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

From mintOC
Jump to: navigation, search
(→‎Mathematical formulation)
Line 19: Line 19:
  
 
== Mathematical formulation ==
 
== Mathematical formulation ==
 +
 +
 +
<p>
 +
<math>
 +
\begin{array}{llcl}
 +
 +
\minimize\limits_{u,v,w}~~ &J(u,v)=\norm{u(\cdot,\cdot,10)}_{2,\Omega}^2 +2\norm{u(\cdot,\cdot,\cdot)}_{2,\Omega\times T}^2+\frac{1}{500}\sum\limits_{l=1}^9\norm{v_l(\cdot)}^2_{2,T} &  & \label{Eq:HeatEqActPlObj}\\
 +
      \st ~~~~ &\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\label{Eq:HeatEqActPlHeat}\\
 +
    & u(x,y,t) =0    &\text{ on } &\partial\Omega\times T \label{Eq:HeatEqActPlDirichlet}\\
 +
    & u(x,y,0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\label{Eq:HeatEqActPlInitial}\\
 +
    & -M w_l(t)\leq v_l(t)\leq M w_l(t) \text{ for all } l\in \{1,\dots,9\} &\text{ in } & T \label{Eq:BigM}\\
 +
    & \sum\limits_{l=1}^9 w_l(t) = 1 &\text{ in } & T \label{Eq:HeatEqActPlBudget}\\
 +
    & w_l(t)\in \{0,1\} \text{ for all } l\in \{1,\dots,9\} &\text{ in } &T.
 +
 +
\end{array}
 +
</math>
 +
</p>
  
 
==Parameters==
 
==Parameters==

Revision as of 16:17, 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. We consider a rectangle \Omega=[0,1]\times[0,2] with the boundary \partial\Omega and the time horizon T=[0,10] as the domains. 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

Failed to parse (unknown function "\minimize"): \begin{array}{llcl} \minimize\limits_{u,v,w}~~ &J(u,v)=\norm{u(\cdot,\cdot,10)}_{2,\Omega}^2 +2\norm{u(\cdot,\cdot,\cdot)}_{2,\Omega\times T}^2+\frac{1}{500}\sum\limits_{l=1}^9\norm{v_l(\cdot)}^2_{2,T} & & \label{Eq:HeatEqActPlObj}\\ \st ~~~~ &\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\label{Eq:HeatEqActPlHeat}\\ & u(x,y,t) =0 &\text{ on } &\partial\Omega\times T \label{Eq:HeatEqActPlDirichlet}\\ & u(x,y,0) = 100 \sin(\pi x)\sin(\pi y) &\text{ in }& \Omega\label{Eq:HeatEqActPlInitial}\\ & -M w_l(t)\leq v_l(t)\leq M w_l(t) \text{ for all } l\in \{1,\dots,9\} &\text{ in } & T \label{Eq:BigM}\\ & \sum\limits_{l=1}^9 w_l(t) = 1 &\text{ in } & T \label{Eq:HeatEqActPlBudget}\\ & w_l(t)\in \{0,1\} \text{ for all } l\in \{1,\dots,9\} &\text{ in } &T. \end{array}

Parameters

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
Retrieved from "https://mintoc.de/index.php?title=Control_of_Heat_Equation_with_Actuator_Placement&oldid=1774"