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?
choose c so that total area is 1, i.e. c = 4-2.a.x1-2.b.(1-x2)