-9495-1. doi: 10.1007/s10884-015-9495-1.  Google Scholar

[9]

R. Johnson and C. Núñez, Remarks on linear-quadratic dissipative control systems, Discr. Cont. Dyn. Sys. B, 20 (2015), 889-914. doi: 10.3934/dcdsb.2015.20.889.  Google Scholar

[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, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-29025-6.  Google Scholar

[11]

W. Kratz, A limit theorem for monotone matrix functions, Linear Algebra Appl., 194 (1993), 205-222. doi: 10.1016/0024-3795(93)90122-5.  Google Scholar

[12]

W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Mathematical Topics 6, Akademie Verlag, Berlin, 1995.  Google Scholar

[13]

W. Kratz, Definiteness of quadratic functionals, Analysis (Munich), 23 (2003), 163-183. doi: 10.1524/anly.2003.23.2.163.  Google Scholar

[14]

Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York, 1972.  Google Scholar

[15]

A. S. Mishchenko, V. E. Shatalov and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator, Springer-Verlag, Berlin, Heidelberg, New York, 1990. doi: 10.1007/978-3-642-61259-6.  Google Scholar

[16]

S. Novo, C. Núñez and R. Obaya, Ergodic properties and rotation number for linear Hamiltonian systems, J. Differential Equations, 148 (1998), 148-185. doi: 10.1006/jdeq.1998.3469.  Google Scholar

[17]

W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences 31, Springer-Verlag, New York, 1980. doi: 10.1007/978-1-4612-6110-0.  Google Scholar

[18]

W. T. Reid, Principal solutions of nonoscillatory linear differential systems, J. Math. Anal. Appl., 9 (1964), 397-423. doi: 10.1016/0022-247X(64)90026-5.  Google Scholar

[19]

P. Šepitka and R. Šimon Hilscher, Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems, J. Dynam. Differential Equations, 26 (2014), 57-91. doi: 10.1007/s10884-013-9342-1.  Google Scholar

[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), 137-175. Google Scholar

[21]

R. Šimon Hilscher, Sturmian theory for linear Hamiltonian systems without controllability, Math. Nachr., 248 (2011), 831-843. doi: 10.1002/mana.201000071.  Google Scholar

[22]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[23]

M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems, International J. Difference Equ., 2 (2007), 221-244.  Google Scholar

[24]

V. A. Yakubovich, A linear-quadratic optimization problem and the frequency theorem for periodic systems. I, Siberian Math. J., 27 (1986), 614-630. doi: 10.1007/bf00969175.  Google Scholar

[25]

V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems. II, Siberian Math. J., 31 (1990), 1027-1039. doi: 10.1007/BF00970068.  Google Scholar

[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), 1039-1047. Google Scholar

show all references

References:
[1]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[2]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.  Google Scholar

[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), 484-502. doi: 10.1007/s00033-003-1068-1.  Google Scholar

[4]

R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non-autonomous control processes, Discrete Contin. Dynam. Systems, Ser. A, 9 (2003), 677-704. doi: 10.3934/dcds.2003.9.677.  Google Scholar

[5]

R. Fabbri, R. Johnson, S. Novo and C. Núñez, On linear-quadratic dissipative control processes with time-varying coefficients, Discrete Contin. Dynam. Systems, Ser. A , 33 (2013), 193-210. doi: 10.3934/dcds.2013.33.193.  Google Scholar

[6]

R. Johnson and M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems, Mem. Amer. Math. Soc., 136 (1998), viii+48 pp. doi: 10.1090/memo/0646.  Google Scholar

[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, Springer Proceedings in Mathematics & Statistics, Springer International Publishing Switzerland, 94 (2014), 131-159. doi: 10.1007/978-3-319-08251-6.  Google Scholar

[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), 1-29, http://link.springer.com/article/10.1007/s10884-015-9495-1. doi: 10.1007/s10884-015-9495-1.  Google Scholar

[9]

R. Johnson and C. Núñez, Remarks on linear-quadratic dissipative control systems, Discr. Cont. Dyn. Sys. B, 20 (2015), 889-914. doi: 10.3934/dcdsb.2015.20.889.  Google Scholar

[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, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-29025-6.  Google Scholar

[11]

W. Kratz, A limit theorem for monotone matrix functions, Linear Algebra Appl., 194 (1993), 205-222. doi: 10.1016/0024-3795(93)90122-5.  Google Scholar

[12]

W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Mathematical Topics 6, Akademie Verlag, Berlin, 1995.  Google Scholar

[13]

W. Kratz, Definiteness of quadratic functionals, Analysis (Munich), 23 (2003), 163-183. doi: 10.1524/anly.2003.23.2.163.  Google Scholar

[14]

Y. Matsushima, Differentiable Manifolds, Marcel Dekker, New York, 1972.  Google Scholar

[15]

A. S. Mishchenko, V. E. Shatalov and B. Yu. Sternin, Lagrangian Manifolds and the Maslov Operator, Springer-Verlag, Berlin, Heidelberg, New York, 1990. doi: 10.1007/978-3-642-61259-6.  Google Scholar

[16]

S. Novo, C. Núñez and R. Obaya, Ergodic properties and rotation number for linear Hamiltonian systems, J. Differential Equations, 148 (1998), 148-185. doi: 10.1006/jdeq.1998.3469.  Google Scholar

[17]

W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences 31, Springer-Verlag, New York, 1980. doi: 10.1007/978-1-4612-6110-0.  Google Scholar

[18]

W. T. Reid, Principal solutions of nonoscillatory linear differential systems, J. Math. Anal. Appl., 9 (1964), 397-423. doi: 10.1016/0022-247X(64)90026-5.  Google Scholar

[19]

P. Šepitka and R. Šimon Hilscher, Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems, J. Dynam. Differential Equations, 26 (2014), 57-91. doi: 10.1007/s10884-013-9342-1.  Google Scholar

[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), 137-175. Google Scholar

[21]

R. Šimon Hilscher, Sturmian theory for linear Hamiltonian systems without controllability, Math. Nachr., 248 (2011), 831-843. doi: 10.1002/mana.201000071.  Google Scholar

[22]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8.  Google Scholar

[23]

M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems, International J. Difference Equ., 2 (2007), 221-244.  Google Scholar

[24]

V. A. Yakubovich, A linear-quadratic optimization problem and the frequency theorem for periodic systems. I, Siberian Math. J., 27 (1986), 614-630. doi: 10.1007/bf00969175.  Google Scholar

[25]

V. A. Yakubovich, Linear-quadratic optimization problem and the frequency theorem for periodic systems. II, Siberian Math. J., 31 (1990), 1027-1039. doi: 10.1007/BF00970068.  Google Scholar

[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), 1039-1047. Google Scholar

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