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On the approximate controllability of some semilinear parabolic boundary-value problems. (English) Zbl 0936.93010

The paper studies the L p -approximate controllability problem in time T for a class of semilinear parabolic systems in a bounded domain Ω in n . Considered are two types of problems: (𝒫 D ) and (𝒫 N ). In each problem, the control functions are fairly general, given by the form v=v(x,t). In (𝒫 D ), the control v(x,t) and the nonlinear term f(y), y being the state of the system, enter the equation under a homogeneous boundary condition of the Dirichlet type, and v, supposed to be nonnegative on Ω×(0,T), belongs to a dense subset 𝒰 of L + p (Ω×(0,T)). In (𝒫 N ), v and f appear on the boundary condition of the Neumann type, and v belong to a dense subset 𝒰 of L p (Ω×(0,T)). For example, (𝒫 N ) is written as

y t-Δy=0,y ν+f(y)| Ω =v,y(·,0)=y 0 ·

In both cases, the problem in the linear case where f(y)=0 is first solved. Based on this result, the semilinear problem is solved as a perturbation of a linear problem cancelling the nonlinear term. The problem is also studied via the Kakutani fixed-point theorem for other semilinear parabolic systems including multivalued coefficients (a case of flux boundary controls).

Reviewer: T.Nambu (Kobe)
MSC:
93B05Controllability
93C20Control systems governed by PDE
35B37PDE in connection with control problems (MSC2000)
35K55Nonlinear parabolic equations