1 Optimal preheating of an espresso cup

1 Optimal preheating of an espresso cup At time t = 0, boiling water (that is, water with temperature 100 ◦C) is poured into an espresso cup; after P seconds (the preheating time), the water is poured out, and espresso, with initial temperature 95 ◦C, is poured in. (You can assume that these pouring operations occur instantaneously.) The espresso is then consumed exactly 15 s later (yes, instantaneously). The problem is to choose the preheating time P in order to maximize the temperature of the espresso when it is consumed. We can model this situation as follows. We take the temperature of the liquid in the cup (either water or espresso) as one state; we model the cup using an n-state finite-element model. The vector x(t) ∈ R gives the temperature distribution at time t: x1(t) is the temperature of the liquid (water or espresso) at time t, and x2(t), . . . , xn+1(t) are the temperatures of the elements in our model of the cup. All temperatures are in ◦C, and t is in seconds. The dynamics are