American Institute of Mathematical Sciences

Advanced
December  2006, 5(4): 861-885. doi: 10.3934/cpaa.2006.5.861

Second order conditions for the controllability of nonlinear systems with drift

Antonio Marigonda 1,

1. 

Dipartimento di Matematica Pura e Applicata, Università di Padova, via Belzoni 7, 35131 Padova, Italy

Received  November 2005 Revised  May 2006 Published  September 2006

In this paper we study controllability of control systems in $\mathbb R^n$ of the form $\dot x=f(x)+\sum_{i=1}^m$ $u_ig_i(x)$ with $u\in\mathcal U$ compact convex subset of $\mathbb R^n$ with a rather general target. The symmetric (driftless) case, i.e. $f=0$, is a very classical topic, and in this case the results on controllability and Hölder continuity of the minimal time function $T$ are related to certain properties of the Lie algebra generated by the $g_i$'s. Here, we want to extend some results on controllability and Hölder continuity of $T$ to some cases where $f\ne 0$.
Keywords: Control theory, Petrov condition., small time controllability.
Mathematics Subject Classification: Primary: 93B2.
Citation: Antonio Marigonda. Second order conditions for the controllability of nonlinear systems with drift. Communications on Pure and Applied Analysis, 2006, 5 (4) : 861-885. doi: 10.3934/cpaa.2006.5.861
[1]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control and Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
[2]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[3]

Laurenz Göllmann, Helmut Maurer. Theory and applications of optimal control problems with multiple time-delays. Journal of Industrial and Management Optimization, 2014, 10 (2) : 413-441. doi: 10.3934/jimo.2014.10.413
[4]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006
[5]

Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control and Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125
[6]

E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
[7]

Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074
[8]

Sergei Avdonin, Jeff Park, Luz de Teresa. The Kalman condition for the boundary controllability of coupled 1-d wave equations. Evolution Equations and Control Theory, 2020, 9 (1) : 255-273. doi: 10.3934/eect.2020005
[9]

Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control and Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004
[10]

Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415
[11]

K. Renee Fister, Jennifer Hughes Donnelly. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences & Engineering, 2005, 2 (3) : 499-510. doi: 10.3934/mbe.2005.2.499
[12]

Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1007-1034. doi: 10.3934/dcds.2005.13.1007
[13]

Shihu Li, Wei Liu, Yingchao Xie. Small time asymptotics for SPDEs with locally monotone coefficients. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4801-4822. doi: 10.3934/dcdsb.2020127
[14]

Mickael Chekroun, Michael Ghil, Jean Roux, Ferenc Varadi. Averaging of time - periodic systems without a small parameter. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 753-782. doi: 10.3934/dcds.2006.14.753
[15]

Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control and Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002
[16]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[17]

Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations and Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039
[18]

Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks and Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014
[19]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143
[20]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control and Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy