Control of Heat Equation with Actuator Placement

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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 regularizes the state over time and the third term regularizes 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

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

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:

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

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.

Discretization

To solve the problem we apply a "first discretize, then optimize" approach and 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 $u_{i,j,k}=u(ih_x,jh_y,kh_t)$,$v_{k ,l}=v_l(kh_t)$, $\kappa_{i,j}=\kappa(ih_x,jh_y)$, with the stepsizes in space $h_x,h_y$, and 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 does on the time discretization. That is why we tagged this problem as containing mesh-independent and mesh-dependent 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