Article
Mathematics
Xiaogang Liu, Kiran Naseem Aslam, Muhammad Shoaib Saleem, Shuili Ren
Summary: In this article, we introduce a general class of convex functions and prove some basic properties. We establish Hermite-Hadamard type inequalities and fractional versions of Hermite-Hadamard type inequalities using the Riemann-Liouville integral operator. Furthermore, we provide applications to special means of real numbers. It is observed from the remarks in this paper that several significant results of ligature can be immediately obtained from our results by selecting suitable involved parameters.
JOURNAL OF MATHEMATICS
(2023)
Article
Computer Science, Artificial Intelligence
Cheng-De Zheng, Zeda Zhang, Yu Lu, Huaguang Zhang
Summary: This paper discusses the stochastic stability of genetic regulatory networks (GRNs) with semi-Markov switching and time-varying delays. By introducing Legendre polynomials and weighted Legendre polynomials, integral inequalities are derived and Lyapunov-Krasovskii functionals are established. Sufficient conditions for stochastically asymptotic stability are proposed using free-weight matrices and the acquired integral inequalities.
NEURAL COMPUTING & APPLICATIONS
(2022)
Article
Mechanics
Sandor Bozoki, Gabor Domokos, Florian Kovacs, Krisztina Regos
Summary: The study focuses on the monostatic property of convex polyhedra, establishing upper and lower bounds for the minimum number of faces and vertices, and improving the lower limit on mono-unstable vertices through an algorithm.
INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES
(2022)
Article
Mathematics, Interdisciplinary Applications
Lei Xu, Shuhong Yu, Tingsong Du
Summary: This study firstly defines the generalized n-polynomial convex mappings as a generalization of convex mappings and explores their properties. It then establishes two Hermite-Hadamard's-type integral inequalities in the frame of fractal space, as well as presents improved integral inequalities for mappings with first-order derivatives in absolute value belonging to the generalized n-polynomial convexity.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
Article
Mathematics, Applied
Muhammad Samraiz, Kanwal Saeed, Saima Naheed, Gauhar Rahman, Kamsing Nonlaopon
Summary: This paper studies a new class of integral inequalities using a strong type of convexity called n-polynomial exponential type s-convex function. These inequalities are established by utilizing the Holder's inequality, which has various applications in optimization theory. Some existing results are obtained from newly explored consequences, and some novel limits for specific means of positive real numbers are shown as applications.
Article
Mathematics
Kristina Krulic Himmelreich, Josip Pecaric, Dora Pokaz, Marjan Praljak
Summary: This paper extends Hardy's type inequalities to convex functions of higher order and provides upper bounds for the generalized Hardy's inequality, along with some applications.
Article
Automation & Control Systems
Shaoxin Sun, Xin Dai, Ruipeng Xi, Yuliang Cai, Xiangpeng Xie, Chunhua Zhang
Summary: This work focuses on studying the problems of stochastic finite-time boundedness (SFTB) and reachable set estimation with input-output finite-time mean square stabilization (IO-FTMSS) for multiple state delayed semi-Markovian jump models subject to uncertainties, stochastic process as well as input constraint. A nonlinear passive control mechanism is designed and utilized to achieve a closed-loop model. The problems of SFTB and IO-FTMSS are then solved through the stochastic Lyapunov function. The stability conditions are provided using linear matrix inequalities and the controller gain matrices can be derived. The feasibility of the proposed strategy is demonstrated through simulation.
INTERNATIONAL JOURNAL OF CONTROL AUTOMATION AND SYSTEMS
(2022)
Article
Mathematics
Kristina Krulic Himmelreich
Summary: In this paper, we use Taylor's formula to prove new Hardy-type inequalities involving convex functions. We introduce new results related to the Hardy-Hilbert inequality, Polya-Knopp inequality, and bounds for the identity related to the Hardy-type functional. Finally, we present mean value theorems of Cauchy type.
MATHEMATICA SLOVACA
(2022)
Article
Computer Science, Interdisciplinary Applications
Mohammad Fiuzy, Saeed Shamaghdari
Summary: This paper presents a H infinity proportional-integral-derivative (PID) control mechanism for structural uncertain fractional order linear systems with convex polytopic and two-norm bounded uncertainties subject to input saturation. The stability analysis and stabilization of the system are investigated using the Gronwall-Bellman lemma and the sector condition of the saturation function. The main strategy is to design a fractional order PID controller under input saturation problem using a non-iterative strategy based on the LMI. The validity and superiority of the proposed method are shown through a numerical example.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Interdisciplinary Applications
Sofia Ramzan, Muhammad Uzair Awan, Silvestru Sever Dragomir, Bandar Bin-Mohsin, Muhammad Aslam Noor
Summary: This paper presents a novel parameterized fractional integral identity and derives a series of fractional variants of certain classical inequalities by using this result and the convexity property of the mapping. It also explores applications of the main outcomes to special means of real numbers and proposes a new numerical scheme for solving non-linear equations, demonstrating its application in numerical analysis.
FRACTAL AND FRACTIONAL
(2023)
Article
Automation & Control Systems
Simon Michalowsky, Carsten Scherer, Christian Ebenbauer
Summary: This research focuses on analysing and designing gradient-based discrete-time optimization algorithms for unconstrained optimization problems with strongly convex objective functions and Lipschitz continuous gradients. By formulating the problem as a robustness analysis issue and utilizing a adaptation of the theory of integral quadratic constraints, a framework has been established to analyze convergence rates and robustness properties of existing algorithms, as well as to design novel robust optimization algorithms with specified guarantees and the ability to exploit additional structure in the objective function.
INTERNATIONAL JOURNAL OF CONTROL
(2021)
Article
Automation & Control Systems
Xiaokai Zhai, Huiling Xu
Summary: This paper addresses the robust stabilisation problem of spatially interconnected systems with uncertainties represented by linear fractional transformations. By solving semidefinite programs, necessary and sufficient conditions for robust stabilising controller existence are derived, and a numerically tractable algorithm is proposed.
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
(2021)
Editorial Material
Automation & Control Systems
Roberto Kawakami Harrop Galvao, Marcelo Carvalho Minhoto Teixeira, Tomasz Szulc, Edvaldo Assuncao, Marco Antonio Leite Beteto
Summary: This note discusses conditions for a set of non-singular matrices, such that any convex combination of these matrices is also non-singular. It points out that the conditions provided in a previous paper are only necessary conditions and may not be sufficient in general. New sufficient conditions are established based on stability results using Linear Matrix Inequalities (LMIs) for a class of fractional order systems. Numerical examples suggest that the new LMI conditions may be less conservative compared to existing tests.
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
(2022)
Article
Automation & Control Systems
Eduardo S. Tognetti, Tassio M. Linhares
Summary: This paper investigates the design of locally stabilizing dynamic output feedback controllers and estimates the domain of attraction for discrete-time Takagi-Sugeno (T-S) fuzzy systems. Considering the saturation effect on the control signal and the incomplete measurement of premise variables, the fuzzy output controller can have a different number of fuzzy rules and membership functions from the T-S model. To obtain local stabilizable conditions, a new approach is proposed by modeling the variation rate of membership functions without using upper bounds. The design conditions are expressed as linear matrix inequality relaxations based on fuzzy Lyapunov functions using slack variables introduced by Finsler's lemma. Numerical examples demonstrate the effectiveness of the proposed approach.
INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE
(2023)
Article
Automation & Control Systems
Abhishek Dhyani, Tushar Jain
Summary: A convex optimization-based iterative algorithm is proposed in this paper to synthesize a minimal-norm state-feedback globally non-overshooting/undershooting (NOUS) tracking controller, and an upper bound on the achievable settling time is provided.
IFAC JOURNAL OF SYSTEMS AND CONTROL
(2022)
Article
Mathematics, Applied
Milan Korda, Didier Henrion, Igor Mezic
Summary: The paper introduces a convex-optimization-based framework for computing invariant measures of polynomial dynamical systems and Markov processes by approximating an infinite-dimensional linear program, leading to the reconstruction of measures including approximating measure support and constructing weakly converging absolutely continuous approximations. The framework also provides a method to certify the nonexistence of an invariant measure and can be adapted to compute eigenmeasures of the Perron-Frobenius operator, serving as a generalization of ergodic optimization method.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Operations Research & Management Science
Yohann De Castro, Fabrice Gamboa, Didier Henrion, Jean Bernard Lasserre
Summary: This short note discusses the application of the Christoffel-Darboux polynomial in approximation theory and data science, as well as its role in the semi-algebraic D-optimal experimental design problem in statistics. The article uses elementary notions of convex analysis and mentions geometric interpretations and algorithmic consequences.
OPTIMIZATION LETTERS
(2021)
Article
Computer Science, Theory & Methods
Matteo Tacchi, Tillmann Weisser, Jean Bernard Lasserre, Didier Henrion
Summary: This study presents a systematic deterministic numerical scheme for approximating the volume of basic semi-algebraic sets by exploiting a correlative sparsity pattern, leading to significantly reduced problem sizes and potential for parallel computations.
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2022)
Article
Computer Science, Information Systems
Jakub Marecek, Stathis Maroulis, Vana Kalogeraki, Dimitrios Gunopulos
Summary: Monitoring of streamed data using low-rank techniques is crucial for detecting abnormal behavior in applications such as Internet of Things. The proposed algorithm successfully recovers the subspace representation and performs event detection, showing promising results in the experimental evaluation using induction-loop data from Dublin, Ireland.
Article
Computer Science, Information Systems
Ramen Ghosh, Jakub Marecek, Wynita M. Griggs, Matheus Souza, Robert N. Shorten
Summary: This article discusses the design of distributed algorithms for social sensing platforms, with a focus on fairness among contributing agents. It introduces iterated function systems as a tool for designing and analyzing such systems, which can deliver predictable quality of service and operate efficiently. The case study of a network of parked vehicles demonstrates the effectiveness of the system and the predictability of agent access to the platform.
IEEE INTERNET OF THINGS JOURNAL
(2022)
Article
Mathematics
Didier Henrion, Jean Bernard Lasserre
Summary: This study investigates a class of moment problems and proposes a method based on linear measurements, namely the first degree moments, to recover the complete information of a measure supported on the graph of a function. The results show that by solving a series of semidefinite relaxations, all the moments can be obtained. The function can be recovered using the optimal solution sequence.
CONSTRUCTIVE APPROXIMATION
(2022)
Article
Automation & Control Systems
Marianne Souaiby, Aneel Tanwani, Didier Henrion
Summary: This article establishes the existence of Lyapunov functions and describes algorithms for their numerical computation. By constructing cone-copositive Lyapunov functions, the stability analysis of state-constrained systems is addressed. The article proves that exponentially stable complementarity systems always admit a continuously differentiable cone-copositive Lyapunov function, and further investigates the approximation methods for this function.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
(2022)
Article
Robotics
Pavel Trutman, Mohab Safey El Din, Didier Henrion, Tomas Pajdla
Summary: This study introduces a globally optimal solution to the IK problem for a 7DOF manipulator with revolute joints and a polynomial objective function, demonstrating that kinematic constraints due to rotations can be generated by second-degree polynomials. The method shows a high success rate in practice.
IEEE ROBOTICS AND AUTOMATION LETTERS
(2022)
Article
Automation & Control Systems
Jakub Marecek, Michal Roubalik, Ramen Ghosh, Robert N. Shorten, Fabian R. Wirth
Summary: In power systems, it is important to regulate the aggregate demand of distributed energy resources (DERs) for predictability and fairness. Traditional controllers, such as the proportional-integral (PI) controller, cannot guarantee this. However, incrementally input-to-state stable (iISS) controllers can guarantee predictability and fairness even considering the non-linearity of the alternating-current model.
Article
Computer Science, Theory & Methods
Matteo Tacchi, Jean Bernard Lasserre, Didier Henrion
Summary: The article discusses the problem of computing the Lebesgue volume of compact basic semi-algebraic sets and presents an improved linear programming method that uses pseudo-moments and polynomials to approximate the volume. The research finds that the convergence speed of this method is significantly accelerated when the set is the smooth super-level set of a single polynomial.
DISCRETE & COMPUTATIONAL GEOMETRY
(2022)
Article
Automation & Control Systems
Marianne Souaiby, Aneel Tanwani, Didier Henrion
Summary: In this study, we investigate the time evolution of a probability measure for a class of state-constrained dynamical systems described by evolution variational inequalities. Unlike smooth ordinary differential equations, the evolution of this probability measure is not necessarily invertible due to the nonsmooth nature of the differential inclusion. Instead, we approximate the original nonsmooth system using Lipschitz approximation and construct a sequence of measures obtained from Liouville equations. This sequence converges to the measure describing the evolution of the distribution of states for the original nonsmooth system, allowing us to numerically approximate the evolution of moments.
Article
Automation & Control Systems
Vyacheslav Kungurtsev, Jakub Marecek, Ramen Ghosh, Robert Shorten
Summary: In many sharing-economy applications and other conventional applications, it is desirable to regulate the behavior of an ensemble of agents while guaranteeing both the regulation of the ensemble as a whole and the revenue or quality of service of each agent. Previous research has established guarantees of unique ergodicity in the presence of linear filters. In this study, we extend these guarantees to systems that include non-linear elements such as non-linear filters.
Article
Automation & Control Systems
Ekaterina Dudkina, Michelangelo Bin, Jane Breen, Emanuele Crisostomi, Pietro Ferraro, Steve Kirkland, Jakub Marecek, Roderick Murray-Smith, Thomas Parisini, Lewi Stone, Serife Yilmaz, Robert Shorten
Summary: This paper reviews classic methods for node ranking and compares their performance in a benchmark network that considers the community-based structure of society. The outcome of the ranking procedure is then used to decide which individuals should be tested, and possibly quarantined, first. Finally, the extension of these ranking methods to weighted graphs is explored, and the importance of weights in a contact network is investigated through a toy model and comparison of node rankings in the context of disease spread.
INTERNATIONAL JOURNAL OF CONTROL
(2023)
Article
Multidisciplinary Sciences
Quan Zhou, Jakub Marecek, Robert Shorten, Ashwani Kumar
Summary: This article proposes a new market-clearing mechanism for two-sided markets, which promotes income equality per hour worked across different subgroups and within each subgroup. It introduces the concept of subgroup fairness (Inter-fairness) combined with other fairness notions (Intra-fairness) and customer utility (Customer-Care) in the market-clearing problem. The study shows that the non-convex problem can be approximated efficiently using semidefinite programming, allowing for the implementation of the market-clearing mechanism.
Article
Computer Science, Artificial Intelligence
Quan Zhou, Jakub Marecek, Robert Shorten
Summary: In machine learning, the behaviour of multiple subgroups of an underlying human population can be captured through training data. However, under-representation bias arises when the training data for the subgroups are not carefully controlled. To address this, two notions of fairness are introduced for timeseries forecasting problems: subgroup fairness and instantaneous fairness. These notions extend predictive parity to the learning of dynamical systems. Globally convergent methods for the fairness-constrained learning problems are also demonstrated using hierarchies of convexifications of non-commutative polynomial optimization problems. The run time of these methods can be significantly reduced by exploiting sparsity in the convexifications, as demonstrated through empirical results on biased data sets.
JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
(2023)