T = 100; p = 1.; δ = 0.05127; ρ = 1./(1. + δ); c = 50.; R0 = 1.;
obj[qlist_ /; VectorQ[qlist, NumericQ]] :=
 Module[{Rlist, obj},
  Rlist = FoldList[(#1 - #2) &, R0, qlist];

If[Min[Rlist] < 0, Return[-1000]]; (* Constraint to keep R>=0 *)

(* This is the objective function *)
Sum[ρ^t*(p*qlist[[t + 1]] -
      0.5 c ((qlist[[t + 1]])^2)/Rlist[[t + 1]]), {t, 0, T - 1}]
  ]

choicevar = Table[Unique[q], {i, 0, T - 1}];
sol = NMaximize[Prepend[Thread[0 <= choicevar], obj[choicevar]],
   choicevar];
sol[[1]]