Hoai-Minh Nguyen

We study the cost of fast controls for a linearized KdV system and a nonlinear KdV system locally, using right Neumann boundary control for non-critical lengths. Since the operator associated with the linearized system is neither self-adjoint nor skew-adjoint, its (known) spectral properties are not directly amenable to the moment method, leaving optimal cost bounds an open problem. We address this difficulty by shifting attention to a related KdV system and deriving the optimal bounds from the new one.

This paper is devoted to the stabilization of a linear control system y' = Ay + Bu and its suitable non-linear variants where ( A , 𝒟( A )) is an infinitesimal generator of a strongly continuous group in a Hilbert space ℍ, and B defined in a Hilbert space U is an admissible control operator with respect to the semigroup generated by A . Let λ ϵ ℝ and assume that, for some positive symmetric, invertible Q = Q( λ ) ϵ ℒ(ℍ), for some non-negative, symmetric R = R (λ) ϵ ℒ(ℍ), and for some non-negative, symmetric W = W (λ) ∈ ℒ(U), it holds AQ + QA ^* − BW B ^* + QRQ + 2 λQ = 0. We then present a new approach to study the stabilization of such a system and its suitable nonlinear variants. Both the stabilization using dynamic feedback controls and the stabilization using static feedback controls in a weak sense are investigated. To our knowledge, the nonlinear case is out of reach previously when B is unbounded for both types of stabilization.

We derive sharp bounds for the boundary control cost of the one-dimensional fractional Schrödinger and heat equations. The analysis of the lower bound is based on the study of the control cost of a related singular boundary control problem in finite time, using tools from complex analysis. The analysis of the upper bound relies on the moment method, involving estimates of the Fourier transform of a class of compactly supported functions.

We derive sharp bounds for the boundary control cost of the one-dimensional fractional Schrödinger and heat equations. The analysis of the lower bound is based on the study of the control cost of a related singular boundary control problem in finite time, using tools from complex analysis. The analysis of the upper bound relies on the moment method, involving estimates of the Fourier transform of a class of compactly supported functions.

We construct a static feedback control in a trajectory sense and a dynamic feedback control to obtain the local rapid boundary stabilization of a KdV system for noncritical lengths using Gramian operators. We also construct a time-varying feedback control in the trajectory sense and a time-varying dynamic feedback control to reach the local finite-time boundary stabilization for the same system.

We investigate the local boundary controllability of the Korteweg-de Vries (KdV) equation with right Neumann boundary controls at critical lengths. We show that the KdV system is not locally null-controllable in small time for all critical lengths for which the unreachable subspace of the linearized system has dimension at least two. This result extends the work of Coron, Koenig, and Nguyen, who established it for a subclass of these lengths. We also obtain a new controllability time for such systems for all but two critical lengths. It is worth noting that the latest results on the controllability time prior to this work date back to the work of Cerpa ( 2007) and Cerpa and Crépéau (2009).

We investigate the local boundary controllability of the Korteweg-de Vries (KdV) equation with right Neumann boundary controls at critical lengths. We show that the KdV system is not locally null-controllable in small time for all critical lengths for which the unreachable subspace of the linearized system has dimension at least two. This result extends the work of Coron, Koenig, and Nguyen, who established it for a subclass of these lengths. We also obtain a new controllability time for such systems for all but two critical lengths. It is worth noting that the latest results on the controllability time prior to this work date back to the work of Cerpa (2007) and Cerpa and Crepeau (2009).

In this paper, we introduce a spectral projection on the target system for the Fredholm backstepping transformation. This removes the main obstruction on the growth of eigenvalues for the convergence of different series involved in the definition of the Fredholm backstepping transformation, thus allowing for applications in higher dimensions.