Approximate controllability for nonlinear degenerate parabolic problems with bilinear control

TL;DR

This paper establishes that nonlinear degenerate parabolic equations can be approximately controlled in the $L^2$ sense using bilinear controls, even without initial sign constraints, by developing new weighted Sobolev space embeddings.

Contribution

It introduces new embedding results for weighted Sobolev spaces and demonstrates approximate controllability for nonlinear degenerate parabolic problems with bilinear controls.

Findings

Systems can be steered in $L^2$ from any nonzero, nonnegative initial state to a neighborhood of any nonnegative target.

Control results hold even when initial sign constraints are relaxed.

New embedding results are crucial for well-posedness and controllability analysis.

Abstract

In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy-Neumann problems. First, we will 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 $L^2$ 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 date.