Rakesh, Paul Sacks
We prove stability for a coefficient determination problem for a two velocity 2x2 system of hyperbolic PDEs in one space dimension.
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We solve the problem of stabilization of a class of linear first-order hyperbolic systems featuring n rightward convecting transport PDEs and m leftward convecting transport PDEs. Using the backstepping approach yields solutions to stabilization in minimal time and observer based output feedback.
It is shown that some aspects of classic KPP-theory of 1937 can be applied to a number of higher-order dispersion, hyperbolic, and other equations.
Smoothing effect and Fredholm property for first-order hyperbolic PDEs
本文研究了一阶双曲型PDEs的平滑效应与Fredholm性质，构造了周期性问题的正则化器并证明其由索引为零的Fredholm算子建模。
Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We radically disprove the obvious conjecture extending this fact to arbitrary singularities: namely, we present examples of diffusion at all non-simple singularity classes of generic wavefronts in odd-dimensional spaces, which are not reducible to diffusion at simple singular points.
In this article we deal with stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasise is on analysing the effect of multiplicative Lévy noise to such problems and establishing wellposedness by developing a suitable weak entropy solution framework. The proof of existence is based on the vanishing viscosity technique. The uniqueness is settled by interpreting Kruzkov's doubling technique in the presence noise.
We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions) of strictly hyperbolic PDE's with constant coefficients in $R^N$ at the parabolic singular points of their wavefronts. These points form the next difficult family of classes of the natural classification of singular points after the so-called simple singularities, studied previously. Also we promote a computer program counting for topologically different morsifications of critical points of smooth functions, and hence also for local components of the complement of a gener…
In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial differential equations. Our analysis leads to new integral equations of non-convolution type, which can be solved by classical methods. We derive solutions of these integral equations, which in turn lead to solutions of the associated Cauchy problems.
Introduction to inverse problems for hyperbolic PDEs
本文介绍了求解波方程逆系数问题的两种主要方法：边界控制法和几何光学方法，重点阐述了边界控制法的原理与应用。
We study the Dirichlet problem for first order hyperbolic quasi-linear functional PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. While the question of existence of a solution has already been answered in its predecessor, the present paper discusses the uniqueness and continuous dependence on the coefficients of the PDE. Under the assumption that the functional dependence is causal, we are able to derive a contraction principle which is the key to proof uniqueness and continuous dependence. Such a causal functional d…
We study stability aspects for the determination of space and time-dependent lower order perturbations of the wave operator in three space dimensions with point sources. The problems under consideration here are formally determined and we establish Lipschitz stability results for these problems. The main tool in our analysis is a modified version of Bukgheim-Klibanov method based on Carleman estimates.
We prove stability for a formally determined inverse problem for a hyperbolic PDE where the coefficients depend on space and time variables. The hyperbolic operator has constant wave speed and we study the recovery of zeroth order and first order coefficients and the space dimension can be one or higher. We use a modification of the Bukhgeim-Klibanov method to obtain our results.
Here we study the Dirichlet problem for first order linear and quasi-linear hyperbolic PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. By means of a Lyapunov function we show the existence of a unique solution in the space of functions of bounded variation and its continuous dependence on all the data of the linear problem. Finally, we conclude the existence of a solution to the quasi-linear case by utilizing the Schauder fixed point theorem. This type of problems considered here appears in applications such as trans…
Output-feedback stabilization of a class of n+m linear hyperbolic ODE-PDE-ODE systems
针对边界耦合 ODE 的 n+m 个异向线性双曲 PDE 系统，论文结合 backstepping 与时滞方法设计状态观测器和状态反馈，进而得到反同位测量下的输出反馈镇定控制律。
This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations on nonsmooth domains. In particular, we study the smoothness in the specific scale $\ B^r_{τ,τ}, \ \frac{1}τ=\frac{r}{d}+\frac{1}{p}\ $ of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.
Homogeneous Dirichlet problem for degenerate parabolic-hyperbolic PDE driven by Levy noise
本文研究了由 Levy 噪声驱动的退化抛物-双曲型 PDE 的齐次 Dirichlet 问题，基于 Kružkov 半熵公式发展了熵解的适定性理论，并采用更简单的方法获得了全局 Kato 不等式。
Intelligent acceleration adaptive control of linear $2\times2$ hyperbolic PDE systems
针对线性$2\ imes2$双曲型PDE系统，提出基于DeepONet的智能加速自适应控制，离线训练后在线快速生成反步核，避免数值求解，实现高达三个数量级的加速并保证稳定性。
Stochastic Solutions for Hyperbolic PDE
针对双曲型 PDE 缺乏随机表示理论的问题，本文提出了一种双曲型 PDE 解的随机表示方法。
We study a modified version of an initial-boundary value problem describing the formation of colony patterns of bacteria \textit{Escherichia Coli}. The original system of three parabolic equations was studied numerically and analytically and gave insights into the underlying mechanisms of chemotaxis. We focus here on the parabolic-elliptic-parabolic approximation and the hyperbolic-elliptic-parabolic limiting system which describes the case of pure chemotactic movement without random diffusion. We first construct local-in-time solutions for the parabolic-elliptic-parabolic system. Then we prov…
State constrained convex Nash equilibrium problems coupled with linear hyperbolic PDEs
研究与线性双曲型PDE耦合的状态约束凸GNEP的均衡存在性，克服集值约束映射正则性问题，给出一阶最优性条件，展示其在波动方程等应用中的相关性。
Ensembles of Hyperbolic PDEs: Stabilization by Backstepping
本文将耦合线性双曲型PDE的backstepping方法从有限集合推广至状态为函数值的无限连续系综，设计了具有空间与系综两维度的指数镇定标量控制律，并将核结果推广至连续参数化的Goursat形式PDE。
Fluorescent Optical Tomography (FOT) is a new bio-medical imaging method with wider industrial applications. It is currently intensely researched since it is very precise and with no side effects for humans, as it uses non-ionising red and infrared light. Mathematically, FOT can be modelled as an inverse parameter identification problem, associated with a coupled elliptic system with Robin boundary conditions. Herein we utilise novel methods of Calculus of Variations in $L^\infty$ to lay the mathematical foundations of FOT which we pose as a PDE-constrained minimisation problem in $L^p$ and $L…
We propose to solve polynomial hyperbolic partial differential equations (PDEs) with convex optimization. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures. The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems. This result is th…
We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the Dirichlet-to-Neumann map measured at a single boundary point, or integrated against a fixed measure. This result is valid even when the Dirichlet data is only given on a small subset of the boundary. We also give related uniqueness results on Riemannian manifolds.
Neumann-Neumann Waveform Relaxation Method for Hyperbolic PDE with Time Delay
针对带时滞双曲偏微分方程，提出Neumann-Neumann波形松弛方法，通过傅里叶分析和拉普拉斯变换证明收敛性，并给出数值示例。
Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions
本文证明了高阶偏微分方程的双曲近似在存在光滑解时的收敛性，并提供了数值验证。
Hyperbolic partial differential equations with complex characteristics on Fourier Lebesgue spaces
本文研究了具有复特征的双曲偏微分方程在傅里叶勒贝格空间上的良好性质，通过调和分析方法证明了相关傅里叶积分算子的有界性，并在特定条件下获得了新结果。
Hybrid inverse problems are based on the interplay of two types of waves, in order to allow for imaging with both high resolution and high contrast. The inversion procedure often consists of two steps: first, internal measurements involving the unknown parameters and some related quantities are obtained, and, second, the unknown parameters have to be reconstructed from the internal data. The reconstruction in the second step requires the solutions of certain PDE to satisfy some non-zero constraints, such as the absence of nodal or critical points, or a non-vanishing Jacobian. In this work, we …
Output-feedback stabilization of an underactuated network of N interconnected n + m hyperbolic PDE systems
针对仅单端驱动、测量位于另一端的N个n+m阶异向双曲PDE子系统链式欠驱动网络，提出含新型积分变换的输出反馈镇定方法，结合子系统性质递归设计控制器与观测器，经仿真验证有效。
An inverse hyperbolic obstacle problem
该研究聚焦双曲型反障碍问题，建立了其Hölder稳定性，探究利用障碍物外围区域边界的两次测量值重构障碍物边界未知函数的问题。
We detail in this article the necessity of a change of paradigm for the delay-robust control of systems composed of two linear first order hyperbolic equations. One must go back to the classical trade-off between convergence rate and delay-robustness. More precisely, we prove that, for systems with strong reflections, canceling the reflection at the actuated boundary will yield zero delay-robustness. Indeed, for such systems, using a backstepping-controller, the corresponding target system should preserve a small amount of this reflection to ensure robustness to a small delay in the loop. This…
Motivated by engineering applications of subsea installation by deepwater construction vessels in oil drilling, and of aid delivery by unmanned aerial vehicles in disaster relief, we develop output-feedback boundary control of heterodirectional coupled hyperbolic PDEs sandwiched between two ODEs, where the measurement is the output state of one ODE and suffers a time delay. After rewriting the time-delay dynamics as a transport PDE of which the left boundary connects with the sandwiched system, a state observer is built to estimate the states of the overall system of ODE-heterodirectional coup…
Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and "damping-type" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. A…
In this article we establish an Adam's Inequality in the Hyperbolic space. As an application we will also prove the asymptotic behaviour of the best constants in the Sobolev inequality and also discuss the solvability of Q curvature type PDE's in the Hyperbolic space.
Small-Gain Stability Analysis of Hyperbolic-Parabolic PDE Loops
本文通过小增益稳定性分析，研究了一维空间中双曲-抛物型PDE环路在空间上范数下的稳定性，针对两种典型情况提出了ISS和指数稳定性的充分条件。
The Evolution of Pointwise Statistics in Hyperbolic Equations with Random Data
本文研究了超几何方程中随机数据的点统计演进，推导了线性和非线性方程的统计演化方程，为随机动态的高效评估提供了方法。
Backstepping Control of Multidimensional Coupled First-Order Hyperbolic PDEs with Collinear Velocities
针对共线速度场的多维耦合一阶双曲系统，通过特征曲线变换将系统转化为连续的一维系统族，设计反步控制器实现有限时间镇定。
Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE
本文针对快双曲PDE耦合慢ODE的多时间尺度系统，基于奇异摄动与谱分析方法证明，当降阶系统和边界层系统均指数稳定时，小参数下全系统指数稳定，改进了需更强假设的Lyapunov方法结果。
Moving-Horizon Estimators for Hyperbolic and Parabolic PDEs in 1-D
针对PDE观测器计算繁重的问题，提出基于反步法的PDE滚动时域估计器，将难求解观测器PDE转换为可显式求解形式，提供双曲与抛物型PDE的显式公式并验证其收敛性。
Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit and their respective rate functions lead to the analysis of the skeleton equation; a degenerate parabolic-hyperbolic PDE with irregular drift. We develop a robust well-posedness theory for such PDEs in energy-critical spaces based on concepts of renormalized solutions and the equation's kinetic form. We establish these properties by proving that renormalized solutions are equivalent to classical weak solutions, extending concepts of [DiPerna, Lions; Ann. Math., 1989], […
On rate of convergence of finite difference scheme for degenerate parabolic-hyperbolic PDE with Levy noise
本文针对带Lévy噪声的退化抛物-双曲型PDE，提出半离散有限差分格式，通过有界变差估计和Kružkov变量加倍变体，证明熵解与近似解$L^1$差期望以$(Δx)^{1/7}$速率收敛。
Robust stabilization of $2 \times 2$ first-order hyperbolic PDEs with uncertain input delay
针对带不确定边界输入时滞的$2\ imes2$一阶双曲PDE系统，提出含仿射Volterra变换的新型反步补偿器，通过特征线法等解决适定性难题，证明对小时滞变化鲁棒，数值算例验证了性能。
We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is based on the construction of global viscosity solutions for the Hamilton-Jacobi equation $H(x,d_xu)=h$. We prove that these solutions are fixed points of the associated Lax-Oleinik semigroup. The presented results can also be viewed as a new application of Marchal's theorem, whose main use in recent literature has been to prove the existence of periodic orbits.
An Analysis of Constraint-Relaxation in PDE-Based Inverse Problems
针对PDE约束优化逆问题的非线性挑战，本文提出其松弛表述的通用分析与离散化框架，证明松弛表述可简化为加权非线性最小二乘问题，权重矩阵为PDE解的Gram矩阵且部分可从测量估计，为统一数据驱动降阶模型提供途径。
Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay
本文将 DeepONet 算子学习扩展到带状态与输入/输出时滞的一阶双曲型 PIDE 反步控制，证明增益算子可任意精度近似并保证闭环、观测器及输出反馈稳定，仿真显示计算量降低两个数量级。
Modeling physical phenomena like heat transport and diffusion is crucially dependent on the numerical solution of partial differential equations (PDEs). A PDE solver finds the solution given coefficients and a boundary condition, whereas an inverse PDE solver goes the opposite way and reconstructs these inputs from an existing solution. In this article, we investigate techniques for solving inverse PDE problems using a gradient-based methodology. Conventional PDE solvers based on the finite element method require a domain meshing step that can be fragile and costly. Grid-free Monte Carlo metho…
Neural Inverse Operators for Solving PDE Inverse Problems
针对算子到函数形式的 PDE 逆问题，提出由 DeepONets 与 FNO 组合而成的 Neural Inverse Operators，实验表明其更准确稳健且速度快几个数量级。
Backstepping Neural Operators for $2\times 2$ Hyperbolic PDEs
本文首次将 DeepONet 用于逼近 $2\ imes 2$ 双曲型耦合 PDE 的 Goursat 型增益核 PDE 系统，证明了逼近的存在性与连续性，并给出基于近似增益核的输出反馈设计，实现半全局实际指数稳定且大幅加速增益计算。
A uniqueness result in the inverse problem for an inhomogeneous hyperbolic system on a real vector bundle over a smooth compact manifold, based on energy measurements for improperly known sources, is established.
Inverse problems for infinite-dimensional transport PDEs on Wasserstein space
本文提出了一种在Wasserstein空间上处理无穷维传输PDE反问题的新框架，通过高阶微积分和变分方法，首次实现了从价值函数数据中识别成本函数和相互作用核。
Fredholm Alternative for Periodic-Dirichlet Problems for Linear Hyperbolic Systems
本文研究了线性双曲系统在周期-狄利克雷边界条件下的Fredholm替代定理，通过构建两个尺度的巴纳赫空间来建模问题。
Sampled-Data Boundary Feedback Control of 1-D Hyperbolic PDEs with Non-Local Terms
本文研究了1-D线性一阶双曲型方程组非局部项的边界反馈控制，采用零阶保持器设计，证明了采样周期足够小可保证闭环指数稳定性，且采样周期越小收敛速度越快。
Analysis and Control of a Non-Standard Hyperbolic PDE Traffic Flow Model
本文研究了非标准双曲型一维非线性交通流模型，证明了其全局解的存在性与唯一性，并提出基于边界反馈的全局稳定控制方法。
Motivated by wave or Dirac equations on noncommutative deformations of Minkowski space, linear integro-differential equations of the form $(D+λW)f=0$ are studied, where $D$ is a normal or prenormal hyperbolic differential operator on ${\mathbb R}^n$, $λ\in\mathbb C$ is a coupling constant, and $W$ is a regular integral operator with compactly supported kernel. In particular, $W$ can be non-local in time, so that a Hamiltonian formulation is not possible. It is shown that for sufficiently small $|λ|$, the hyperbolic character of $D$ is essentially preserved. Unique advanced/retarded fundamental…
On inverse problems for several coupled PDE systems arising in mathematical biology
本文研究数学生物学中耦合PDE系统的反问题，通过测量部分边界平均通量数据识别未知系数，提出基于单调性性质的新方案，通过控制源项注入实现唯一可识别性，并联系实际生物学应用。
Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs
针对PDE数据同化与唯一延拓反问题，论文建立抽象框架并利用条件稳定性证明PINN泛化误差估计，辅以四类线性PDE示例和数值实验验证。
Delay-Adaptive Control of First-order Hyperbolic PIDEs
针对具有未知输入时滞的一阶双曲型偏积分微分方程，提出一种基于输运PDE表示和无穷维backstepping的时滞自适应控制方法，实现全局稳定并通过仿真验证。
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