The optimal control: existence and characterization

Our aim is to minimize the number of diabetics without complication individuals and the cost of vaccination program. Mathematically, it can be interpreted by optimisation of the objective functional

where A>0, B>0 and G>0 are the cost coefficients. They are selected to weigh the relative importance of u(t), v(t) and w(t) at time t. T is the final time.

In other words, we seek the optimals controls u∗, v∗ and w∗ such that

where U is the set of admissible controls defined by U={(u,v,w)/ umin≤u(t)≤umax, umin≤v(t)≤umax and wmin≤w(t)≤wmax/t∈[0,T]}