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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. Google Scholar |
[2] |
A. Altarovici, O. Bokanowski and H. Zidani, A general Hamilton-Jacobi framework for nonlinear state-constrained control problems,, ESAIM: Control, 19 (2013), 337.
doi: 10.1051/cocv/2012011. |
[3] |
B. Aulbach, Asymptotic stability regions via extensions of Zubov's method. I and II,, Nonlinear Anal., 7 (1983), 1431.
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, (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.
doi: 10.1137/090762075. |
[6] |
R. W. Brockett, Asymptotic stability and feedback stabilization,, in Differential Geometric Control Theory (eds. R. W. Brockett, (1983), 181.
|
[7] |
F. Camilli, A. Cesaroni, L. Grüne and F. Wirth, Stabilization of controlled diffusions and Zubov's method,, Stoch. Dyn., 6 (2006), 373.
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.
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, (2000), 277.
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.
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.
doi: 10.1137/06065129X. |
[12] |
F. Camilli and P. Loreti, A Zubov method for stochastic differential equations,, NoDEA, 13 (2006), 205.
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.
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.
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.
doi: 10.1007/s002110050241. |
[17] |
L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,, Lecture Notes in Mathematics, (1783).
doi: 10.1007/b83677. |
[18] |
L. Grüne and O. S. Serea, Differential games and Zubov's method,, SIAM J. Control Optim., 49 (2011), 2349.
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.
|
[20] |
H. M. Soner, Optimal control problems with state-space constraint I,, SIAM J. Cont. Optim., 24 (1986), 552.
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.
|
[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.
|
[23] |
V. I. Zubov, Methods of A.M. Lyapunov and Their Application,, P. Noordhoff, (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. Google Scholar |
[2] |
A. Altarovici, O. Bokanowski and H. Zidani, A general Hamilton-Jacobi framework for nonlinear state-constrained control problems,, ESAIM: Control, 19 (2013), 337.
doi: 10.1051/cocv/2012011. |
[3] |
B. Aulbach, Asymptotic stability regions via extensions of Zubov's method. I and II,, Nonlinear Anal., 7 (1983), 1431.
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, (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.
doi: 10.1137/090762075. |
[6] |
R. W. Brockett, Asymptotic stability and feedback stabilization,, in Differential Geometric Control Theory (eds. R. W. Brockett, (1983), 181.
|
[7] |
F. Camilli, A. Cesaroni, L. Grüne and F. Wirth, Stabilization of controlled diffusions and Zubov's method,, Stoch. Dyn., 6 (2006), 373.
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.
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, (2000), 277.
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.
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.
doi: 10.1137/06065129X. |
[12] |
F. Camilli and P. Loreti, A Zubov method for stochastic differential equations,, NoDEA, 13 (2006), 205.
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.
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.
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.
doi: 10.1007/s002110050241. |
[17] |
L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,, Lecture Notes in Mathematics, (1783).
doi: 10.1007/b83677. |
[18] |
L. Grüne and O. S. Serea, Differential games and Zubov's method,, SIAM J. Control Optim., 49 (2011), 2349.
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.
|
[20] |
H. M. Soner, Optimal control problems with state-space constraint I,, SIAM J. Cont. Optim., 24 (1986), 552.
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.
|
[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.
|
[23] |
V. I. Zubov, Methods of A.M. Lyapunov and Their Application,, P. Noordhoff, (1964).
|
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