Recent zbMATH articles in MSC 93D09https://zbmath.org/atom/cc/93D092021-06-15T18:09:00+00:00WerkzeugRobust stability and boundedness of stochastic differential equations with delay driven by \(G\)-Brownian motion.https://zbmath.org/1460.930772021-06-15T18:09:00+00:00"Ren, Yong"https://zbmath.org/authors/?q=ai:ren.yong"Sakthivel, Rathinasamy"https://zbmath.org/authors/?q=ai:sakthivel.rathinasamy"Sun, Guozheng"https://zbmath.org/authors/?q=ai:sun.guozhengIn this paper, sufficient conditions are given for the robust stability and boundedness of some stochastic differential equations with delay driven by $G$-Brownian motion.
Reviewer: Jin Liang (Shanghai)Optimization and control of dynamic systems. Foundations, main developments, examples and challenges.https://zbmath.org/1460.930022021-06-15T18:09:00+00:00"Górecki, Henryk"https://zbmath.org/authors/?q=ai:gorecki.henrykThis is a book about optimization and control of dynamic systems, which introduces many optimization and polyoptimization methods and contains many practical examples,
The book consists of 17 chapters: 1. Introduction; 2. Logics; 3. Some Fundamental Mathematical Models; 4. Fundamental Properties and Requirements of Control Systems; 5. Unconstrained Extrema of Functions; 6. Extrema Subject to Equality Constraints; 7. Extrema Subject to Equality and Inequality Constraints; 8. Parametric Optimization of Continuous Linear Dynamic Systems; 9. Elements of Variational Calculus; 10. Dynamic Optimization of Systems; 11. Maximum Principle; 12. Dynamic Programming; 13. Linear Quadratic Problems; 14. Optimization of Discrete-Continuous Hybrid Systems; 15. Elements of Multicriteria Optimization; 16. Mathematical Model of a Bicycle and Its Stability Analysis; 17. Concluding Remarks.
Chapters 1--4 present a classification of mathematical models of optimization, basic concepts of formal logic, basic mathematical models, and basic limitations in the analysis of dynamic systems respectively.
Chapters 5--7, 9 and 15 concern a basic and important problem of searching for a maximum or minimum (extremum, in general) of functions, where Chapter 5 is about the real functions in the interiors of their domains without any additional constrains, Chapter 6 is about real functions in some subsets of their domains, Chapter 7 is about the functions of several variables, Chapter 9 is about the functions of infinitely many variables, and Chapter 15 is about the polyoptimization problems, i.e, the problems of searching for extremes of vector functions of many variables.
Chapters 8, 10 and 11 address the parametric optimization of dynamic systems, the modern problems and methods of dynamic optimization respectively.
In Chapters 12 and 13, a general method of dynamic programming and an application to the optimization of linear non-stationary systems are described.
In Chapters 14 and 16, a chain system (that is, a ladder system) occurring in telecommunications and a mathematical model of the bike are well investigated.
From the Preface, we learn that this book is the outcome of a selection of the author's lectures for Ph.D. students in Electrical Engineering, Automation, Computer Science and Biomedical Engineering. I think it is an interesting book, and I would like to recommend it to students and researchers interested in optimization and control theory.
Reviewer: Jin Liang (Shanghai)On the superstability of an interval family of differential-algebraic equations.https://zbmath.org/1460.930782021-06-15T18:09:00+00:00"Shcheglova, A. A."https://zbmath.org/authors/?q=ai:shcheglova.alla-arkadevnaSummary: We consider an interval family of differential-algebraic equations (DAE) under assumptions that guarantee the coincidence of the structure of the general solution of each of the systems in this family with the structure of the general solution of the nominal system. The analysis is based on transforming the interval family of DAE to a form in which the differential and algebraic parts are separated. This transformation includes the inversion of an interval matrix. An estimate for the stability radius is found assuming the superstability of the differential subsystem of nominal DAE. Sufficient conditions for the robust stability are obtained based on the superstability condition for the differential part of the interval family.
Reviewer: Reviewer (Berlin)Mixed robustness: analysis of systems with uncertain deterministic and random parameters by the example of linear systems.https://zbmath.org/1460.930252021-06-15T18:09:00+00:00"Tremba, A. A."https://zbmath.org/authors/?q=ai:tremba.a-aSummary: The robustness of linear systems with constant coefficients is considered. There are methods and tools for analyzing the stability of systems with random or deterministic uncertainties. At the same time, there are no approaches to the analysis of systems containing both types of parametric uncertainty. The article classifies the types of robustness and introduces a new type -- ``mixed parametric robustness'' -- which includes several variations. The proposed statements of mixed robustness problems can be viewed as intermediate versions between the classical deterministic and probabilistic approaches to robustness. Several cases are listed in which the problems are easy to solve. In the general case, stability tests based on the scenario approach can be applied to robust systems; however, these tests can be computationally costly. A simple graph-analytical approach based on robust \(D \)-decomposition (robust \(D \)-partition) is proposed to calculate the desired stability probability. This method is suitable for the case of a small number of random parameters. The final stability probability estimate is calculated in a deterministic way and can be found with arbitrary precision. Approximate methods for solving the above problems are described. Examples and a generalization of mixed robustness to other types of systems are given.
Reviewer: Reviewer (Berlin)