# 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] 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 [2] Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001 [9] Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021001 [10] 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 [11] 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 [12] Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172 [13] Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021011 [14] 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 [15] 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 [16] 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 [17] 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 [18] Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 [19] 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 [20] Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399

2019 Impact Factor: 1.338