Abstract
In this chapter we study controllability properties of the 1-D semilinear heat equation with a sublinear reaction term in the case when it is controlled by a static degenerate spatially averaged actuator of type ( 1.29). We will show that, if the actuator at hand ensures the approximate controllability of the truncated linear equation in L 2(0, 1), then the original semilinear equation is exactly controllable in any finite-dimensional subspace spanned by the eigenfunctions of the associated linear spectral problem. We will also suggest a topology in which this semilinear heat equation is globally approximately controllable at any positive time. Extensions to the case of several spatial dimensions are also discussed.
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