Control of Heat Equation with Actuator Placement

Control of Heat Equation with Actuator Placement
State dimension:	1
Differential states:	1
Continuous control functions:	9
Discrete control functions:	9
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. The objective function is quadratic, its first term captures the desired final state $\overline{u}\equiv 0$, 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. Originally, the problem formulation was non-convex. We overcome this issue by substitution of $v(t)w_l(t)$ by $v_l(t)$ and adding the Big $M$ formulation.

Mathematical formulation

$\begin{array}{llc}\min {}_{u,v,w}& J(u,v)=||u(\cdot ,\cdot ,t_f)|{|}_{2,\mathrm{\Omega }}^2+2||u(\cdot ,\cdot ,\cdot )|{|}_{2,\mathrm{\Omega }\times T}^2+\frac{1}{500}\sum _{l=1}^L||v_l(\cdot )|{|}_{2,T}^2& \\ \text{ s.t.}& \frac{\partial u}{\partial t}(x,y,t)-\kappa \mathrm{\Delta }u(x,y,t)=\sum _{l=1}^9{v}_l(t)f_l(x,y)& \text{ in }& \mathrm{\Omega }\times T\\ & u(x,y,t)=0& \text{ on }& \partial \mathrm{\Omega }\times T\\ & u(x,y,0)=100\sin (\pi x)\sin (\pi y)& \text{ in }& \mathrm{\Omega }\\ & -M{w}_l(t)\le v_l(t)\le M{w}_l(t)\text{ for all }l\in \{1,\dots ,L\}& \text{ in }& T\\ & \sum _{l=1}^L{w}_l(t)\le W& \text{ in }& T\\ & w_l(t)\in \{0,1\}\text{ for all }l\in \{1,\dots ,L\}& \text{ in }& T.\end{array}$

Parameters

We define the source term for all locations $l\in \{1,\dots ,L\}$ and a fix parameter $\epsilon \in {\mathbb{ℝ}}_{+}$: $f_l(x,y)=\frac{1}{\sqrt{2\pi \epsilon }}{e}^{\frac{-((x_l-x{)}^2+(y_l-y{)}^2)}{2\epsilon }}$ where $(x_l,y_l)$ is the coordinate of the mesh point of the $l$th possible actuator location.

The parameters used are:

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

The parameter $\kappa$ describes the thermal dissipativity of the material in the domain $\mathrm{\Omega }$, 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 $W$ and $t_f$ denotes the final time.

Insert non-formatted text here==Discretization== To solve the problem we apply a "first discretize, then optimize" approach an discretize the components of the problem. For the heat equation, we use a five-point-stencil in space and the implicit euler in time. For $i=1,\dots ,N-1$, $j=1,\dots ,M-1$, and $k=0,\dots ,T_n-1$, this yields:

$\frac{1}{h_t}(u_{i,j,k}-u_{i,j,k-1})-\frac{{\kappa }_{i,j}}{h_x{h}_y}(u_{i-1,j,k}+u_{i+1,j,k}+u_{i,j-1,k}+u_{i,j+1,k}-4u_{i,j,k})=\sum _{l=1}^L(v_{k+1,l}+v_{k,l})f_l(i{h}_x,j{h}_y),$

with $u_{i,j,k}=u(i{h}_x,j{h}_y,k{h}_t)$, the stepsizes $h_x,h_y$ in space, and the stepsize in time $h_t$, respectively.

It holds for the source buget with the discretized binary controls $w_{l,k}$ for all $k\in \{0,\dots ,T_n\}$: $\sum _{l=1}^L{w}_{k,l}\le W.$

This condition is called SOS-$W$ conditon.

We remark that the number of discretized binary variables does not depend on the space discretization but it depends on the time discretization. Thats why we taged this problem containing mesh-independend and as mesh-dependend integer variables.

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