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A class of singular first order differential equations with applications in reaction-diffusion
On linear-quadratic dissipative control processes with time-varying coefficients
1. | Dipartimento di Sistemi e Informatica, Università di Firenze, Via di Santa Marta 3, 50139 Firenze, Italy |
2. | Dipartimento di Sistemi e Informatica, Università di Firenze, Facolta' di Ingegneria, Via di Santa Marta 3, 50139 Firenze, Italy |
3. | Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain, Spain |
References:
[1] |
F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser, Basel, 2000. |
[2] |
W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York, 629 (1978). |
[3] |
R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non autonomous control processes, Discrete Contin. Dyn. Syst., Ser. A, 9 (2003), 677-704.
doi: 10.3934/dcds.2003.9.677. |
[4] |
R. Fabbri, R. Johnson, S. Novo and C. Núñez, Some remarks concerning weakly disconjugate linear Hamiltonian systems, J. Math. Anal. Appl., 380 (2011), 853-864.
doi: 10.1016/j.jmaa.2010.11.036. |
[5] |
D. J. Hill, Dissipative nonlinear systems: basic properies and stability analysis, Proc. 31st IEEE Conference on Decision and Control, Vol., 4 (1992), 3259-3264. |
[6] |
D. J. Hill and P. J. Moylan, Dissipative dynamical systems: Basic input-output and state properties, J. Franklin Inst., 309 (1980), 327-357.
doi: 10.1016/0016-0032(80)90026-5. |
[7] |
R. Johnson and M. Nerurkar, "Controllability, Stabilization and the Regulator Problem for Random Differential Systems," Mem. Amer. Math. Soc., 646 (1998). |
[8] |
R. Johnson, S. Novo and R. Obaya, Ergodic properties and Weyl $M$-functions for linear Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 130A (2000), 1045-1079.
doi: 10.1017/S0308210500000573. |
[9] |
E. Lee and B. Markus, "Foundation of Optimal Control Theory," John Wiley & Sons, New York, 1967. |
[10] |
I. G. Polushin, Stability results for quasidissipative systems, Proc. 3rd Eur. Control Conference ECC'95, (1995), 681-686. |
[11] |
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. |
[12] |
A. V. Savkin and I. R. Petersen, Structured dissipativeness and absolute stability of nonlinear systems, Internat. J. Control, 62 (1995), 443-460.
doi: 10.1080/00207179508921550. |
[13] |
H. L. Trentelman and J. C. Willems, "Storage Functions for Dissipative Linear Systems Are Quadratic State Functions," Proc. 36th IEEE Conf. Decision and Control, (1997), 42-49. |
[14] |
J. C. Willems, Dissipative dynamical systems. Part I: General theory. Part II: Linear systems with quadratic supply rates, Arch. Rational Mech. Anal., 45 (1972), 321-393.
doi: 10.1007/BF00276493. |
[15] |
V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. I, Siberian Math. J., 27 (1986), 614-630.
doi: 10.1007/BF00969175. |
[16] |
V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. II, Siberian Math. J., 31 (1990), 1027-1039.
doi: 10.1007/BF00970068. |
[17] |
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.
doi: 10.1109/TAC.2007.899013. |
show all references
References:
[1] |
F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser, Basel, 2000. |
[2] |
W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York, 629 (1978). |
[3] |
R. Fabbri, R. Johnson and C. Núñez, On the Yakubovich Frequency Theorem for linear non autonomous control processes, Discrete Contin. Dyn. Syst., Ser. A, 9 (2003), 677-704.
doi: 10.3934/dcds.2003.9.677. |
[4] |
R. Fabbri, R. Johnson, S. Novo and C. Núñez, Some remarks concerning weakly disconjugate linear Hamiltonian systems, J. Math. Anal. Appl., 380 (2011), 853-864.
doi: 10.1016/j.jmaa.2010.11.036. |
[5] |
D. J. Hill, Dissipative nonlinear systems: basic properies and stability analysis, Proc. 31st IEEE Conference on Decision and Control, Vol., 4 (1992), 3259-3264. |
[6] |
D. J. Hill and P. J. Moylan, Dissipative dynamical systems: Basic input-output and state properties, J. Franklin Inst., 309 (1980), 327-357.
doi: 10.1016/0016-0032(80)90026-5. |
[7] |
R. Johnson and M. Nerurkar, "Controllability, Stabilization and the Regulator Problem for Random Differential Systems," Mem. Amer. Math. Soc., 646 (1998). |
[8] |
R. Johnson, S. Novo and R. Obaya, Ergodic properties and Weyl $M$-functions for linear Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 130A (2000), 1045-1079.
doi: 10.1017/S0308210500000573. |
[9] |
E. Lee and B. Markus, "Foundation of Optimal Control Theory," John Wiley & Sons, New York, 1967. |
[10] |
I. G. Polushin, Stability results for quasidissipative systems, Proc. 3rd Eur. Control Conference ECC'95, (1995), 681-686. |
[11] |
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. |
[12] |
A. V. Savkin and I. R. Petersen, Structured dissipativeness and absolute stability of nonlinear systems, Internat. J. Control, 62 (1995), 443-460.
doi: 10.1080/00207179508921550. |
[13] |
H. L. Trentelman and J. C. Willems, "Storage Functions for Dissipative Linear Systems Are Quadratic State Functions," Proc. 36th IEEE Conf. Decision and Control, (1997), 42-49. |
[14] |
J. C. Willems, Dissipative dynamical systems. Part I: General theory. Part II: Linear systems with quadratic supply rates, Arch. Rational Mech. Anal., 45 (1972), 321-393.
doi: 10.1007/BF00276493. |
[15] |
V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. I, Siberian Math. J., 27 (1986), 614-630.
doi: 10.1007/BF00969175. |
[16] |
V. A. Yakubovich, A linear-quadratic problem of optimization and the frequency theorem for periodic systems. II, Siberian Math. J., 31 (1990), 1027-1039.
doi: 10.1007/BF00970068. |
[17] |
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.
doi: 10.1109/TAC.2007.899013. |
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