ϕ-Null controllability of a semi-discrete linear KdV equation with localized interior control
Abstract
This article studies the ϕ(h)-null controllability of a space-discrete approximation of the linear Korteweg-de Vries (KdV) equation posed on a bounded domain, with a localized interior control. The notion of ϕ(h)-null controllability is a suitable analogue of null controllability in the context of discretized control systems. We employ a duality approach, reducing the controllability problem to an observability inequality for the associated discrete adjoint system. To this end, we first establish a suitable discrete Carleman estimate for the underlying discrete dispersive operator with a localized observation term. As a consequence, we derive a relaxed observability inequality, which allows us to establish the ϕ(h)-null controllability of the space-discrete control system. Furthermore, we show that the controls obtained for the discrete system converge, as h tends to 0, to a control driving the associated continuous KdV control system to zero, thereby recovering the null controllability of the continuous system as well.
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BibTeX
@misc{kumar2026ϕnull,
title = {{ϕ-Null controllability of a semi-discrete linear KdV equation with localized interior control}},
author = {Manish Kumar and Rajib Dutta},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05768404v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - ϕ-Null controllability of a semi-discrete linear KdV equation with localized interior control
AU - Manish Kumar
AU - Rajib Dutta
PY - 2026
DA - 2026/09/29
PB - HAL
UR - https://hal.science/hal-05768404v1
AB - This article studies the ϕ(h)-null controllability of a space-discrete approximation of the linear Korteweg-de Vries (KdV) equation posed on a bounded domain, with a localized interior control. The notion of ϕ(h)-null controllability is a suitable analogue of null controllability in the context of discretized control systems. We employ a duality approach, reducing the controllability problem to an observability inequality for the associated discrete adjoint system. To this end, we first establish a suitable discrete Carleman estimate for the underlying discrete dispersive operator with a localized observation term. As a consequence, we derive a relaxed observability inequality, which allows us to establish the ϕ(h)-null controllability of the space-discrete control system. Furthermore, we show that the controls obtained for the discrete system converge, as h tends to 0, to a control driving the associated continuous KdV control system to zero, thereby recovering the null controllability of the continuous system as well.
ER -