Approximate controllability for nonlinear degenerate parabolic problems with bilinear control

In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy–Neumann problems. First, we obtain embedding results for weighted Sobolev spaces, that have proved decisive in reaching well-posedness for nonlinear degenerate problems. Then, we show that the above systems can be steered in L²$L^2$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear piecewise static controls. Moreover, we extend the above result relaxing the sign constraint on the initial data.