ABSTRACT
The objective of the optimization is to find the control u (t) that restores homeostasis in the shortest time, with the constraint that the average of the control is fixed at a small value η · umax, where umax is the upper bound of the control and η is small. That is,
minT subject to u (t) 1 T
u (t)dt = ηumax. (A.1)
To incorporate this constraint into the analysis, a new state variable W (t) is introduced that satisfies
dW dt
= u − η · umax (A.2) with the condition W (0) = W (T ) = 0. Note that Equation (A.2) is equivalent to Equation (A.1). Indeed, the integration of Equation (A.2) yields W (T ) − W (0) =∫ T
0 u (t) dt − ηumaxT , which is exactly Equation (A.1) by the condition W (0) = W (T ) = 0.
