In this article, we consider the rolling (or development) of two Riemannian connected manifolds $(M,g)$ and $(\hat{M},\hat{g})$ of dimensions $2$ and $3$ respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of \cite{MortadaKokkonenChitour}, which modelled the general setting of the rolling of two Riemannian connected manifolds with different dimensions as a driftless control affine system on a fibered space $Q$, with an emphasis on understanding the local structure of the rolling orbits, i.e., the reachable sets in $Q$. In this paper, the state space $Q$ has dimension eight and we show that the possible dimensions of non open rolling orbits belong to the set $\{2,5,6,7\}$. We describe the structures of orbits of dimension $2$, the possible local structures of rolling orbits of dimension $5$ and some of dimension $7$.
In this paper, we prove an approximate controllability result for the linearized Boussinesq system around a fluid at rest, in a two dimensional channel, when the control acts only on the temperature, through the upper boundary.
This paper deals with strong structural controllability of structured networks. A structured network is a family of structured systems (called node systems) that are interconnected by means of a structured interconnection law. The node systems and their structured interconnection law are given by pattern matrices. It is shown that a structured network is strongly structurally controllable if and only if an associated structured system is. This structured system will in general have a very large state space dimension, and therefore existing tests for verifying strong structural controllability …
Exact controllability of two-dimensional hydroelastic waves
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）
We describe a new method which allows us to obtain a result of exact controllability to trajectories of multidimensional conservation laws in the context of entropy solutions and under a mere non-degeneracy assumption on the flux and a natural geometric condition.
In this paper, we study approximate and exact controllability of the linear difference equation $x(t) = \sum\_{j=1}^N A\_j x(t - Λ\_j) + B u(t)$ in $L^2$, with $x(t) \in \mathbb C^d$ and $u(t) \in \mathbb C^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable matrix coefficients. When $Λ\_1, \dotsc, Λ\_N$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllabili…
We consider the null-controllability problem for the generalized Baouendi-Grushin equation $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_ωu$ on a rectangular domain. Sharp controllability results already exist when the control domain $ω$ is a vertical strip, or when $q(x) = x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $ω$. In some geometries for $ω$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability. O…
Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions
研究了取值于可分实希尔伯特空间的线性随机控制系统的近似能控性。第一步证明了对偶线性倒向随机微分方程解的存在唯一性。该方程的特殊性在于，除了一个无界算子作用于解的Y分量外，还有另一个无界算子作用于解的Z分量。借助于这个对偶方程，我们推导出了近似能控性与能观性之间的对偶性。最后，在作用于向前方程状态过程的无界算子是指数稳定半群的无穷小生成元的假设下，证明了广义Hautus检验为近似能控性提供了一个必要条件.本文将Buckdahn，Quincampoix and Tessitore（2006）和Goraac（2007）的结果从有限维推广到无限维。
In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the corresponding Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev.
The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal $2$-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the $2$-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all six linearly independent directions in the configuration space; every direction and orientation can be achieved by op…
This paper is devoted to study exact controllability of two one-dimensional coupled wave equations with first-order coupling terms with coefficients depending on space and time. We give a necessary and sufficient condition for both exact controllability in high frequency in the general case and the unique continuation in the cascade case.
In this paper, we prove the local-controllability to positive constant trajectories of a nonlinear system of two coupled ODE equations, posed in the one-dimensional spatial setting, with nonlocal spatial nonlinearites, and using only one localized control with a moving support. The model we deal with is derived from the well-known nonlinear reaction-diffusion Gray-Scott model when the diffusion coefficient of the first chemical species $d_u$ tends to $0$ and the diffusion coefficient of the second chemical species ${d_v}$ tends to $+ \infty$. The strategy of the proof consists in two main step…
This letter examines the controllability of consensus dynamics on matrix-weighed networks from a graph-theoretic perspective. Unlike the scalar-weighted networks, the rank of weight matrix introduces additional intricacies into characterizing the dimension of controllable subspace for such networks. Specifically, we investigate how the definiteness of weight matrices influences the dimension of the controllable subspace. In this direction, graph-theoretic characterizations of the lower and upper bounds on the dimension of the controllable subspace are provided by employing, respectively, dista…
