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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 and 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.

[2]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 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), 484-502. 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, Ser. A, 9 (2003), 677-704. 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, Ser. A , 33 (2013), 193-210. 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), viii+48 pp. 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, Springer Proceedings in Mathematics & Statistics, Springer International Publishing Switzerland, 94 (2014), 131-159. 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), 1-29, http://link.springer.com/article/10.1007/s10884-015-9495-1. 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), 889-914. 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, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-29025-6.

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

R. Šimon Hilscher, Sturmian theory for linear Hamiltonian systems without controllability, Math. Nachr., 248 (2011), 831-843. 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), 320-358. 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), 221-244.

[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.

[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.

[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.

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.

[2]

R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 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), 484-502. 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, Ser. A, 9 (2003), 677-704. 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, Ser. A , 33 (2013), 193-210. 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), viii+48 pp. 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, Springer Proceedings in Mathematics & Statistics, Springer International Publishing Switzerland, 94 (2014), 131-159. 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), 1-29, http://link.springer.com/article/10.1007/s10884-015-9495-1. 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), 889-914. 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, Springer, Switzerland, 2016. doi: 10.1007/978-3-319-29025-6.

[11]

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

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

R. Šimon Hilscher, Sturmian theory for linear Hamiltonian systems without controllability, Math. Nachr., 248 (2011), 831-843. 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), 320-358. 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), 221-244.

[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.

[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.

[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.

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