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

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This problem is governed by the heat equation and is adapted from Iftime and Demetriou (<bib id="Iftime2009"/>).
 
This problem is governed by the heat equation and is adapted from Iftime and Demetriou (<bib id="Iftime2009"/>).
 
Its goal is to choose a place to apply an actuator in a given area depending on time.
 
Its goal is to choose a place to apply an actuator in a given area depending on time.
−
We consider a rectangle <math>\Omega=[0,1]\times[0,2]</math> with the boundary <math>\partial\Omega</math> and the time horizon <math>T=[0,10]</math> as the domains.
 
 
The objective function is quadratic, its first term captures the desired final state <math>\bar{u}\equiv 0</math>, the second term regularize the state over time and the third term regularize the continuous controls.  
 
The objective function is quadratic, its first term captures the desired final state <math>\bar{u}\equiv 0</math>, 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.
 
The constraints are a source budget, which limits the quantity of placed actuators, and the two-dimensional heat equation with some source function.
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\begin{array}{llcl}
 
\begin{array}{llcl}
  
−
\min\limits_{u,v,w}~~ &J(u,v)=||u(\cdot,\cdot,10)||_{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]
+
\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]
 
     \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,t) =0    &\text{ on } &\partial\Omega\times T \\[10pt]
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<math>
 
<math>
 
\begin{array}{rcl}
 
\begin{array}{rcl}
 +
\Omega &=& [0,1] \times [0,2],\\
 
L &=& 9, \\
 
L &=& 9, \\
 
\kappa &=& 0.01,\\
 
\kappa &=& 0.01,\\

Revision as of 17:44, 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) = 1 &\text{ in } & T\\[10pt] & w_l(t)\in \{0,1\} \text{ for all } l\in \{1,\dots,L\} &\text{ in } &T. \end{array}

Parameters

These fixed values are used within the model.

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

The parameter \kappa describes the thermal dissipativity of the material in the domain \Omega , it may vary in space. The parameter L indicates the number of possible actuator locations. They are distributed as indecated in the picture.

Reference solution

Source Code

References

[Iftime2009]Orest V. Iftime; Michael A. Demetriou (2009): {O}ptimal control of switched distributed parameter systems with spatially scheduled actuators . {A}utomatica, 45, 312--323Link to Google Scholar
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