Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: a qualitative approach

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

“…In the recent work [18] (written after the current paper was submitted) we obtained a different non-negative controllability result for a system like in (1.1) in several space dimensions with the terms f which can be superlinear at infinity but are not necessarily superlinear near the origin. The result in [18] always requires at least three "large" static bilinear controls (whose magnitude increases as the precision of steering increases) applied subsequently for very short times.…”

“…The result in [18] always requires at least three "large" static bilinear controls (whose magnitude increases as the precision of steering increases) applied subsequently for very short times. Unlike the present paper, based on the use of the dynamics imposed by the diffusion-reaction term like y xx + α(x)y in (1.3) in the first place, the method of [18] focuses on the "suppression" of the effect of the diffusion term like the above-mentioned y xx . It makes use of the part of the dynamics of (1.1) which can be approximated by the trajectories of the ordinary differential equation dz/dt = α(x)z in L 2 (Ω).…”

“…It makes use of the part of the dynamics of (1.1) which can be approximated by the trajectories of the ordinary differential equation dz/dt = α(x)z in L 2 (Ω). Accordingly, the method of [18] does not apply to obtain the results of Theorem 1.1, allowing the use of "small" single static controls, and of Corollary 1.3, allowing the principal extension of the set of non-negative initial states to the set described in conditions (1.5).…”

Controllability of the Parabolic System Via Bilinear Control

“…There are few results of controllability of bilinear distributed parameter systems. In [12], Khapalov discussed the nonnegative approximate controllability of the parabolic system with superlinear term governed by a bilinear control and in [11], he also discussed the bilinear null-controllability of the parabolic system with the reaction term satisfying Newton's law. In [14,19], the authors obtained the exact controllability of parabolic equations for some particular targets.…”

Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping

“…• In recent works [7][8][9][10][11] several global approximate controllability results were obtained for rather general semilinear parabolic equation with nonlinear terms admitting superlinear growth.…”