I have differential equations f11, f12, f21, and f22 with respect to t with parameters A and B. I've defined and solved these equations using NDSolve in Mathematica as follows:
A = 2;
B = 6;
NDSolve[
{f11'[t] == -(1/2) I A f12[t] + 1/2 I A f21[t],
f12'[t] == -(1/2) I A f11[t] - I B f12[t] + 1/2 I A f22[t],
f21'[t] == 1/2 I A f11[t] + I B f21[t] - 1/2 I A f22[t],
f22'[t] == 1/2 I A f12[t] - 1/2 I A f21[t],
f22[0] == 0,f11[0] == 1, f12[0] == 0, f21[0] == 0},
{f11[t], f12[t], f21[t], f22[t]}, {t, 0, 1}]
Plotting the solution for f22[t] produces a graph :
I want to adjust the parameters A and B to maximize the value of f22[t]. I just want the maximum value that f22[t] can achieve in any t. In this example, there will be several solutions available, I am interested in understand how I can approach the problem in Mathematica using this example.
Can you advise how to optimize A, B (and t) to achieve this maximum? I'm open to using numerical optimization methods or machine learning algorithms if necessary. Any guidance would be much appreciated.