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Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations
Stabilizability in optimization problems with unbounded data
1. | Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Scarpa, 16, Roma 00181, Italy |
2. | Dipartimento di Matematica "Tullio Levi-Civita", Università di Padova, Via Trieste, 63, Padova 35121, Italy |
In this paper we extend the notions of sample and Euler stabilizability to a set of a control system to a wide class of systems with unbounded controls, which includes nonlinear control-polynomial systems. In particular, we allow discontinuous stabilizing feedbacks, which are unbounded approaching the target. As a consequence, sampling trajectories may present a chattering behaviour and Euler solutions have in general an impulsive character. We also associate to the control system a cost and provide sufficient conditions, based on the existence of a special Lyapunov function, which allow for the existence of a stabilizing feedback that keeps the cost of all sampling and Euler solutions starting from the same point below the same value, in a uniform way.
References:
[1] |
Z. Artstein,
Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173.
doi: 10.1016/0362-546X(83)90049-4. |
[2] |
A. Bressan and F. Rampazzo,
Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141.
doi: 10.1007/s00205-009-0237-6. |
[3] |
R. W. Brockett,
Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 27 (1983), 181-191.
|
[4] |
P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston, Inc., Boston, MA, 2004. |
[5] |
L. Cesari, Optimization Theory and Applications, Problems with ordinary differential equations. Applications of Mathematics, (New York), 17. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[6] |
F. H. Clarke, Yu. S. Ledyaev, L. Rifford and R. J. Stern,
Feedback stabilization and Lyapunov functions, SIAM J. Control Optim., 39 (2000), 25-48.
doi: 10.1137/S0363012999352297. |
[7] |
F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin,
Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control, 42 (1997), 1394-1407.
doi: 10.1109/9.633828. |
[8] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998. |
[9] |
J.-M. Coron and L. Rosier,
A relation between continuous time-varying and discontinuous feedback stabilization, J. Math. Systems Estim. Control, 4 (1994), 67-84.
|
[10] |
S. N. Dashkovskiĭ, D. V. Efimov and È. D. Sontag,
Input-to-state stability and related properties of systems, Autom. Remote Control, 72 (2011), 1579-1614.
doi: 10.1134/S0005117911080017. |
[11] |
H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York, 1969. |
[12] |
D. Yu. Karamzin, V. A. de Oliveira, F. L. Pereira and G. N. Silva,
On some extension of optimal control theory, Eur. J. Control, 20 (2014), 284-291.
doi: 10.1016/j.ejcon.2014.09.003. |
[13] |
C. M. Kellett and A. R. Teel,
Uniform asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function, Proceedings of the 39th IEEE Conference on Decision and Control, 4 (2000), 3994-3999.
doi: 10.1109/CDC.2000.912339. |
[14] |
N. N. Krasovskiĭ and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988. |
[15] |
A. C. Lai and M. Motta, Stabilizability in optimal control, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 41, 32 pp.
doi: 10.1007/s00030-020-00647-7. |
[16] |
A. C. Lai and M. Motta, Stabilizability in impulsive optimization problems, Proceedings of the 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS, Vienna, IFAC-PapersOnLine, 52 (2019), 352–357. Google Scholar |
[17] |
A. C. Lai, M. Motta and F. Rampazzo,
Minimum restraint functions for unbounded dynamics: General and control-polynomial systems, Pure and Applied Functional Analysis, 1 (2016), 583-612.
|
[18] |
M. Malisoff, L. Rifford and E. Sontag,
Global asymptotic controllability implies input-to-state stabilization, SIAM J. Control Optim., 42 (2004), 2221-2238.
doi: 10.1137/S0363012903422333. |
[19] |
M. Motta and F. Rampazzo,
Asymptotic controllability and optimal control, Journal of Differential Equations, 254 (2013), 2744-2763.
doi: 10.1016/j.jde.2013.01.006. |
[20] |
M. Motta and C. Sartori,
On asymptotic exit-time control problems lacking coercivity, ESAIM Control Optim. Calc. Var., 20 (2014), 957-982.
doi: 10.1051/cocv/2014003. |
[21] |
M. Motta and C. Sartori,
Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dyn. Syst., 21 (2008), 513-535.
doi: 10.3934/dcds.2008.21.513. |
[22] |
F. Rampazzo and C. Sartori,
Hamilton-Jacobi-Bellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077.
doi: 10.1512/iumj.2000.49.1736. |
[23] |
L. Rifford,
Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM J. Control Optim., 39 (2000), 1043-1064.
doi: 10.1137/S0363012999356039. |
[24] |
L. Rifford,
Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM J. Control Optim., 41 (2002), 659-681.
doi: 10.1137/S0363012900375342. |
[25] |
E. P. Ryan,
On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback, SIAM J. Control Optim., 32 (1994), 1597-1604.
doi: 10.1137/S0363012992235432. |
[26] |
E. D. Sontag,
A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Optim., 21 (1983), 462-471.
doi: 10.1137/0321028. |
[27] |
E. Sontag and H. J. Sussmann,
Nonsmooth control-Lyapunov functions, Proceedings of the IEEE Conference on Decision and Control, 3 (1995), 2799-2805.
doi: 10.1109/CDC.1995.478542. |
[28] |
E. Sontag and Y. Wang, Various results concerning set input-to-state stability, Proc. 34th IEEE Conf. Decision and Control, New Orleans, December, (1995), 1330–1335. Google Scholar |
[29] |
R. Vinter, Optimal Control, Birkhäuser, Boston, 2000. |
show all references
References:
[1] |
Z. Artstein,
Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173.
doi: 10.1016/0362-546X(83)90049-4. |
[2] |
A. Bressan and F. Rampazzo,
Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141.
doi: 10.1007/s00205-009-0237-6. |
[3] |
R. W. Brockett,
Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 27 (1983), 181-191.
|
[4] |
P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston, Inc., Boston, MA, 2004. |
[5] |
L. Cesari, Optimization Theory and Applications, Problems with ordinary differential equations. Applications of Mathematics, (New York), 17. Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[6] |
F. H. Clarke, Yu. S. Ledyaev, L. Rifford and R. J. Stern,
Feedback stabilization and Lyapunov functions, SIAM J. Control Optim., 39 (2000), 25-48.
doi: 10.1137/S0363012999352297. |
[7] |
F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin,
Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control, 42 (1997), 1394-1407.
doi: 10.1109/9.633828. |
[8] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998. |
[9] |
J.-M. Coron and L. Rosier,
A relation between continuous time-varying and discontinuous feedback stabilization, J. Math. Systems Estim. Control, 4 (1994), 67-84.
|
[10] |
S. N. Dashkovskiĭ, D. V. Efimov and È. D. Sontag,
Input-to-state stability and related properties of systems, Autom. Remote Control, 72 (2011), 1579-1614.
doi: 10.1134/S0005117911080017. |
[11] |
H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York, 1969. |
[12] |
D. Yu. Karamzin, V. A. de Oliveira, F. L. Pereira and G. N. Silva,
On some extension of optimal control theory, Eur. J. Control, 20 (2014), 284-291.
doi: 10.1016/j.ejcon.2014.09.003. |
[13] |
C. M. Kellett and A. R. Teel,
Uniform asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function, Proceedings of the 39th IEEE Conference on Decision and Control, 4 (2000), 3994-3999.
doi: 10.1109/CDC.2000.912339. |
[14] |
N. N. Krasovskiĭ and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988. |
[15] |
A. C. Lai and M. Motta, Stabilizability in optimal control, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 41, 32 pp.
doi: 10.1007/s00030-020-00647-7. |
[16] |
A. C. Lai and M. Motta, Stabilizability in impulsive optimization problems, Proceedings of the 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS, Vienna, IFAC-PapersOnLine, 52 (2019), 352–357. Google Scholar |
[17] |
A. C. Lai, M. Motta and F. Rampazzo,
Minimum restraint functions for unbounded dynamics: General and control-polynomial systems, Pure and Applied Functional Analysis, 1 (2016), 583-612.
|
[18] |
M. Malisoff, L. Rifford and E. Sontag,
Global asymptotic controllability implies input-to-state stabilization, SIAM J. Control Optim., 42 (2004), 2221-2238.
doi: 10.1137/S0363012903422333. |
[19] |
M. Motta and F. Rampazzo,
Asymptotic controllability and optimal control, Journal of Differential Equations, 254 (2013), 2744-2763.
doi: 10.1016/j.jde.2013.01.006. |
[20] |
M. Motta and C. Sartori,
On asymptotic exit-time control problems lacking coercivity, ESAIM Control Optim. Calc. Var., 20 (2014), 957-982.
doi: 10.1051/cocv/2014003. |
[21] |
M. Motta and C. Sartori,
Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dyn. Syst., 21 (2008), 513-535.
doi: 10.3934/dcds.2008.21.513. |
[22] |
F. Rampazzo and C. Sartori,
Hamilton-Jacobi-Bellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077.
doi: 10.1512/iumj.2000.49.1736. |
[23] |
L. Rifford,
Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM J. Control Optim., 39 (2000), 1043-1064.
doi: 10.1137/S0363012999356039. |
[24] |
L. Rifford,
Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM J. Control Optim., 41 (2002), 659-681.
doi: 10.1137/S0363012900375342. |
[25] |
E. P. Ryan,
On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback, SIAM J. Control Optim., 32 (1994), 1597-1604.
doi: 10.1137/S0363012992235432. |
[26] |
E. D. Sontag,
A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Optim., 21 (1983), 462-471.
doi: 10.1137/0321028. |
[27] |
E. Sontag and H. J. Sussmann,
Nonsmooth control-Lyapunov functions, Proceedings of the IEEE Conference on Decision and Control, 3 (1995), 2799-2805.
doi: 10.1109/CDC.1995.478542. |
[28] |
E. Sontag and Y. Wang, Various results concerning set input-to-state stability, Proc. 34th IEEE Conf. Decision and Control, New Orleans, December, (1995), 1330–1335. Google Scholar |
[29] |
R. Vinter, Optimal Control, Birkhäuser, Boston, 2000. |
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