Difference between revisions of "Control of Heat Equation with Actuator Placement"

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(Parameters)
(Parameters)
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These fixed values are used within the model.
+
We define the source term for all locations <math>
 +
l\in \{1,\dots,L\} <\math> and a
 +
fix parameter <math>\varepsilon\in \R_+</math>:
 +
<math>
 +
f_l(x,y) = \frac{1}{\sqrt{2\pi\varepsilon}}e^{\frac{-((x_l-x)^2+(y_l-y)^2)}{2\varepsilon}}
 +
<\math>
 +
where <math>(x_l,y_l)</math> is the coordinate of the mesh point of the $l$th possible actuator location.
 +
 
 +
The parameters used are:
  
 
<math>
 
<math>
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The  parameter  <math> \kappa </math> describes the thermal dissipativity of the material in the domain <math> \Omega </math>, it may vary in space.
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The  parameter  <math> \kappa </math> describes the thermal dissipativity of the material in the domain <math> \Omega </math>, it may vary in space: <math> \kappa(x,y) </math>.
 
The parameter  <math> L </math> indicates the number of possible actuator locations. They are distributed as indicated in the picture.
 
The parameter  <math> L </math> indicates the number of possible actuator locations. They are distributed as indicated in the picture.
 
+
The source budget is limited by <math>W<\math>.
 
+
We define the source term for all locations $l\in \{1,\dots,L\}$ and a
+
fix parameter <math>\varepsilon\in \R_+</math>:
+
<math>
+
f_l(x,y) &=& \frac{1}{\sqrt{2\pi\varepsilon}}e^{\frac{-((x_l-x)^2+(y_l-y)^2)}{2\varepsilon}}
+
<\math>
+
where <math>(x_l,y_l)</math> is the coordinate of the mesh point of the $l$th possible actuator location.
+
  
 
==Reference solution==
 
==Reference solution==

Revision as of 17:08, 23 February 2016

Control of Heat Equation with Actuator Placement
State dimension: 1
Differential states: 1
Continuous control functions: L
Discrete control functions: L
Path constraints: 3
Interior point equalities: 2


This problem is governed by the heat equation and is adapted from Iftime and Demetriou ([Iftime2009]Author: Orest V. Iftime; Michael A. Demetriou
Journal: {A}utomatica
Number: 2
Pages: 312--323
Title: {O}ptimal control of switched distributed parameter systems with spatially scheduled actuators
Volume: 45
Year: 2009
Link to Google Scholar
). Its goal is to choose a place to apply an actuator in a given area depending on time. The objective function is quadratic, its first term captures the desired final state \bar{u}\equiv 0, the second term regularize the state over time and the third term regularize the continuous controls. The constraints are a source budget, which limits the quantity of placed actuators, and the two-dimensional heat equation with some source function. Additionally, we assume Dirichlet boundary conditions and initial conditions.


Mathematical formulation


\begin{array}{llcl}

\min\limits_{u,v,w}~~ &J(u,v)=||u(\cdot,\cdot,t_f)||_{2,\Omega}^2 +2||u(\cdot,\cdot,\cdot)||_{2,\Omega\times T}^2+\frac{1}{500}\sum\limits_{l=1}^L||v_l(\cdot)||^2_{2,T} &  \\[10pt]
     \text{ s.t.} ~~~~ &\frac{\partial u}{\partial t}(x,y,t)- \kappa \Delta u(x,y,t)=\sum\limits_{l=1}^9 v_l(t) f_l(x,y) &\text{ in }&\Omega\times T\\[10pt]
     & u(x,y,t) =0    &\text{ on } &\partial\Omega\times T \\[10pt]
     & u(x,y,0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\\[10pt]
     & -M w_l(t)\leq v_l(t)\leq M w_l(t) \text{ for all } l\in \{1,\dots,L\} &\text{ in } & T \\[10pt]
     & \sum\limits_{l=1}^L w_l(t) = W &\text{ in } & T\\[10pt]
     & w_l(t)\in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T.

\end{array}

Parameters

We define the source term for all locations Failed to parse (unknown function "\math"): l\in \{1,\dots,L\} <\math> and a fix parameter <math>\varepsilon\in \R_+

Failed to parse (unknown function "\math"): f_l(x,y) = \frac{1}{\sqrt{2\pi\varepsilon}}e^{\frac{-((x_l-x)^2+(y_l-y)^2)}{2\varepsilon}} <\math> where <math>(x_l,y_l)

is the coordinate of the mesh point of the $l$th possible actuator location.

The parameters used are:


\begin{array}{rcl}
\Omega &=& [0,1] \times [0,2],\\
L &=& 9, \\
\kappa &=& 0.01,\\
t_f &=&10,\\
W &=& 1.
\end{array}


The parameter  \kappa describes the thermal dissipativity of the material in the domain  \Omega , it may vary in space:  \kappa(x,y) . The parameter  L indicates the number of possible actuator locations. They are distributed as indicated in the picture. The source budget is limited by Failed to parse (unknown function "\math"): W<\math>. ==Reference solution== ==Source Code== ==References== <biblist/> <!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here --> [[Category:MIOCP]]