= Solution
Impose the target temperature on the post-jump <exponential decay>:
$$
\left[\left(\frac\alpha2-1\right)T_0+\beta\right]e^{-k(t_2-t_1)}=(\alpha-1)T_0.
$$
Solving for the jump and eliminating $t_1$ using $e^{kt_1}=2(\alpha-1)/(\alpha-2)$ gives
$$
\boxed{\beta=(\alpha-1)T_0e^{k(t_2-t_1)}-\frac{\alpha-2}{2}T_0=\frac{\alpha-2}{2}T_0\bigl(e^{kt_2}-1\bigr).}
$$
\b[This is the unique required temperature increase, and it satisfies the stipulated lower bound.] Indeed $t_2>t_1$ implies
$$
\beta-\frac{\alpha T_0}{2}=(\alpha-1)T_0\bigl(e^{k(t_2-t_1)}-1\bigr)>0.
$$
Thus no additional restriction on the prescribed time is needed.
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