Global non-negative controllability of the semilinear parabolic equation governed by bilinear control.

*(English)*Zbl 1024.93026Summary: 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.

Reviewer: S.K.Ntouyas (Ioannina)

##### MSC:

93C20 | Control/observation systems governed by partial differential equations |

93B03 | Attainable sets, reachability |

##### Keywords:

semilinear parabolic equation; global approximate controllability; bilinear system; reachability
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\textit{A. Y. Khapalov}, ESAIM, Control Optim. Calc. Var. 7, 269--283 (2002; Zbl 1024.93026)

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