Local Exact Controllability to Stationary Solutions of a Semilinear Parabolic Equation

This paper establishes the local exact controllability of the quasilinear porous media equation with Dirichlet boundary condition.\\ Consider the equation aligned &yₜ - Δ a(y) = mu+f on Q\\ &y(0)=y₀,\ y Σ = 0 aligned on the $n+1$-dimensional cylinder Q = Ω × (0, T) with lateral boundary Σ = ∂ Ω × (0, T). The exact controllability in finite time is proved when \|y₀ - yₛ\|_W1, n₀(Ω) ∩ C( Ω) is sufficiently small, $n > 1$, for every stationary solution yₛ such that a(yₛ) ∈ W2, q(Ω), where $q>n$. It is assumed that Ω is a bounded open set with C² boundary and that a ∈ C²( R), $a'>0$.