Zubov's equation for state-constrained perturbed nonlinear systems

Previous Article
Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition
MCRF Home
This Issue
Next Article
Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result

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

Abstract
Related Papers

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.

Keywords: Zubov's approach, Hamilton-Jacobi equations, viscosity solution., state-constrained nonlinear systems, Domain of attraction.

Mathematics Subject Classification: 93D09, 35F21, 49L2.

Citation: Lars Grüne, Hasnaa Zidani. Zubov's equation for state-constrained perturbed nonlinear systems. Mathematical Control & 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	R. W. Brockett, Asymptotic stability and feedback stabilization,, in Differential Geometric Control Theory* (eds. R. W. Brockett*, (1983), 181. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	F. Camilli and P. Loreti, A Zubov method for stochastic differential equations,, NoDEA, 13 (2006), 205. doi: 10.1007/s00030-005-0036-1. Google Scholar
[13]	F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer, (1998). Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation,, Numer. Math., 75 (1997), 319. doi: 10.1007/s002110050241. Google Scholar
[17]	L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,, Lecture Notes in Mathematics, (1783). doi: 10.1007/b83677. Google Scholar
[18]	L. Grüne and O. S. Serea, Differential games and Zubov's method,, SIAM J. Control Optim., 49 (2011), 2349. doi: 10.1137/100787829. Google Scholar
[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. Google Scholar
[20]	H. M. Soner, Optimal control problems with state-space constraint I,, SIAM J. Cont. Optim., 24 (1986), 552. doi: 10.1137/0324032. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	V. I. Zubov, Methods of A.M. Lyapunov and Their Application,, P. Noordhoff, (1964). Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	R. W. Brockett, Asymptotic stability and feedback stabilization,, in Differential Geometric Control Theory* (eds. R. W. Brockett*, (1983), 181. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	F. Camilli and P. Loreti, A Zubov method for stochastic differential equations,, NoDEA, 13 (2006), 205. doi: 10.1007/s00030-005-0036-1. Google Scholar
[13]	F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Springer, (1998). Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation,, Numer. Math., 75 (1997), 319. doi: 10.1007/s002110050241. Google Scholar
[17]	L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization,, Lecture Notes in Mathematics, (1783). doi: 10.1007/b83677. Google Scholar
[18]	L. Grüne and O. S. Serea, Differential games and Zubov's method,, SIAM J. Control Optim., 49 (2011), 2349. doi: 10.1137/100787829. Google Scholar
[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. Google Scholar
[20]	H. M. Soner, Optimal control problems with state-space constraint I,, SIAM J. Cont. Optim., 24 (1986), 552. doi: 10.1137/0324032. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	V. I. Zubov, Methods of A.M. Lyapunov and Their Application,, P. Noordhoff, (1964). Google Scholar

[1]	Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389
[2]	Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure & Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793
[3]	Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649
[4]	Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic & Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255
[5]	Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[6]	Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331
[7]	Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176
[8]	Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
[9]	Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363
[10]	Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461
[11]	Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917
[12]	Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : ⅰ-ⅲ. doi: 10.3934/dcdss.201805i
[13]	Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[14]	Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385
[15]	Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647
[16]	Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855
[17]	Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167
[18]	Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 447-459. doi: 10.3934/cpaa.2003.2.447
[19]	Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[20]	David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences