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March  2015, 5(1): 55-71. doi: 10.3934/mcrf.2015.5.55

## 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

Received  October 2013 Revised  February 2014 Published  January 2015

The paper gives a characterization of the uniform robust domain of attraction for a finite non-linear controlled system subject to perturbations and state constraints. We extend the Zubov approach to characterize this domain by means of the value function of a suitable infinite horizon state-constrained control problem which at the same time is a Lyapunov function for the system. We provide associated Hamilton-Jacobi-Bellman equations and prove existence and uniqueness of the solutions of these generalized Zubov equations.
Citation: Lars Grüne, Hasnaa Zidani. Zubov's equation for state-constrained perturbed nonlinear systems. Mathematical Control and Related Fields, 2015, 5 (1) : 55-71. doi: 10.3934/mcrf.2015.5.55
##### 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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##### 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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