Why are Bezier curves numerically less stable for a larger number of control points? I think the question is quite straightforward. Why are Bezier curves with more control points numerically more unstable. Can someone give me clear substantiated reason(s)? And with this the notion of numerical (un)stability? Does it maybe have to do with optimization or interpolation of the control points?
A Bezier curve is a polynomial parametric curve and the shape of the curve is based on the position of the control points.
 A: Well, Bezier curves are just polynomials, and interpolation with high degree polynomials has a bad reputation. It may be that the papers you're reading are just passing along the folklore. Experts whom I trust (Trefethen) say that the folklore is dubious -- the alleged problems are poorly articulated and the bad reputation is not deserved.
As one of the comments said, the numerical stability (or instability) depends on what particular computation you're doing. If you're doing interpolation, then you have to be careful what polynomial basis you use. The power basis is very bad; Bernstein and Chebyshev bases are much better.
The standard references in this area are two papers by Farouki and Rajan:
On the numerical condition of polynomials in Bernstein form
Computer Aided Geometric Design
Volume 4, Issue 3, November 1987, Pages 191-216
Algorithms for polynomials in Bernstein form
Computer Aided Geometric Design
Volume 5, Issue 1, June 1988, Pages 1-26
I don't see any problem in using Bezier control points as the independent variables in an optimization process. If you displace a control point by some vector $V$, then each point on the Bezier curve will move by $kV$, where $k \le 1$, since the Bernstein polynomials are $\le 1$ on the interval $[0,1]$. Seems like that's the sort of stability you're seeking.
