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Global non-negative controllability of the semilinear parabolic equation governed by bilinear control. (English) Zbl 1024.93026
Summary: We study the global approximate controllability of the one-dimensional semilinear convection-diffusion-reaction equation governed in a bounded domain via the coefficient (bilinear control) in the additive reaction term. Clearly, even in the linear case, due to the maximum principle, such a system is not globally or locally controllable in any reasonable linear space. It is also well known that for the superlinear terms admitting a power growth at infinity the global approximate controllability by traditional additive controls of localized support is out of question. However, we will show that a system like that can be steered in \(L^{2}(0,1)\) from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static (\(x\)-dependent only) above-mentioned bilinear controls, applied subsequently in time, while only one such control is needed in the linear case.

MSC:
93C20 Control/observation systems governed by partial differential equations
93B03 Attainable sets, reachability
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