  Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup
\{0\}$ is called a configuration of pebbles on the vertices of $G$ and the
weight of $f$ is $w(f)=\sum_{u\in V}f(u)$ which is just the total number of
pebbles assigned to vertices. A pebbling step from a vertex $u$ to one of its
neighbors $v$ reduces $f(u)$ by two and increases $f(v)$ by one. A pebbling
configuration $f$ is said to be solvable if for every vertex $ v $, there
exists a sequence (possibly empty) of pebbling moves that results in a pebble
on $v$. A pebbling configuration $f$ is a $t$-restricted pebbling configuration
(abbreviated $t$RPC) if $f(v)\leq t$ for all $v\in V$. The $t$-restricted
optimal pebbling number $\pi_t^*(G)$ is the minimum weight of a solvable $t$RPC
on $G$. Chellali et.al. [Discrete Appl. Math. 221 (2017) 46-53] characterized
connected graphs $G$ having small $2$-restricted optimal pebbling numbers and
characterization of graphs $G$ with $\pi_2^*(G)=5$ stated as an open problem.
In this paper, we solve this problem. We improve the upper bound of the
$2$-restricted optimal pebbling number of trees of order $n$. Also, we study
$2$-restricted optimal pebbling number of some grid graphs, corona and
neighborhood corona of two specific graphs.
