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In this paper, we study the global approximate multiplicative controllabilityfor nonlinear degenerate parabolic Cauchy-Neumann problems. First, we willobtain embedding results for weighted Sobolev spaces, that have proved decisivein reaching well-posedness for nonlinear degenerate problems. Then, we showthat the above systems can be steered in $L^2$ from any nonzero, nonnegativeinitial state into any neighborhood of any desirable nonnegative target-stateby bilinear piecewise static controls. Moreover, we extend the above resultrelaxing the sign constraint on the initial date.

Approximate controllability for nonlinear degenerate parabolic problems with bilinear control