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Gengsheng Wang

Publications and source records attributed to Gengsheng Wang.

At least 19 recordsLinked to original sources
arXiv2026

Endpoint Asymptotics of Optimal Stabilization Prefactors

Consider the stabilizable finite-dimensional linear control system $\dot x=Ax+Bu$. For a prescribed decay rate $δ>0$, we define the optimal stabilization prefactor $$ \hat{C}(δ) := \inf_{K\in\mathbb{R}^{m\times n}} \sup_{t\ge0}e^{δt}\|e^{(A+BK)t}\|. $$ We determine its asymptotic behavior as $δ$ approaches the right endpoint of the achievable decay-rate range. If $(A,B)$ is controllable and $μ$ is its largest controllability index, then $\hat{C}(δ)\asympδ^{μ-1}$ as $δ\to+\infty$; whereas if $(A,B)$ is stabilizable but not controllable, then $\hat{C}(δ)\asymp(δ_*-δ)^{-(q-1)}$ as $δ\uparrowδ_*$, where $δ_*$ is the supremal achievable decay rate and $q$ is the largest size of a Jordan block of the uncontrollable part associated with the spectral boundary $\operatorname{Re}λ=-δ_*$. Thus, in both cases, the endpoint asymptotic order of $\hat{C}(δ)$ is determined by the corresponding structural invariant---$μ$ in the controllable case and $q$ in the noncontrollable case---and conversely this order recovers that invariant.

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arXiv2026

Controllable subspaces and real-part observability for Schrödinger equations on $\mathbb T^d$

We study internal controllability of Schrödinger equations on tori with controls acting only through their real parts. This real-part constraint leads to a real-linear control problem for which full null controllability may fail. We characterize the maximal null-controllable subspace and show that the uncontrollable directions are precisely given by the purely imaginary stationary modes. These results provide an explicit description of controllable and uncontrollable directions for Schrödinger equations with real-part controls. The characterization is obtained through observability inequalities with sharp stationary correction terms. For cylindrical open control regions, we prove such an estimate by a real-part compactness--uniqueness argument. For admissible measurable product control regions in the shifted free case, we establish the corresponding estimate for rough spacetime control sets. The proof combines complex-valued observability inputs with a double-spectrum analysis arising from the coupling of the forward and backward Schrödinger evolutions in the real-part observation.

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arXiv2026

Observability from measurable sets for strongly coupled parabolic systems via single-component observation

We establish an observability inequality from space-time measurable sets for a class of strongly coupled parabolic systems consisting of two equations, where the observation acts on a single-component. The model is motivated by parabolic equations with complex coefficients and serves as a prototypical example of strongly coupled systems. The main difficulty lies in the fact that, unlike in the scalar and weakly coupled cases, pointwise-in-time interpolation observability estimates fail, as the observed component may exhibit high-frequency oscillatory cancellations induced by the coupling. To overcome this difficulty, we develop a new integral-type interpolation observability inequality based on a Remez-type inequality. With the aid of this integral-type interpolation observability inequality and the strategy developed in [Phung and Wang, JEMS, (2013), 681--703] and [Apraiz, Escauriaza,Wang and Zhang, JEMS, (2014), 2433--2475] for deriving observability from measurable sets, we obtain the desired observability inequality.

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arXiv2025

Periodic propagation of singularities for heat equations with time delay

This paper presents two remarkable phenomena associated with the heat equation with a time delay: namely, the propagation of singularities and periodicity. These are manifested through a distinctive mode of propagation of singularities in the solutions. Precisely, the singularities of the solutions propagate periodically in a bidirectional fashion along the time axis. Furthermore, this propagation occurs in a stepwise manner. More specifically, when propagating in the positive time direction, the order of the joint derivatives of the solution increases by 2 for each period; conversely, when propagating in the reverse time direction, the order of the joint derivatives decreases by 2 per period. Additionally, we elucidate the way in which the initial data and historical values impact such a propagation of singularities. The phenomena we have discerned not only corroborate the pronounced differences between heat equations with and without time delay but also vividly illustrate the substantial divergence between the heat equation with a time delay and the wave equation, especially when viewed from the point of view of singularity propagation.

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arXiv2025

Quantitative observability for the Schrödinger equation with an anharmonic oscillator

This paper studies the observability inequalities for the Schrödinger equation associated with an anharmonic oscillator $H=-\frac{\d^2}{\d x^2}+|x|$. We build up the observability inequality over an arbitrarily short time interval $(0,T)$, with an explicit expression for the observation constant $C_{obs}$ in terms of $T$, for some observable set that has a different geometric structure compared to those discussed in \cite{HWW}. We obtain the sufficient conditions and the necessary conditions for observable sets, respectively. We also present counterexamples to demonstrate that half-lines are not observable sets, highlighting a major difference in the geometric properties of observable sets compared to those of Schrödinger operators $H=-\frac{\d^2}{\d x^2}+|x|^{2m}$ with $m\ge 1$. Our approach is based on the following ingredients: First, the use of an Ingham-type spectral inequality constructed in this paper; second, the adaptation of a quantitative unique compactness argument, inspired by the work of Bourgain-Burq-Zworski \cite{Bour13}; third, the application of the Szegö's limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for proving counterexamples of observability inequalities.

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arXiv2024

Observability inequality, log-type Hausdorff content and heat equations

This paper studies observability inequalities for heat equations on both bounded domains and the whole space $\mathbb{R}^d$. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension $s$ for any $s\in (d-1,d]$, but also for certain sets of Hausdorff dimension $d-1$. On the whole space $\mathbb{R}^d$, we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality. To obtain these observability inequalities, we use the adapted Lebeau-Robiano strategy from \cite{Duyckaerts2012resolvent}. For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: We establish a spectral inequality/a Logvinenko-Sereda uncertainty principle; we set up a quantitative propagation of smallness of analytic functions; we build up a Remez' inequality; and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate in Lubinsky \cite{Lubinsky1997small}, which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations.

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arXiv2024

Sampling Observability for Heat Equations with Memory

This paper studies the sampling observability for the heat equations with memory in the lower-order term, where the observation is conducted at a finite number of time instants and on a small open subset at each time instant. We present a two-sided sampling observability inequality and give a sharp sufficient condition to ensure the aforementioned inequality. We also provide a method to select the time instants and then to design the observation regions, based on a given memory kernel, such that the above-mentioned inequality holds for these time instants and observation regions. Additionally, we demonstrate that the positions of these time instants depend significantly on the memory kernel.

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arXiv2023

Observability for heat equations with time-dependent analytic memory

This paper presents a complete analysis of the observability property of heat equations with time-dependent real analytic memory kernels. More precisely, we characterize the geometry of the space-time measurable observation sets ensuring sharp observability inequalities, which are relevant both for control and inverse problems purposes. Despite the abundant literature on the observation of heat-like equations, existing methods do not apply to models involving memory terms. We present a new methodology and observation strategy, relying on the decomposition of the flow, the time-analyticity of solutions and the propagation of singularities. This allows us to obtain a sufficient and necessary geometric condition on the measurable observation sets for sharp two-sided observability inequalities. In addition, some applications to control and relevant open problems are presented.

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arXiv2022

Stabilizability of linear systems with discrete observation mode

For linear control systems, the usual state feedback stabilizability has two components: one is a continuous observation mode (i.e., to observe solutions continuously in time), and the other is a class of feedback laws (which is usually the space of all of the linear and bounded operators from a state space to a control space). This paper studies the stabilizability for abstract linear control systems, with a discrete observation mode (i.e., to observe solutions discretely in time) and two different classes of feedback laws. We first characterize these types of stabilizabilities via some weak observability inequalities for the dual systems. Then, we use these characterizations to reveal the connections between these types of stabilizabilities and those with continuous observation mode. Finally, we show some applications of the aforementioned weak observability inequalities.

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arXiv2022

Feedback law to stabilize linear infinite-dimensional systems

We design a new feedback law to stabilize a linear infinite-dimensional control system, where the state operator generates a C0-group and the control operator is unbounded. Our feedback law is based on the integration of a mutated Gramian operator-valued function. In the structure of the aforementioned mutated Gramian operator, we utilize the weak observability inequality in [21, 14] and borrow some idea used to construct generalized Gramian operators in [11, 23, 24]. Unlike most related works where the exact controllability is required, we only assume the above-mentioned weak observability inequality which is equivalent to the stabilizability of the system.

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arXiv2022

Characterizations of complete stabilizability

We present several characterizations, via some weak observability inequalities, on the complete stabilizability for a control system $[A,B]$, i.e., $y'(t)=Ay(t)+Bu(t)$, $t\geq 0$, where $A$ generates a $C_0$-semigroup on a Hilbert space $X$ and $B$ is a linear and bounded operator from another Hilbert space $U$ to $X$. We then extend the aforementioned characterizations in two directions: first, the control operator $B$ is unbounded; second, the control system is time-periodic. We also give some sufficient conditions, from the perspective of the spectral projections, to ensure the weak observability inequalities. As applications, we provide several examples, which are not null controllable, but can be verified, via the weak observability inequalities, to be completely stabilizable.

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arXiv2021

Flow decomposition for heat equations with memory

We build up a decomposition for the flow generated by the heat equation with a real analytic memory kernel. It consists of three components: The first one is of parabolic nature; the second one gathers the hyperbolic component of the dynamics, with null velocity of propagation; the last one exhibits a finite smoothing effect. This decomposition reveals the hybrid parabolic-hyperbolic nature of the flow and clearly illustrates the significant impact of the memory term on the parabolic behavior of the system in the absence of memory terms.

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arXiv2021

Unique continuation inequalities for the parabolic-elliptic chemotaxis system

This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain. This system is a simplified version of the chemotaxis model introduced by Keller and Segel. With the aid of priori L^infty-estimates (for solutions of the system) built up in this paper, we treat the semi-linear parabolic equation in the system as a linear parabolic equation, and then use the frequency function method and the localization technique to build up two unique continuation inequalities for the system. As a consequence of the above-mentioned two inequalities, we have the following qualitative unique continuation property: if one component of a solution vanishes in a nonempty open subset at some time T>0, then the solution is identically zero.

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arXiv2020

Switching properties of time optimal controls for systems of heat equations coupled by constant matrices

This paper studies the time optimal control problem for systems of heat equations coupled by a pair of constant matrices. The control constraint is of the ball-type, while the target is the origin of the state space. We obtain an upper bound for the number of switching points of the optimal control over each interval with a fixed length. Also, we prove that at each switching point, the optimal control jump from one direction to the reverse direction.

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arXiv2020

Characterizations of stabilizable sets for some parabolic equations in $\mathbb{R}^n$

We consider the parabolic type equation in $\mathbb{R}^n$: \begin{align}\label{equ-0} (\partial_t+H)y(t,x)=0,\,\,\, (t,x)\in (0,\infty)\times\mathbb{R}^n;\;\; \quad y(0,x)\in L^2(\mathbb{R}^n), \end{align} where $H$ can be one of the following operators: (i) a shifted fractional Laplacian; (ii) a shifted Hermite operator; (iii) the Schrödinger operator with some general potentials. We call a subset $E\subset \mathbb{R}^n$ as a stabilizable set for the above equation, if there is a linear bounded operator $K$ on $L^2(\mathbb{R}^n)$ so that the semigroup $\{e^{-t(H-χ_EK)}\}_{t\geq 0}$ is exponentially stable. (Here, $χ_E$ denotes the characteristic function of $E$, which is treated as a linear operator on $L^2(\mathbb{R}^n)$.) This paper presents different geometric characterizations of the stabilizable sets for the above equation with different $H$. In particular, when $H$ is a shifted fractional Laplacian, $E\subset \mathbb{R}^n$ is a stabilizable set if and only if $E\subset \mathbb{R}^n$ is a thick set, while when $H$ is a shifted Hermite operator, $E\subset \mathbb{R}^n$ is a stabilizable set for if and only if $E\subset \mathbb{R}^n$ is a set of positive measure. Our results, together with the results on the observable sets for the above equation obtained in \cite{AB,Ko,Li,M09}, reveal such phenomena: for some $H$, the class of stabilizable sets contains strictly the class of observable sets, while for some other $H$, the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for the above equation where $H$ is the Schrödinger operator with some general potentials.

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arXiv2020

Observable sets, potentials and Schrödinger equations

We characterize observable sets for 1-dim Schrödinger equations in $\mathbb{R}$: $i \partial_t u = (-\partial_x^2+x^{2m})u$ (with $m\in \mathbb{N}:=\{0,1,\dots\}$). More precisely, we obtain what follows: First, when $m=0$, $E\subset\mathbb{R}$ is an observable set at some time if and only if it is thick, namely, there is $γ>0$ and $L>0$ so that $$ \left|E \bigcap [x, x+ L]\right|\geq γL\;\;\mbox{for each}\;\;x\in \mathbb{R}; $$ Second, when $m=1$ ($m\geq 2$ resp.), $E$ is an observable set at some time (at any time resp. ) if and only if it is weakly thick, namely $$ \varliminf_{x \rightarrow +\infty} \frac{|E\bigcap [-x, x]|}{x} >0. $$ From these, we see how potentials $x^{2m}$ affect the observability (including the geometric structures of observable sets and the minimal observable time). Besides, we obtain several supplemental theorems for the above results, in particular, we find that a half line is an observable set at time $T>0$ for the above equation with $m=1$ if and only if $T>\fracπ{2}$.

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arXiv2019

On switching properties of time optimal controls for linear ODEs

In this paper, we present some properties of time optimal controls for linear ODEs with the ball-type control constraint. More precisely, for an optimal control, we build up an upper bound for the number of its switching points; show that it jumps from one direction to the reverse direction at each switching point; give its dynamic behaviour between two consecutive switching points; and study its switching directions.

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