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), 805-823. 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), 65-78. 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, John Wiley and Sons, 1967.  Google Scholar

[6]

J. L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. 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), Collection Travaux en cours, Hermann, Paris, 2005. Google Scholar

[8]

L. Pandolfi, Linear control systems: Controllability with constrained control, J. Optim. Theory Appl., 19 (1976), 577-585. 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), 327-345. doi: 10.1137/0318025.  Google Scholar

[10]

E. Sontag, An algebraic approach to bounded controllability of linear systems, Internat. J. Control, 39 (1984), 181-188. doi: 10.1080/00207178408933158.  Google Scholar

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E. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, Vol. 6, Texts in Applied Mathematics, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[12]

E. Zuazua, Switching control, J. Eur. Math. Soc., 13 (2011), 85-117. 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), 805-823. 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), 65-78. 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, John Wiley and Sons, 1967.  Google Scholar

[6]

J. L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. 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), Collection Travaux en cours, Hermann, Paris, 2005. Google Scholar

[8]

L. Pandolfi, Linear control systems: Controllability with constrained control, J. Optim. Theory Appl., 19 (1976), 577-585. 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), 327-345. doi: 10.1137/0318025.  Google Scholar

[10]

E. Sontag, An algebraic approach to bounded controllability of linear systems, Internat. J. Control, 39 (1984), 181-188. doi: 10.1080/00207178408933158.  Google Scholar

[11]

E. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, Vol. 6, Texts in Applied Mathematics, Springer-Verlag, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[12]

E. Zuazua, Switching control, J. Eur. Math. Soc., 13 (2011), 85-117. doi: 10.4171/JEMS/245.  Google Scholar

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