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.

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