This is a tough one. In probabilistic speak, you want to construct a distribution on [0,1] with a continuous density function that is prescribed at 0 and 1.
I take it you want the mass to shift smoothly around [0,1] as the parameters vary. What about making the density a piecewise linear function, e.g. with a W shape?
u'(0)=a
u'(x1)=0
u'(1/4)=0
u'(1/2)=c
u'(3/4)=0
u'(x2)=0
u'(1)=b
x1=min(1/4,1/a)
x2=max(3/4,1/b)
choose c so that total area is 1, i.e. c = 42.a.x12.b.(1x2)
