On optimal thermal control of an idealized room including a max-control cost term

Abstract

We study an open-loop optimal control problem for energy efficient cooling of a simple room.
The goal is to provide insights for the development of an implementable control scheme based on a Model Predictive Control
strategy. The cost function includes a non-standard Mayer-term
$
  F(\max_t u(t))  ,
$
that is, a term depending on the peak value of the control.
Necessary conditions  for an optimal control problem are formulated and it is
observed that candidates of optimality can be obtained as solutions of a related
two-point boundary value problem (TPBVP). Due to numerical issues a finite dimensional approximation of the optimal control problem is considered to provide approximations to the solutions
of the TPBVP. The procedure is then illustrated on a numerical example.