Consider the production and inventory storage of steel, in a steel factory. The state variable I (t) measures the number of tons of steel that we have in inventory at time t [0, T ]. There is a demand rate D(t) tons of steel per day at time t [0, T ], and we must choose the production rate P(t) tons of steel per day at time t [0, T ], (Note: This is our control variable).

h(I ) is the cost of holding inventory I in dollars per day, and c(P) is the cost of producing steel at rate P, also in dollars per day, The state variable constraint, I (t) 0, is imposed so that the demand is satisfied for all t. In other words, backlogging of demand is not permitted.

The constraints on the control variable are keeping the production rate P(t) between a specified lower bound P_min and a specified upper bound P_max. Finally, the terminal constraint I (T ) I _min is imposed so that the

terminal inventory is at least I _min.

Let the functions and parameters of the production- inventory model be given by:

h(I ) = 10I, c(P) = 20P, T = 10,

P_min = 600, P_max = 1200, I _min = 800, D(t) = 10t+900

(a) Given the initial inventory of I ₀ = 1, 000 tons of steel on hand at t = 0, write the state equation

which describes how the steel inventory changes over time.

(b) Write the objective function J, minimizing the sum of holding and production costs.

(c) Set P(t) = 1000 for 0 t 10. Determine whether this control is feasible; if it is feasible, compute the value J of the objective function.(Note a feasible control is the one which does not violate state constraints and terminal constraint.

(d) If P(t) = 800, show that the terminal constraint is violated and hence the control is infeasible.

(e) If P(t) = P_min for 0 t 6 and P(t) = P_max for 6 < t 10, show that the control is infeasible because the state constraint is violated.