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Zubov's equation for state-constrained perturbed nonlinear systems
1. | Mathematisches Institute, Universität Bayreuth, 95440 Bayreuth |
2. | Mathematics Department - UMA, ENSTA ParisTech, 91762 Palaiseau, France |
References:
[1] |
M. Abu Hassan and C. Storey, Numerical determination of domains of attraction for electrical power systems using the method of Zubov, Int. J. Control, 34 (1981), 371-381. |
[2] |
A. Altarovici, O. Bokanowski and H. Zidani, A general Hamilton-Jacobi framework for nonlinear state-constrained control problems, ESAIM: Control, Optimisation, and Calculus of Variations., 19 (2013), 337-357.
doi: 10.1051/cocv/2012011. |
[3] |
B. Aulbach, Asymptotic stability regions via extensions of Zubov's method. I and II, Nonlinear Anal., Theory Methods Appl., 7 (1983), 1431-1440, 1441-1454.
doi: 10.1016/0362-546X(83)90010-X. |
[4] |
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997.
doi: 10.1007/978-0-8176-4755-1. |
[5] |
O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control Optim., 48 (2010), 4292-4316.
doi: 10.1137/090762075. |
[6] |
R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. R. W. Brockett, R. S. Millman and H. J. Sussmann), Birkhäuser, Boston, 1983, 181-191. |
[7] |
F. Camilli, A. Cesaroni, L. Grüne and F. Wirth, Stabilization of controlled diffusions and Zubov's method, Stoch. Dyn., 6 (2006), 373-393.
doi: 10.1142/S0219493706001803. |
[8] |
F. Camilli and L. Grüne, Characterizing attraction probabilities via the stochastic Zubov equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 457-468.
doi: 10.3934/dcdsb.2003.3.457. |
[9] |
F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in Nonlinear Control in the Year 2000, Volume 1 (eds. A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek), Lecture Notes in Control and Information Sciences, 258, Springer-Verlag, London, 2000, 277-289.
doi: 10.1007/BFb0110220. |
[10] |
F. Camilli, L. Grüne and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim., 40 (2001), 496-515.
doi: 10.1137/S036301299936316X. |
[11] |
F. Camilli, L. Grüne and F. Wirth, Control Lyapunov functions and Zubov's method, SIAM J. Control Optim., 47 (2008), 301-326.
doi: 10.1137/06065129X. |
[12] |
F. Camilli and P. Loreti, A Zubov method for stochastic differential equations, NoDEA, 13 (2006), 205-222.
doi: 10.1007/s00030-005-0036-1. |
[13] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. |
[14] |
N. Forcadel, Z. Rao and H. Zidani, State-constrained optimal control problems of impulsive differential equations, Applied Mathematics & Optimization, 68 (2013), 1-19.
doi: 10.1007/s00245-013-9193-5. |
[15] |
R. Genesio, M. Tartaglia and A. Vicino, On the estimation of asymptotic stability regions: State of the art and new proposals, IEEE Trans. Autom. Control, 30 (1985), 747-755.
doi: 10.1109/TAC.1985.1104057. |
[16] |
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337.
doi: 10.1007/s002110050241. |
[17] |
L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Lecture Notes in Mathematics, Vol. 1783, Springer, Heidelberg, 2002.
doi: 10.1007/b83677. |
[18] |
L. Grüne and O. S. Serea, Differential games and Zubov's method, SIAM J. Control Optim., 49 (2011), 2349-2377.
doi: 10.1137/100787829. |
[19] |
N. E. Kirin, R. A. Nelepin and V. N. Bajdaev, Construction of the attraction region by Zubov's method, Differ. Equations, 17 (1981), 1347-1361. |
[20] |
H. M. Soner, Optimal control problems with state-space constraint I, SIAM J. Cont. Optim., 24 (1986), 552-561.
doi: 10.1137/0324032. |
[21] |
P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I. Equations of unbounded and degenerate control problems without uniqueness, Adv. Differential Equations, 4 (1999), 275-296. |
[22] |
P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints, Differential Integral Equations, 12 (1999), 275-293. |
[23] |
V. I. Zubov, Methods of A.M. Lyapunov and Their Application, P. Noordhoff, Groningen, 1964. |
show all references
References:
[1] |
M. Abu Hassan and C. Storey, Numerical determination of domains of attraction for electrical power systems using the method of Zubov, Int. J. Control, 34 (1981), 371-381. |
[2] |
A. Altarovici, O. Bokanowski and H. Zidani, A general Hamilton-Jacobi framework for nonlinear state-constrained control problems, ESAIM: Control, Optimisation, and Calculus of Variations., 19 (2013), 337-357.
doi: 10.1051/cocv/2012011. |
[3] |
B. Aulbach, Asymptotic stability regions via extensions of Zubov's method. I and II, Nonlinear Anal., Theory Methods Appl., 7 (1983), 1431-1440, 1441-1454.
doi: 10.1016/0362-546X(83)90010-X. |
[4] |
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997.
doi: 10.1007/978-0-8176-4755-1. |
[5] |
O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM J. Control Optim., 48 (2010), 4292-4316.
doi: 10.1137/090762075. |
[6] |
R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (eds. R. W. Brockett, R. S. Millman and H. J. Sussmann), Birkhäuser, Boston, 1983, 181-191. |
[7] |
F. Camilli, A. Cesaroni, L. Grüne and F. Wirth, Stabilization of controlled diffusions and Zubov's method, Stoch. Dyn., 6 (2006), 373-393.
doi: 10.1142/S0219493706001803. |
[8] |
F. Camilli and L. Grüne, Characterizing attraction probabilities via the stochastic Zubov equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 457-468.
doi: 10.3934/dcdsb.2003.3.457. |
[9] |
F. Camilli, L. Grüne and F. Wirth, A regularization of Zubov's equation for robust domains of attraction, in Nonlinear Control in the Year 2000, Volume 1 (eds. A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek), Lecture Notes in Control and Information Sciences, 258, Springer-Verlag, London, 2000, 277-289.
doi: 10.1007/BFb0110220. |
[10] |
F. Camilli, L. Grüne and F. Wirth, A generalization of Zubov's method to perturbed systems, SIAM J. Control Optim., 40 (2001), 496-515.
doi: 10.1137/S036301299936316X. |
[11] |
F. Camilli, L. Grüne and F. Wirth, Control Lyapunov functions and Zubov's method, SIAM J. Control Optim., 47 (2008), 301-326.
doi: 10.1137/06065129X. |
[12] |
F. Camilli and P. Loreti, A Zubov method for stochastic differential equations, NoDEA, 13 (2006), 205-222.
doi: 10.1007/s00030-005-0036-1. |
[13] |
F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. |
[14] |
N. Forcadel, Z. Rao and H. Zidani, State-constrained optimal control problems of impulsive differential equations, Applied Mathematics & Optimization, 68 (2013), 1-19.
doi: 10.1007/s00245-013-9193-5. |
[15] |
R. Genesio, M. Tartaglia and A. Vicino, On the estimation of asymptotic stability regions: State of the art and new proposals, IEEE Trans. Autom. Control, 30 (1985), 747-755.
doi: 10.1109/TAC.1985.1104057. |
[16] |
L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337.
doi: 10.1007/s002110050241. |
[17] |
L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Lecture Notes in Mathematics, Vol. 1783, Springer, Heidelberg, 2002.
doi: 10.1007/b83677. |
[18] |
L. Grüne and O. S. Serea, Differential games and Zubov's method, SIAM J. Control Optim., 49 (2011), 2349-2377.
doi: 10.1137/100787829. |
[19] |
N. E. Kirin, R. A. Nelepin and V. N. Bajdaev, Construction of the attraction region by Zubov's method, Differ. Equations, 17 (1981), 1347-1361. |
[20] |
H. M. Soner, Optimal control problems with state-space constraint I, SIAM J. Cont. Optim., 24 (1986), 552-561.
doi: 10.1137/0324032. |
[21] |
P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I. Equations of unbounded and degenerate control problems without uniqueness, Adv. Differential Equations, 4 (1999), 275-296. |
[22] |
P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints, Differential Integral Equations, 12 (1999), 275-293. |
[23] |
V. I. Zubov, Methods of A.M. Lyapunov and Their Application, P. Noordhoff, Groningen, 1964. |
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