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 $\mathrm{\Omega }=[0,1]\times [0,2]$ with the boundary $\partial \mathrm{\Omega }$ and the time horizon $T=[0,10]$ as the domains. 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.

Mathematical formulation

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

Parameters

These fixed values are used within the model.

$\begin{array}{rcl}L& =& 9,\\ \kappa & =& 0.01.\end{array}$\\ The parameter $\kappa$ may vary in $\mathrm{\Omega }$.

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