k=2;
n=3;
T=1;
x0={1,0,0};
A1={{-2.5,0,0},{1,0,-0.5},{1.5,0,0.5}};
A2={{0,0.5,1},{0,-0.5,1.5},{0,0,-2.5}};

runExperiment[T0_]:=MatrixExp[(T-T0)*A2].MatrixExp[T0*A1].x0

Plot[runExperiment[i][[k]],{i,0,1}]

MaximalBy[#,First]&@Table[{runExperiment[i][[k]],i},{i,0,1,0.01}]

f[t_]:=Piecewise[{{MatrixExp[t*A1].x0,t<=0.61},{MatrixExp[(t-0.61)*A2].MatrixExp[0.61*A1].x0,t>0.61}}]

Plot[{f[i][[1]],f[i][[2]],f[i][[3]]},{i,0,1}]