HOD in inner models with Woodin cardinals

We analyze the hereditarily ordinal definable sets $\mathrm{HOD}$ in $M_n(x)[g]$ for a Turing cone of reals $x$, where $M_n(x)$ is the canonical inner model with $n$ Woodin cardinals build over $x$ and $g$ is generic over $M_n(x)$ for the L'evy collapse up to its bottom inaccessible cardinal. We prove that assuming ${\mathbf{\Pi }}_{n+2}^1$-determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x)[g]} = M_n($\mathcal{M}${\infty} | \kappa\infty, \Lambda),$where$$\mathcal{M}$_\infty$ is a direct limit of iterates of $M_{n+1}$, ${\delta }_{\mathrm{\infty }}$ is the least Woodin cardinal in ${\mathcal{M}}_{\mathrm{\infty }}$, ${\kappa }_{\mathrm{\infty }}$ is the least inaccessible cardinal in ${\mathcal{M}}_{\mathrm{\infty }}$ above ${\delta }_{\mathrm{\infty }}$, and $\mathrm{\Lambda }$ is a partial iteration strategy for ${\mathcal{M}}_{\mathrm{\infty }}$. It will also be shown that under the same hypothesis ${\mathrm{HOD}}^{M_n(x)[g]}$ satisfies $\mathrm{GCH}$.