Switching control

Zuazua, Enrique. "Switching control." Journal of the European Mathematical Society 013.1 (2011): 85-117. <http://eudml.org/doc/277355>.

@article{Zuazua2011,
abstract = {We analyze the problem of switching controls for control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to the other in a systematic way so that, in each instant of time, only one actuator is active. We first address a finite-dimensional model and show that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. To do this we introduce a new variational principle building a new functional based on the adjoint system whose minimizers yield the switching controls. When the above rank condition fails the same variational strategy applies but the controls obtained this way fail to be of switching form since they may be, for some instants of time, convex combinations of both controllers. We then address the same issue for the 1-d heat equation endowed by two pointwise controls. We show that, due to the time-analyticity of solutions, under suitable conditions on the location of the controllers, switching control strategies exist. We also show that the controls we obtain are optimal in the sense that, for instance, for two scalar valued controls, they are of minimal $L^2(0,T;\mathbb \{R\}^2)$-norm, the space $\mathbb \{R\}^2$ being endowed with the $\ell ^1$-norm. We also discuss some possible extensions to multi-dimensional heat equations which require a premilinary analysis of generic properties of the spectrum that, as far as we know, are not yet well understood.},
author = {Zuazua, Enrique},
journal = {Journal of the European Mathematical Society},
keywords = {switching control; vatiational approach; rank conditions; finite-dimensional system; heat equation; genericity; switching control; variational approach; rank conditions; finite-dimensional system; heat equation; genericity},
language = {eng},
number = {1},
pages = {85-117},
publisher = {European Mathematical Society Publishing House},
title = {Switching control},
url = {http://eudml.org/doc/277355},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Zuazua, Enrique
TI - Switching control
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 1
SP - 85
EP - 117
AB - We analyze the problem of switching controls for control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to the other in a systematic way so that, in each instant of time, only one actuator is active. We first address a finite-dimensional model and show that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. To do this we introduce a new variational principle building a new functional based on the adjoint system whose minimizers yield the switching controls. When the above rank condition fails the same variational strategy applies but the controls obtained this way fail to be of switching form since they may be, for some instants of time, convex combinations of both controllers. We then address the same issue for the 1-d heat equation endowed by two pointwise controls. We show that, due to the time-analyticity of solutions, under suitable conditions on the location of the controllers, switching control strategies exist. We also show that the controls we obtain are optimal in the sense that, for instance, for two scalar valued controls, they are of minimal $L^2(0,T;\mathbb {R}^2)$-norm, the space $\mathbb {R}^2$ being endowed with the $\ell ^1$-norm. We also discuss some possible extensions to multi-dimensional heat equations which require a premilinary analysis of generic properties of the spectrum that, as far as we know, are not yet well understood.
LA - eng
KW - switching control; vatiational approach; rank conditions; finite-dimensional system; heat equation; genericity; switching control; variational approach; rank conditions; finite-dimensional system; heat equation; genericity
UR - http://eudml.org/doc/277355
ER -