arXiv subjects
Publications and source records attributed to Hoai-Minh Nguyen.
We establish several variants of the multilinear multiplier theorem of Coifman and Meyer. We also present examples that are not covered by existing theories. Our motivation comes from applications to the definition of the Jacobian and Hessian determinant in the distributional sense.
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.
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.
This is a brief introduction to control theory in finite-dimensional spaces. The material is partly based on my lectures for the Master 1 program in Mathématiques et applications at Sorbonne University, delivered over the past few years. The aim is to provide a concise overview of the subject, primarily focusing on the linear setting. Proofs are presented in detail and are selected to allow for extensions to the infinite-dimensional case in many situations. Topics covered include the Kalman rank condition, the Hautus test, observability, stability, detectability and dynamic observers, the Pole Shifting Theorem, the linear test for controllability, linear-quadratic optimal control over finite and infinite horizons, and stabilization via Gramians.
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).
We propose a method to establish the rapid stabilization of the bilinear Schrödinger control system and its linearized system, and the finite time stabilization of the linearized system using the Grammian operators. The analysis of the rapid stabilization involves a new quantity (variable) which is inspired by the adjoint state in the optimal control theory and is proposed in our recent work on control systems associated with strongly continuous group. The analysis of the finite time stabilization follows the strategy introduced by Coron and Nguyen in the study of the finite time stabilization of the heat equation and incorporate a new ingredient involving the estimate of the cost of controls of the linearized system in small time derived in this paper.
We briefly discuss the contribution of Haïm Brezis and his co-authors on the characterizations of the Sobolev norms and the total variation using non-local functionals. Some ideas of the analysis are given and new results are presented.
This paper is devoted to the local null controllability of the Burgers control system $y_t - y_{xx} + y y_x = u(t)$ on a bounded interval imposed by the zero Dirichlet boundary condition. It is known from the work of Marbach that this control system is not locally null controllable in small time. In this paper, we prove that the system is not locally null controllable in finite time as well. Our approach is inspired by the works of Coron, Koenig, and Nguyen, and Nguyen on the controllability of the KdV system and is different from the one of Marbach.
We establish the turnpike property for linear quadratic control problems for which the control operator is admissible and may be unbounded, under quite general and natural assumptions. The turnpike property has been well studied for bounded control operators, based on the theory of differential and algebraic Riccati equations. For unbounded control operators, there are only few results, limited to some special cases of hyperbolic systems in dimension one or to analytic semigroups. Our analysis is inspired by the pioneering work of Porretta and Zuazua \cite{PZ13}. We start by approximating the admissible control operator with a sequence of bounded ones. We then prove the convergence of the approximate problems to the initial one in a suitable sense. Establishing this convergence is the core of the paper. It requires to revisit in some sense the linear quadratic optimal control theory with admissible control operators, in which the roles of energy and adjoint states, and the connection between infinite-horizon and finite-horizon optimal control problems with an appropriate final cost are investigated.
This paper is devoted to the exponential stability for one-dimensional linear wave equations with in-domain localized damping and several types of Wentzell (or dynamic) boundary conditions. In a quite general boundary setting, we establish the exponential decay of solutions towards the corresponding steady states. The results are obtained either by the multiplier method or spectral analysis in an $L^2$-functional framework, and then with input-to-state technics in an $L^p$-functional framework for $p \in (2,\infty)$.
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 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.
This paper is devoted to the stabilization of a linear control system $y' = A y + B u$ and its suitable non-linear variants where $(A, \cD(A))$ is an infinitesimal generator of a strongly continuous {\it group} in a Hilbert space $\mH$, and $B$ defined in a Hilbert space $\mU$ is an admissible control operator with respect to the semigroup generated by $A$. Let $λ\in \mR$ and assume that, for some {\it positive} symmetric, invertible $Q = Q(λ) \in \cL(\mH)$, for some {\it non-negative}, symmetric $R = R(λ) \in \cL(\mH)$, and for some {\it non-negative}, symmetric $W = W(λ) \in \cL(\mU)$, it holds $$ A Q + Q A^* - B W B^* + Q R Q + 2 λQ = 0. $$ 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 extend the range of parameters associated with the Gagliardo-Nirenberg interpolation inequalities in the fractional Coulomb-Sobolev spaces for radial functions. We also study the optimality of this newly extended range of parameters.
We establish the exponential decay of the solutions of the damped wave equations in one-dimensional space where the damping coefficient is a nowhere-vanishing function of space. The considered PDE is associated with several dynamic boundary conditions, also referred to as Wentzell/Ventzel boundary conditions in the literature. The analysis is based on the determination of appropriate Lyapunov functions and some further analysis. This result is associated with a regulation problem inspired by a real experiment with a proportional-integral control. Some numerical simulations and additional results on closed wave equations are also provided.
The Korteweg-de Vries (KdV) equation with the right Dirichlet control was initially investigated more than twenty years ago. It was shown that this system is small time, locally, exactly controllable for all non-critical lengths and its linearized system is not controllable for {\it all} critical lengths. Even though the controllability of the KdV system has been studied extensively in the last two decades, the local controllability of this system for critical lengths remains an open question. In this paper, we give a definitive answer to this question. First, we characterize all critical lengths and the corresponding unreachable space for the linearized system. In particular, we show that the unreachable space is always of dimension 1. Second, we prove that the KdV system with the right Dirichlet control is not locally null controllable in small time. Third, we give a criterion to determine whether the system is locally exactly controllable in finite time or {\it not} locally null controllable in any positive time for {\it all} critical lengths. Consequently, we show that there exist critical lengths such that the system is not locally null controllable in small time but is locally exactly controllable in finite time. These facts are surprising and distinct in comparison with related known results. First, it is known that the corresponding KdV system with the right zero Dirichlet is locally exactly controllable in small time using internal controls. Second, the unreachable space of the linearized system of the corresponding KdV system with the right Neumann control might be of arbitrary dimension. Third, the KdV system with the right Neumann control is locally exactly controllable in small time if the corresponding unreachable space is of dimension 1.
The transmission eigenvalue problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for a system without complementing conditions, i.e., the two equations of the system have the same coefficients for the second order terms, and thus being degenerate. These coefficients are allowed to be anisotropic and are assumed to be of class $C^2$. One of the keys of the analysis is to establish the well-posedness and the regularity in $L^p$-scale for such a system. As a result, we largely extend and rediscover known results for which the coefficients for the second order terms are required to be isotropic and of class $C^\infty$ using a new approach.
We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based on the study of the corresponding hyperbolic systems associated with the Riemann invariants. The key ingredient in the study of these systems is the use of the internal dissipation energy to estimate the difference of solutions with their mean values in an average sense.
We establish the full range Gagliardo-Nirenberg and the Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Sobolev-Coulomb spaces for the (fractional) derivative $0 \leq s \leq 1$. As a result, we rediscover known Gaglairdo-Nirenberg interpolation type inequalities associated with Sobolev-Coulomb spaces which were previously established in the scale of $H^{s}$ with $0 < s \leq 1$ and extend them for the full range $W^{s, p}$ with $0\leq s \leq 1$ and $1 < p < + \infty$. Using these newly established weighted inequalities, we derive a new family of one body Hardy-Lieb-Thirring inequalities and use it to establish a new family of many body Hardy-Lieb-Thirring inequalities with a strong repulsive interaction term in $L^p$ scale.