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 and 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.

[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.

[3]

K. M. Griordaris and V. Kapila, Actuator Saturation Control, Marcel Dekker, 2002.

[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.

[5]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, SIAM Series in Applied Mathematics, John Wiley and Sons, 1967.

[6]

J. L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.

[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.

[8]

L. Pandolfi, Linear control systems: Controllability with constrained control, J. Optim. Theory Appl., 19 (1976), 577-585. doi: 10.1007/BF00934656.

[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.

[10]

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

[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.

[12]

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

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.

[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.

[3]

K. M. Griordaris and V. Kapila, Actuator Saturation Control, Marcel Dekker, 2002.

[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.

[5]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, SIAM Series in Applied Mathematics, John Wiley and Sons, 1967.

[6]

J. L. Lions, Exact controllability, stabilizability and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.

[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.

[8]

L. Pandolfi, Linear control systems: Controllability with constrained control, J. Optim. Theory Appl., 19 (1976), 577-585. doi: 10.1007/BF00934656.

[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.

[10]

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

[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.

[12]

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

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