Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms

Khapalov, Alexander Y.

doi:10.1201/9780203904190.ch7

Cited by 21 publications

(13 citation statements)

References 14 publications

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“…1.1 is out of question as well.) On the positive global controllability results for semilinear parabolic equations with superlinear terms and additive controls of localized support we refer to [1,6,8,9,[13][14][15] (see also the discussion on [16] in the beginning of Sect. 4).…”

Section: Resultsmentioning

confidence: 99%

“…non-negative. Indeed, if α(x) ≤ 0 this is true with b = 0 [16] (in this case (2.6) becomes the standard generalized maximum principle). Otherwise, equation (2.6) follows from the maximum principle applied to the function z(x, t) = e −bt y(x, t), satisfying…”

Section: Preliminariesmentioning

confidence: 99%

“…Remark 3.1. In [16] we derived the estimates of type (3.1) and (3.2) (in somewhat more general situation) for the case when α is a positive constant. They are as follows:…”

Section: Proof Of Theorem 12: Preliminariesmentioning

confidence: 99%

“…In [16] we proved the global approximate controllability in L 2 (0, 1) at any time T > 0 of the following Dirichlet boundary problem…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

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

Khapalov

2002

ESAIM: COCV

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Abstract.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 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.Mathematics Subject Classification. 93, 35.

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Section: Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…Remark 3.1. In [16] we derived the estimates of type (3.1) and (3.2) (in somewhat more general situation) for the case when α is a positive constant. They are as follows:…”

Section: Proof Of Theorem 12: Preliminariesmentioning

confidence: 99%

“…In [16] we proved the global approximate controllability in L 2 (0, 1) at any time T > 0 of the following Dirichlet boundary problem…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

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

Khapalov

2002

ESAIM: COCV

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“…• 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.…”

Section: Remark 14 (More References On Bilinear Controllability)mentioning

confidence: 99%

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

Khapalov

2006

ESAIM: COCV

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Abstract.We show that the set of nonnegative equilibrium-like states, namely, like (y d , 0) of the semilinear vibrating string that can be reached from any non-zero initial state (y0, y1) ∈ H 1 0 (0, 1) × L 2 (0, 1), by varying its axial load and the gain of damping, is dense in the "nonnegative" part of the subspace L 2 (0, 1) × {0} of L 2 (0, 1) × H −1 (0, 1). Our main results deal with nonlinear terms which admit at most the linear growth at infinity in y and satisfy certain restriction on their total impact on (0, ∞) with respect to the time-variable.Mathematics Subject Classification. 93, 35.

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Global Non-negative Approximate Controllability of Parabolic Equations with Singular Potentials

Vancostenoble

2019

Trends in Control Theory and Partial Differential Equations

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In this work, we consider the linear 1 − d heat equation with some singular potential (typically the so-called inverse square potential). We investigate the global approximate controllability via a multiplicative (or bilinear) control. Provided that the singular potential is not super-critical, we prove that any non-zero and non-negative initial state in L 2 can be steered into any neighborhood of any non-negative target in L 2 using some static bilinear control in L ∞ . Besides the corresponding solution remains non-negative at all times.

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Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms

Cited by 21 publications

References 14 publications

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

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

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

Global Non-negative Approximate Controllability of Parabolic Equations with Singular Potentials

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