  We investigate the complexity of $r$-Approval control problems in $k$-peaked
elections, where at most $k$ peaks are allowed in each vote with respect to an
order of the candidates. We show that most NP-hardness results in general
elections also hold in k-peaked elections even for $k=2,3$. On the other hand,
we derive polynomial-time algorithms for some problems for $k=2$. All our
NP-hardness results apply to Approval and sincere-strategy preference-based
Approval as well. Our study leads to many dichotomy results for the problems
considered in this paper, with respect to the values of $k$ and $r$. In
addition, we study $r$-Approval control problems from the viewpoint of
parameterized complexity and achieve both fixed-parameter tractability results
and W-hardness results, with respect to the solution size. Along the way
exploring the complexity of control problems, we obtain two byproducts which
are of independent interest. First, we prove that every graph of maximum degree
3 admits a specific 2-interval representation where every 2-interval
corresponding to a vertex contains a trivial interval and, moreover,
2-intervals may only intersect at the endpoints of the intervals. Second, we
develop a fixed-parameter tractable algorithm for a generalized $r$-Set Packing
problem with respect to the solution size, where each element in the given
universal set is allowed to occur in more than one r-subset in the solution.
