Control of Heat Equation with Actuator Placement

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
). 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 "\norm"): \begin{array}{llcl} \min\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}

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--323