  Consider a periodic, mean-reverting Ornstein-Uhlenbeck process
$X=\{X_t,t\geq0\}$ of the form $d X_{t}=\left(L(t)+\alpha X_{t}\right) d t+
dB^H_{t}, \quad t \geq 0$, where $L(t)=\sum_{i=1}^{p}\mu_i\phi_i (t)$ is a
periodic parametric function, and $\{B^H_t,t\geq0\}$ is a fractional Brownian
motion of Hurst parameter $\frac12\leq H<1$. In the "ergodic" case $\alpha<0$,
the parametric estimation of $(\mu_1,\ldots,\mu_p,\alpha)$ based on
continuous-time observation of $X$ has been considered in Dehling et al.
\cite{DFK}, and in Dehling et al. \cite{DFW} for $H=\frac12$, and
$\frac12<H<1$, respectively. In this paper we consider the "non-ergodic" case
$\alpha>0$, and for all $\frac12\leq H<1$. We analyze the strong consistency
and the asymptotic distribution for the estimator of
$(\mu_1,\ldots,\mu_p,\alpha)$ when the whole trajectory of $X$ is observed.
