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I have a function $p(\theta,z,t)$ defined at the end of this question. I am trying to do two things here:

  1. Find the global maximum and the global minimum of $p$, for some specific values of $a$ and $d$, over all possible $0 < \theta \le 2\pi$, $0 \le z \le 100$ and $2000 \ge t \ge 0$.
  2. Incorporate the method from 1 into nested loops with a brute-force method - to find the values of $a$ and $d$ which will give the greatest global maximum of $p$, and the smallest global minimum of $p$ over the ranges of $\theta,z,t$, looping over a numerical range such as $-1\le a \le 1$ and $-1\le d \le 1$.

$a$ and $d$ are non-zero. Here is what I have tried for task 1, trying e.g. $a=1/200$ and $d=-1/200$:

FindMaximum[{p,100 >= z >= 0 && 2000>=t>=0 && 0 <\[Theta]<=2*Pi && a==1/200 && d==-1/200} , {\[Theta],t,z}]
(* {8.34524*10^-7, {\[Theta] -> 2.60554, t -> 143.986, z -> 15.5302}} *)

However this provides an answer for a specific value of $\theta,z,t$ (local maximum) and not a global maximum over all the ranges of $\theta,z,t$.

I tried Maximize but that simply returns the same input that I provided.

Maximize[{p,100 >= z>= 0 && 2000>=t>=0 && 0 <\[Theta]<=2*Pi && a==1/200 && d==-1/200} , {a,d}]

Also, for task 2 I am thinking that maybe iterating over a sort of grid might work, but I am not sure how to approach this / proceed, since this would be a grid in 3 dimensions ($\theta,z,t$)?

P.S. This is a simpler component of an existing question with an open bounty and no answers yet.

And here is $p$, in case it matters. Thanks.

p = -(1/2) a E^(-2 d^2 t) (4 E^(
    d^2 t) (E^(a z)
        Cos[\[Theta]] Cos[
        a Cos[\[Theta]] + d Sin[\[Theta]]] (a Cos[\[Theta]] + 
         d Sin[\[Theta]])^2 - 
      E^(a Sin[\[Theta]])
        Sin[\[Theta]] (a d Cos[a z + d Cos[\[Theta]]] Sin[
           2 \[Theta]] + (-d^2 Cos[\[Theta]]^2 + 
            a^2 Sin[\[Theta]]^2) Sin[a z + d Cos[\[Theta]]]) - 
      a^2 E^(a Cos[\[Theta]])
        Sin[d z + 3 \[Theta] + a Sin[\[Theta]]]) + 
   a (2 a E^(2 a Cos[\[Theta]]) Cos[\[Theta]] + 
      d E^(a (Cos[\[Theta]] + Sin[\[Theta]]))
        Cos[\[Theta]] Cos[
        a z - d z + d Cos[\[Theta]] - a Sin[\[Theta]]] + 
      d E^(a (Cos[\[Theta]] + Sin[\[Theta]]))
        Cos[\[Theta]] Cos[(a + d) z + d Cos[\[Theta]] + 
         a Sin[\[Theta]]] + 
      a E^(a (z + Cos[\[Theta]]))
        Cos[\[Theta]] Cos[
        d z + a Cos[\[Theta]] + (a + d) Sin[\[Theta]]] + 
      2 a E^(2 a Sin[\[Theta]]) Sin[\[Theta]] - 
      a E^(a (Cos[\[Theta]] + Sin[\[Theta]]))
        Cos[a z - d z + d Cos[\[Theta]] - 
         a Sin[\[Theta]]] Sin[\[Theta]] + 
      a E^(a (Cos[\[Theta]] + Sin[\[Theta]]))
        Cos[(a + d) z + d Cos[\[Theta]] + 
         a Sin[\[Theta]]] Sin[\[Theta]] + 
      a E^(a (z + Cos[\[Theta]]))
        Cos[d z + 
         a Cos[\[Theta]] + (a + d) Sin[\[Theta]]] Sin[\[Theta]] + 
      d E^(a (z + Cos[\[Theta]]))
        Cos[d z + 
         a Cos[\[Theta]] + (a + d) Sin[\[Theta]]] Sin[\[Theta]] + 
      E^(a (z + Sin[\[Theta]]))
        Cos[a z - a Cos[\[Theta]] + d Cos[\[Theta]] - 
         d Sin[\[Theta]]] ((a - d) Cos[\[Theta]] + d Sin[\[Theta]]) - 
      E^(a (z + Cos[\[Theta]]))
        Cos[d z - 
         a Cos[\[Theta]] + (a - 
            d) Sin[\[Theta]]] (a Cos[\[Theta]] + (-a + 
            d) Sin[\[Theta]]) + 
      a E^(a (Cos[\[Theta]] + Sin[\[Theta]]))
        Cos[\[Theta]] Sin[
        a z - d z + d Cos[\[Theta]] - a Sin[\[Theta]]] + 
      a E^(a (Cos[\[Theta]] + Sin[\[Theta]]))
        Sin[\[Theta]] Sin[
        a z - d z + d Cos[\[Theta]] - a Sin[\[Theta]]] + 
      a E^(a (Cos[\[Theta]] + Sin[\[Theta]]))
        Cos[\[Theta]] Sin[(a + d) z + d Cos[\[Theta]] + 
         a Sin[\[Theta]]] + 
      a E^(a (Cos[\[Theta]] + Sin[\[Theta]]))
        Sin[\[Theta]] Sin[(a + d) z + d Cos[\[Theta]] + 
         a Sin[\[Theta]]] + 
      a E^(a (z + Cos[\[Theta]]))
        Cos[\[Theta]] Sin[
        d z - a Cos[\[Theta]] + (a - d) Sin[\[Theta]]] + 
      E^(a (z + 
          Sin[\[Theta]])) (Cos[
           a z + (a + d) Cos[\[Theta]] + 
            d Sin[\[Theta]]] ((a + d) Cos[\[Theta]] + 
            d Sin[\[Theta]]) + 
         a Sin[\[Theta]] (-Sin[
              a z - a Cos[\[Theta]] + d Cos[\[Theta]] - 
               d Sin[\[Theta]]] + 
            Sin[a z + (a + d) Cos[\[Theta]] + d Sin[\[Theta]]])) + 
      a E^(a (z + Cos[\[Theta]]))
        Cos[\[Theta]] Sin[
        d z + a Cos[\[Theta]] + (a + d) Sin[\[Theta]]]))
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