June  2015, 4(2): 159-175. doi: 10.3934/eect.2015.4.159

Constrained controllability for lumped linear systems

1. 

Mohammed V University, Ecole Normale Supérieure de Rabat, BP 5118, Rabat, Morocco

Received  May 2014 Revised  September 2014 Published  May 2015

We consider linear lumped control systems of the form $y'(t)=Ay(t)+Bu(t)$ where $A \in \mathbb{R}^{m\times m}$, $B \in \mathbb{R}^{m\times p}$. Taking into account eventual control constraint (such as saturation), we study the problem of controllability by using a general variational approach. The results are applied to the following saturation constraints on the control $u(t)=(u_{1}(t), ..., u_{p}(t))$: (i) the quadratic one specified by $\underset{j=1}{\overset{p}\sum}\left|u_{j}(t)\right|^{2} \leq 1$ for all $0\leq t\leq T$ and (ii) the polyhedral one characterized by $\underset{1 \leq j \leq p}{\max}\left|u_{j}(t)\right| \leq 1$ for all $0\leq t\leq T$.
Citation: Larbi Berrahmoune. Constrained controllability for lumped linear systems. Evolution Equations & Control Theory, 2015, 4 (2) : 159-175. doi: 10.3934/eect.2015.4.159
References:
[1]

L. Berrahmoune, A variational approach to constrained controllability for distributed systems,, J. Math. Anal. Appl., 416 (2014), 805.  doi: 10.1016/j.jmaa.2014.03.004.  Google Scholar

[2]

G. Garcia, A. H. Glattfelder and S. Tarbourierch, Advanced Strategies in Control Systems with Input and Output Constraints,, Springer-Verlag, (2007).  doi: 10.1007/978-3-540-37010-9.  Google Scholar

[3]

K. M. Griordaris and V. Kapila, Actuator Saturation Control,, Marcel Dekker, (2002).   Google Scholar

[4]

T. Hu, Z. Lin and L. Qiu, An explicit description of null controllable regions of linear systems with saturating actuators,, Systems Contr. Lett., 47 (2002), 65.  doi: 10.1016/S0167-6911(02)00176-7.  Google Scholar

[5]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory,, SIAM Series in Applied Mathematics, (1967).   Google Scholar

[6]

J. L. Lions, Exact controllability, stabilizability and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar

[7]

S. Micu and E. Zuazua, An introduction to the controllability of partial differential equations,, in Quelques Questions de Théorie du Contrôle (ed. T. Sari), (2005).   Google Scholar

[8]

L. Pandolfi, Linear control systems: Controllability with constrained control,, J. Optim. Theory Appl., 19 (1976), 577.  doi: 10.1007/BF00934656.  Google Scholar

[9]

W. E. Schmitendorf and B. R. Barmish, Null controllability of linear systems with constrained control,, Siam J. Control Optim., 18 (1980), 327.  doi: 10.1137/0318025.  Google Scholar

[10]

E. Sontag, An algebraic approach to bounded controllability of linear systems,, Internat. J. Control, 39 (1984), 181.  doi: 10.1080/00207178408933158.  Google Scholar

[11]

E. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems,, Vol. 6, (1998).  doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[12]

E. Zuazua, Switching control,, J. Eur. Math. Soc., 13 (2011), 85.  doi: 10.4171/JEMS/245.  Google Scholar

show all references

References:
[1]

L. Berrahmoune, A variational approach to constrained controllability for distributed systems,, J. Math. Anal. Appl., 416 (2014), 805.  doi: 10.1016/j.jmaa.2014.03.004.  Google Scholar

[2]

G. Garcia, A. H. Glattfelder and S. Tarbourierch, Advanced Strategies in Control Systems with Input and Output Constraints,, Springer-Verlag, (2007).  doi: 10.1007/978-3-540-37010-9.  Google Scholar

[3]

K. M. Griordaris and V. Kapila, Actuator Saturation Control,, Marcel Dekker, (2002).   Google Scholar

[4]

T. Hu, Z. Lin and L. Qiu, An explicit description of null controllable regions of linear systems with saturating actuators,, Systems Contr. Lett., 47 (2002), 65.  doi: 10.1016/S0167-6911(02)00176-7.  Google Scholar

[5]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory,, SIAM Series in Applied Mathematics, (1967).   Google Scholar

[6]

J. L. Lions, Exact controllability, stabilizability and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  doi: 10.1137/1030001.  Google Scholar

[7]

S. Micu and E. Zuazua, An introduction to the controllability of partial differential equations,, in Quelques Questions de Théorie du Contrôle (ed. T. Sari), (2005).   Google Scholar

[8]

L. Pandolfi, Linear control systems: Controllability with constrained control,, J. Optim. Theory Appl., 19 (1976), 577.  doi: 10.1007/BF00934656.  Google Scholar

[9]

W. E. Schmitendorf and B. R. Barmish, Null controllability of linear systems with constrained control,, Siam J. Control Optim., 18 (1980), 327.  doi: 10.1137/0318025.  Google Scholar

[10]

E. Sontag, An algebraic approach to bounded controllability of linear systems,, Internat. J. Control, 39 (1984), 181.  doi: 10.1080/00207178408933158.  Google Scholar

[11]

E. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems,, Vol. 6, (1998).  doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[12]

E. Zuazua, Switching control,, J. Eur. Math. Soc., 13 (2011), 85.  doi: 10.4171/JEMS/245.  Google Scholar

[1]

Jianlin Jiang, Shun Zhang, Su Zhang, Jie Wen. A variational inequality approach for constrained multifacility Weber problem under gauge. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1085-1104. doi: 10.3934/jimo.2017091

[2]

Venkatesan Govindaraj, Raju K. George. Controllability of fractional dynamical systems: A functional analytic approach. Mathematical Control & Related Fields, 2017, 7 (4) : 537-562. doi: 10.3934/mcrf.2017020

[3]

Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164

[4]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105

[5]

Monika Muszkieta. A variational approach to edge detection. Inverse Problems & Imaging, 2016, 10 (2) : 499-517. doi: 10.3934/ipi.2016009

[6]

Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004

[7]

Weigao Ge, Li Zhang. Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4925-4943. doi: 10.3934/dcds.2016013

[8]

Shige Peng, Mingyu Xu. Constrained BSDEs, viscosity solutions of variational inequalities and their applications. Mathematical Control & Related Fields, 2013, 3 (2) : 233-244. doi: 10.3934/mcrf.2013.3.233

[9]

Christiane Pöschl, Jan Modersitzki, Otmar Scherzer. A variational setting for volume constrained image registration. Inverse Problems & Imaging, 2010, 4 (3) : 505-522. doi: 10.3934/ipi.2010.4.505

[10]

John E. Lagnese. Controllability of systems of interconnected membranes. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 17-33. doi: 10.3934/dcds.1995.1.17

[11]

D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure & Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163

[12]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[13]

Firdaus E. Udwadia, Thanapat Wanichanon. On general nonlinear constrained mechanical systems. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 425-443. doi: 10.3934/naco.2013.3.425

[14]

Andrei Halanay, Luciano Pandolfi. Lack of controllability of thermal systems with memory. Evolution Equations & Control Theory, 2014, 3 (3) : 485-497. doi: 10.3934/eect.2014.3.485

[15]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183

[16]

Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012

[17]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013

[18]

Sergio Amat, Pablo Pedregal. On a variational approach for the analysis and numerical simulation of ODEs. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1275-1291. doi: 10.3934/dcds.2013.33.1275

[19]

Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233

[20]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]