American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result
  • MCRF Home
  • This Issue
  • Next Article
    Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition
March  2015, 5(1): 55-71. doi: 10.3934/mcrf.2015.5.55

Zubov's equation for state-constrained perturbed nonlinear systems

Lars Grüne 1, and  Hasnaa Zidani 2,

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.
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]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[2]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405
[3]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021007
[4]

Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed bvp for the variable-viscosity compressible stokes pdes. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021009
[5]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055
[6]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002
[7]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101
[8]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[9]

Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018
[10]

Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65
[11]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[12]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[13]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[14]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398
[15]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113
[16]

Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021004
[17]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162
[18]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103
[19]

Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031
[20]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences