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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 |
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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