Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set (FAS) if T\F contains no cycles (directed). A collection F of FASs (with repetition allowed) is called an FAS packing if each arc e is used at most w(e) times by the members of F. The purpose of this paper is to give a characterization of all tournaments T=(V,A) with the property that, for every nonnegative integral weight function w defined on A, the minimum total weight of a cycle is equal to the maximum size of an FAS packing.
Publication: Mathematics of Operations Research, Published Online: 23 Feb, 2023, https://doi.org/10.1287/moor.2023.1352
Author: Xujin Chen Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Email: xchen@amss.ac.cn Guoli Ding Mathematics Department, Louisiana State University, Baton Rouge, LA 70803, USA Wenan Zang Department of Mathematics, The University of Hong Kong, Hong Kong, China Qiulan Zhao Department of Mathematics, Nanjing University, Nanjing 210093, China
Input design is an important problem for system identification and has been well studied for the classical system identification, i.e., the maximum likelihood/prediction error method. For the emerging regularized system identification, the study on input design has just started, and it is often formulated as a non-convex optimization problem minimizing a scalar measure of the Bayesian mean squared error matrix subject to certain constraints. Among the state-of-art input design techniques for regularized system identification is the so-called quadratic mapping and inverse embedding (QMIE) method. Based on the quadratic mapping between the input and its covariance, the QMIE method is first to obtain the optimal autocovariance by solving a transformed convex optimization problem and then to find all the inputs corresponding to the optimal autocovariance by the time domain inverse embedding (TDIE). In this paper, we report some new results on the embeddings/inverse embeddings of the QMIE method. Firstly, we present a general result on the frequency domain inverse embedding (FDIE) that is to find the inverse of the quadratic mapping described by the discrete-time Fourier transform. Then we show the relation between the TDIE and the FDIE from a graph signal processing perspective. Finally, motivated by this perspective, we further propose a graph induced embedding and its inverse, which include the previously introduced embeddings as special cases. This deepens our understanding of input design from a broader perspective beyond the time domain and frequency domain viewpoints.
Publication: Automatica, Volume 147, January 2023, 110673
Author: Biqiang Mu Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Email: bqmu@amss.ac.cn Tianshi Chen School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen 518172, China He Kong Shenzhen Key Laboratory of Biomimetic Robotics and Intelligent Systems, Department of Mechanical and Energy Engineering, Southern University of Science and Technology, Shenzhen, 518055, China Guangdong Provincial Key Laboratory of Human-Augmentation and Rehabilitation Robotics in Universities, Southern University of Science and Technology, Shenzhen, 518055, China Bo Jiang Key Laboratory for NSLSCS of Jiangsu Province, School of Mathematical Sciences, Nanjing Normal University, Nanjing, China Lei Wang College of Control Science and Engineering, Zhejiang University, Hangzhou, China Junfeng Wu School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen 518172, China College of Control Science and Engineering, Zhejiang University, Hangzhou, China
In this article, we study the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. We explore the relationship between the large N limit of the constant in the logarithmic Sobolev inequality (LSI) for the N-particle system and the presence or absence of phase transitions for the mean field limit. We show that the non-degeneracy of the LSI constant implies uniform-in-time propagation of chaos and Gaussianity of the fluctuations at equilibrium. As byproducts of our analysis, we provide concise and, to our knowledge, new proofs of a generalised form of Talagrand’s inequality and of quantitative propagation of chaos by employing techniques from the theory of gradient flows, specifically the Riemannian calculus on the space of probability measures.
Publication: Communications in Mathematical Physics (2023). https://doi.org/10.1007/s00220-023-04659-z
Author: Matías G. Delgadino Department of Mathematics, The University of Texas at Austin, Austin, USA Rishabh S. Gvalani Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany Grigorios A. Pavliotis Department of Mathematics, Imperial College London, London, UK Scott A. Smith Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, China Email: ssmith@amss.ac.cn
We establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier--Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most -1/2-\kappa for any \kappa>0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions.Our result applies to any divergence free initial condition in L^{2}\cup B^{-1+\kappa}_{\infty,\infty}, \kappa>0, and implies also non-uniqueness in law.
Publication: Archive for Rational Mechanics and Analysis volume 247, Article number: 46 (2023)
Author: Martina Hofmanová Fakult？t für Mathematik, Universit？t Bielefeld Rongchan Zhu Department of Mathematics, Beijing Institute of Technology XiangchanZhu Academy of Mathematics and Systems Science, Chinese Academy of Sciences Email: zhuxiangchan@126.com
We are concerned with the three-dimensional incompressible Navier–Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in $L^{2}$ we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in particular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.
Publication: The Annals of Probability, 51(2): 524-579 (March 2023). DOI: 10.1214/22-AOP1607
Author: Martina Hofmanová Fakult？t für Mathematik, Universit？t Bielefeld Rongchan Zhu Department of Mathematics, Beijing Institute of Technology XiangchanZhu Academy of Mathematics and Systems Science, Chinese Academy of Sciences Email: zhuxiangchan@126.com
This paper develops a singularity-free adaptive tracking control scheme for a general class of multi-input and multi-output uncertain discrete-time nonlinear systems with non-canonical control gain matrices. The estimation of the control gain matrices, especially in some non-canonical forms, may be singular during parameter adaptation, which leads to the singularity problems of the adaptive control laws. This paper employs the matrix decomposition technique to solve the problem under a linearly parameterized adaptive control framework. The state and output feedback cases are addressed, respectively, to ensure closed-loop stability and asymptotic output tracking. Compared with the existing results, the features of the proposed adaptive control scheme include: (i) the proposed control laws do not involve the high-gain issue commonly encountered in robust control methods; (ii) two different filtered tracking error signals are introduced for the state and output feedback cases, respectively. These filters are crucial to avoid causality contradiction of the adaptive control laws commonly encountered in adaptive control of discrete-time systems; and (iii) a future time signal estimation-based adaptive control law is developed to ensure asymptotic output tracking for the output feedback case without requiring the high-gain observer. Finally, an illustrative example is given to verify the validity of the proposed control scheme.
Publication: Automatica, Volume 153, July 2023, 111054
Author: Yuchun Xu Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Mathematics Sciences, University of Chinese Academy of Sciences, Bejing 100149, China Yanjun Zhang School of Automation, Beijing Institute of Technology, Beijing 100081, China Ji-Feng Zhang Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Mathematics Sciences, University of Chinese Academy of Sciences, Bejing 100149, China Email: jif@iss.ac.cn
This paper is concerned with the adaptive consensus tracking control problem for strict-feedback nonlinear multiagent systems with parameter uncertainty under both fixed and switching topologies. When the topology is fixed, we first propose a novel distributed fusion least-squares algorithm without regressor filtering, which has a clear advantage that the estimate of each agent converges to the true parameter value under a weak cooperative persistent excitation condition. Then, we design a new adaptive consensus tracking control law to guarantee that each agent can asymptotically track the reference trajectory. After that, we generalize the corresponding results to the switching topology case. Finally, two examples are given to demonstrate the theoretical results. Publication: SIAM Journal on Control and Optimization, Vol. 60, No. 5 (2022), 10.1137/21M1419763
Author: Ying Wang Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Wuquan Li School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China Ji-Feng Zhang Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Email: jif@iss.ac.cn
In this paper, we are concerned with the optimal control problems for a class of systems with fast-slow processes. The problem under consideration is to minimize a functional subject to a system described by a two-time scaled McKean--Vlasov stochastic differential equation whose coefficients depend on state components and their probability distributions. Firstly, we establish the existence and uniqueness of the invariant probability measure for the fast process. Then, by using the relaxed control representation and the martingale method, we prove the weak convergence of the slow process and control process in the original problem, and we obtain an associated limit problem in which the coefficients are determined by the average of those of the original problem with respect to the invariant probability measure. Finally, by establishing the nearly optimal control of the limit problem, we obtain the near optimality of the original problem. Publication: SIAM Journal on Control and Optimization, Vol. 60, Iss. 5 (2022), 10.1137/21M1466177
Author: Yun Li Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing 100149, China Fuke Wu School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R. China Ji-Feng Zhang Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing 100149, China Email: jif@iss.ac.cn
This paper is devoted to establishing global Carleman estimates for refined stochastic beam equations. First, by establishing a fundamental weighted identity, two Carleman estimates are derived with different weight functions for the refined stochastic beam equation, which is a coupled system consisting of a stochastic ordinary differential equation and a stochastic partial differential equation. As applications of these Carleman estimates, the exact controllability of the refined system is proved by the least controls in some sense. Different from classical stochastic beam equations, the refined one is exactly controllable at any time. Meanwhile, the uniqueness of an inverse source problem for refined stochastic beam equations is obtained without any requirement on the initial and terminal data. Publication: SIAM Journal on Control and Optimization, Vol. 60, Iss. 5(2022), 10.1137/21M1463513
Author: Yongyi Yu Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Ji-Feng Zhang Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing 100149, China Email: jif@iss.ac.cn
In this paper, we consider optimization problems over closed embedded submanifolds of \mathbb{R}^n, which are defined by the constraints c(x)=0. We propose a class of constraint dissolving approaches for these Riemannian optimization problems. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint dissolving function named CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and ？ojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint dissolving approaches.
Publication: Mathematics of Operations Research, Published Online: 8 Mar 2023, https://doi.org/10.1287/moor.2023.1360
Author: Nachuan Xiao The Institute of Operations Research and Analytics, National University of Singapore, Singapore Xin Liu State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China Email: liuxin@lsec.cc.ac.cn Kim-Chuan Toh Department of Mathematics, and Institute of Operations Research and Analytics, National University of Singapore, Singapore
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