American Institute of Mathematical Sciences

• Previous Article
A generalization of the moment problem to a complex measure space and an approximation technique using backward moments
• DCDS Home
• This Issue
• Next Article
Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations
April  2011, 30(1): 169-186. doi: 10.3934/dcds.2011.30.169

Upper and lower estimates for invariance entropy

 1 Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Received  November 2009 Revised  November 2010 Published  February 2011

Invariance entropy for continuous-time control systems measures how often open-loop control functions have to be updated in order to render a subset of the state space invariant. In the present paper, we derive upper and lower bounds for the invariance entropy of control systems on smooth manifolds, using differential-geometric tools. As an example, we compute these bounds explicitly for projected bilinear control systems on the unit sphere. Moreover, we derive a formula for the invariance entropy of a control set for one-dimensional control-affine systems with a single control vector field.
Citation: Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 169-186. doi: 10.3934/dcds.2011.30.169
References:
 [1] V. A. Boichenko and G. A. Leonov, The direct Lyapunov method in estimates for topological entropy,, J. Math. Sci., 91 (1998), 3370.   Google Scholar [2] V. A. Boichenko, G. A. Leonov and V. Reitmann, "Dimension Theory for Ordinary Differential Equations,", Teubner Texts in Mathematics, (2005).   Google Scholar [3] F. Colonius and C. Kawan, Invariance entropy for control systems,, SIAM J. Control Optim., 48 (2009), 1701.  doi: 10.1137/080713902.  Google Scholar [4] F. Colonius and W. Kliemann, "The Dynamics of Control,", Birkhäuser-Verlag, (2000).   Google Scholar [5] S. Gallot, D. Hulin and J. Lafontaine, "Riemannian Geometry,", Springer-Verlag, (1987).   Google Scholar [6] K. A. Grasse and H. J. Sussmann, Global controllability by nice controls,, in, 133 (1990), 33.   Google Scholar [7] F. Ito, An estimate from above for the entropy and the topological entropy of a $C^1$-diffeomorphism,, Proc. Japan Acad., 46 (1970), 226.  doi: 10.3792/pja/1195520395.  Google Scholar [8] C. Kawan, "Invariance Entropy for Control Systems,", Ph.D thesis, (2009).   Google Scholar [9] C. Kawan, Invariance entropy of control sets,, to appear in SIAM J. Control Optim., ().   Google Scholar [10] G. N. Nair, R. J. Evans, I. M. Y. Mareels and W. Moran, Topological feedback entropy and nonlinear stabilization,, IEEE Trans. Automat. Control, 49 (2004), 1585.  doi: 10.1109/TAC.2004.834105.  Google Scholar [11] A. Noack, "Dimension and Entropy Estimates and Stability Investigations for Nonlinear Systems on Manifolds (Dimensions- und Entropieabschätzungen sowie Stabilitätsuntersuchungen für nichtlineare Systeme auf Mannigfaltigkeiten),", Ph.D thesis (German), (1998).   Google Scholar

show all references

References:
 [1] V. A. Boichenko and G. A. Leonov, The direct Lyapunov method in estimates for topological entropy,, J. Math. Sci., 91 (1998), 3370.   Google Scholar [2] V. A. Boichenko, G. A. Leonov and V. Reitmann, "Dimension Theory for Ordinary Differential Equations,", Teubner Texts in Mathematics, (2005).   Google Scholar [3] F. Colonius and C. Kawan, Invariance entropy for control systems,, SIAM J. Control Optim., 48 (2009), 1701.  doi: 10.1137/080713902.  Google Scholar [4] F. Colonius and W. Kliemann, "The Dynamics of Control,", Birkhäuser-Verlag, (2000).   Google Scholar [5] S. Gallot, D. Hulin and J. Lafontaine, "Riemannian Geometry,", Springer-Verlag, (1987).   Google Scholar [6] K. A. Grasse and H. J. Sussmann, Global controllability by nice controls,, in, 133 (1990), 33.   Google Scholar [7] F. Ito, An estimate from above for the entropy and the topological entropy of a $C^1$-diffeomorphism,, Proc. Japan Acad., 46 (1970), 226.  doi: 10.3792/pja/1195520395.  Google Scholar [8] C. Kawan, "Invariance Entropy for Control Systems,", Ph.D thesis, (2009).   Google Scholar [9] C. Kawan, Invariance entropy of control sets,, to appear in SIAM J. Control Optim., ().   Google Scholar [10] G. N. Nair, R. J. Evans, I. M. Y. Mareels and W. Moran, Topological feedback entropy and nonlinear stabilization,, IEEE Trans. Automat. Control, 49 (2004), 1585.  doi: 10.1109/TAC.2004.834105.  Google Scholar [11] A. Noack, "Dimension and Entropy Estimates and Stability Investigations for Nonlinear Systems on Manifolds (Dimensions- und Entropieabschätzungen sowie Stabilitätsuntersuchungen für nichtlineare Systeme auf Mannigfaltigkeiten),", Ph.D thesis (German), (1998).   Google Scholar
 [1] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 [2] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [3] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [4] Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011 [5] Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 [6] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [7] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [8] Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379 [9] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [10] Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254 [11] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [12] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [13] Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 [14] Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012 [15] Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347 [16] M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014 [17] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [18] 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 [19] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [20] Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

2019 Impact Factor: 1.338