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 $\bar{u}\equiv 0$, 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. 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

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

Parameters

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

The parameters used are:

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

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 $W$ 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:

$\frac{1}{h_t}(u_{i,j,k}-u_{i,j,k-1}) -  \frac{\kappa_{i,j}}{h_x h_y}(u_{i-1,j,k}+u_{i+1,j,k}+u_{i,j-1,k}+u_{i,j+1,k}-4u_{i,j,k}) =   \sum\limits_{l=1}^L(v_{k+1,l}+v_{k  ,l} )f_l(ih_x,jh_y),$

with $u_{i,j,k}=u(ih_x,jh_y,kh_t)$, 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 $k\in \{0,\dots,T_n\}$: $\sum\limits_{l=1}^L w_{k,l}\leq W.$

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