Abstract
In this paper a review of the results on sparse controls for partial differential equations is presented. There are two different approaches to the sparsity study of control problems. One approach consists of taking functions to control the system, putting in the cost functional a convenient term that promotes the sparsity of the optimal control. A second approach deals with controls that are Borel measures and the norm of the measure is involved in the cost functional. The use of measures as controls allows to obtain optimal controls supported on a zero Lebesgue measure set, which is very interesting for practical implementation. If the state equation is linear, then we can carry out a complete analysis of the control problem with measures. However, if the equation is nonlinear the use of measures to control the system is still an open problem, in general, and the use of functions to control the system seems to be more appropriate.
Highlights
In the control of distributed parameter systems, those formulated by partial differential equations, usually we cannot put control devices at every point of the domain
It was observed that the use of the L1 norm of the control in the cost functional leads to the sparsity of the solution
In this paper we present the results obtained in the analysis of sparse control problems, both taking functions or measures as controls
Summary
In the control of distributed parameter systems, those formulated by partial differential equations, usually we cannot put control devices at every point of the domain. We have to determine the power of the controllers as well These controls are called sparse because they are not zero only in a small region of the domain. It was observed that the use of the L1 norm of the control in the cost functional leads to the sparsity of the solution This introduces some mathematical difficulties in the problem due to the lack of differentiability of this functional. Taking a further step in this direction, we find that many times it is even desirable to put the controllers only in finitely many points of the domain, or along a line (in two dimensions), or on a surface (in three dimensions) In these cases we need to use controllers that are localized in a zero Lebesgue measure set. Though some results are indicated and references are provided, we have not considered the numerical approximation of the control problems because this would lead to a very long paper
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