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Set-Valued Maps and Inclusion Problems inModular Metric Spaces
Published in Dhananjay Gopal, Praveen Agarwal, Poom Kumam, Metric Structures and Fixed Point Theory, 2021
As we have evolved into the realm of set-valued maps, the operator equation is also generalized. The inclusion problem Ax ∋ y is then brought into attention. This form of problem arises extensively in optimal control as in the differential inclusion problem, and is central in game theory, where the Kakutani fixed point theorem is central.
Polyhedral optimization of discrete and partial differential inclusions of parabolic type
Published in Optimization, 2023
This paper is concerned with polyhedral optimization problems for parabolic-type discrete and differential inclusions in a significant class of partial differential inclusions in optimization theory. We consider the optimization of the discrete polyhedral problem, with new discrete results which are crucial in determining optimality conditions for the partial derivatives of the inclusion. In the general case, the discrete problem is reduced to solving a convex programming problem in a higher-order space by using the specificity of the polyhedrality of the problem posed on minimization of a function for the intersection of a finite number of the set. Our idea is to make a discrete-approximate method for optimal control problems using differential inclusions, which addresses both numerical and qualitative features of optimization theory. We formulate sufficient conditions of optimality for the first mixed boundary value problems for partial differential inclusions of the parabolic type. The main tools of our investigations are based on the extremal principle and its modifications together with generalized differential calculus. Finally, the numerical example is demonstrated to verify the fundamental theory.
A sliding mode estimation method for fluid flow fields using a differential inclusions-based analysis
Published in International Journal of Control, 2021
Krishna Bhavithavya Kidambi, William MacKunis, Sergey V. Drakunov, Vladimir Golubev
Many researchers have paved the path for analysing the theory of differential inclusions for various engineering applications: Robotics – (Paden & Sastry, 1987); Hybrid systems – (Goebel, Sanfelice, & Teel, 2009); Under water Vehicle – (Fischer, Hughes, Walters, Schwartz, & Dixon, 2014); Switched nonlinear systems - (Teel, Nešić, Lee, & Tan, 2016); Neural networks – (Matusik, Nowakowski, Plaskacz, & Rogowski, 2018; Shi et al., 2019; Wang, Shi, Huang, Zhong, & Zhang, 2018); Chua circuits (Shi et al., 2018). While the theory of differential inclusions has been widely investigated to analyse the behaviour of discontinuous systems, there remains a need for new theoretical tools to rigorously analyse the performance of discontinuous SMO.
Observer for differential inclusion systems with incremental quadratic constraints
Published in International Journal of Systems Science, 2020
Lin Yang, Jun Huang, Min Zhang, Ming Yang
Researchers hope to establish a more accurate mathematical model for the practical control system, so differential inclusion (DI) has gradually become one of the research hotspots in the academic community. As a kind of simple DI systems, linear DI systems firstly attracted widespread attention in the field of control. T. Hu and Lin (2003) proposed the concept of a convex hull Lyapunov function and applied it to linear DI systems (T. Hu, 2007). Using the theory of T. Hu and Lin (2003), Cai, Liu, Huang et al. have addressed the control problem for linear DI systems with bounded inputs (2009), linear time-delay DI systems (Liu et al., 2010), and linear stochastic DI systems (Huang et al., 2010). As we know, Lur'e systems were considered as a simple type of nonlinear systems, and the study of nonlinear DI systems started from Lur'e DI systems naturally. Lur'e DI systems are widely used in practical engineering, such as mechanical systems with friction (Mihajlovic et al., 2006), neural network with discontinuous activation (Forti & Nistri, 2003), discontinuous dynamical systems using the concept of generalised solutions (Van der Schaft & Schumacher, 1996) and so on. A fundamental problem in system synthesis is determining the state of a system from its measurable input and output. One solution to this problem is to design an observer to recover the original state (Che et al., 2020; Huang et al., 2019, 2020). Many scholars have made remarkable achievements in the investigation on observer design for Lur'e DI systems. Osorio and Moreno presented the observer for Lur'e DI systems whose set-valued mappings were upper semi-continuous, dissipative, bounded, convex and closed (2006). In Doris et al. (2008), the observer of Lur'e DI systems was designed under the condition that the set-valued mappings conformed to be upper semi-continuous, bounded, convex, monotone and closed. In Brogliato and Heemels (2009), the set-valued mappings were supposed to be maximal monotone, and an appropriate observer, as well as its well posedness, was given. Huang et al. (2011) constructed the reduced-order observer for the Lur'e DI system by decomposition method. While Tanwani et al. (2014) addressed the observer design problem for Lur'e DI systems when the set-valued mappings were neither time-invariant nor monotone. Huang et al. (2017) studied the observer design approach for singular Lur'e DI systems. Recently, the interesting works on Lur'e DI systems can be found in Guiver et al. (2019) and Le (2019).