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An optimal control problem is considered for a system described by a singular equation of parabolic type. The study bases on a special regularization method. We establish existence of a solution to the regularized problem, as well as the corresponding necessary optimality conditions. The results enable us to find an approximate solution to the original problem even in the absence of solvability. [ABSTRACT FROM AUTHOR]

The dynamic input-output model is well known in economic theory and practice. In this paper, the asymptotic stability and balanced growth solutions of the dynamic input-output system are considered. Under some natural assumptions which do not require the technical coefficient matrix to be indecomposable, it has been proved that the dynamic input-output system is not asymptotically stable and the closed dynamic input-output model has a balanced growth solution. [ABSTRACT FROM AUTHOR]

This work develops hybrid models for large-scale singular differential system and analyzes their asymptotic properties. To take into consideration the discrete shifts in regime across which the behavior of the corresponding dynamic systems is markedly different, our goals are to develop hybrid systems in which continuous dynamics are intertwined with discrete events under random-jump disturbances and to reduce complexity of large-scale singular systems via singularly perturbed Markov chains. To reduce the complexity of large-scale hybrid singular systems, two-time scale is used in the formulation. Under general assumptions, limit behavior of the underlying system is examined. Using weak convergence methods, it is shown that the systems can be approximated by limit systems in which the coefficients are averaged out with respect to the quasi-stationary distributions. Since the limit systems have fewer states, the complexity is much reduced. [ABSTRACT FROM AUTHOR]

Recently, Calvetti et al. have published an interesting paper [Linear Algebra Appl. 316 (2000) 157–169] concerning the least-squares solution of a singular system by using the so-called range restricted GMRES (RRGMRES) method. However, one of the main results (cf. [loc. cit., Theorem 3.3]) seems to be incomplete. As a complement of paper [loc. cit.], in this note we first make an example to show the incompleteness of that theorem, then we give a modified result. [Copyright &y& Elsevier]

In this paper, we study the Generalized Minimal Residual (GMRES) method for solving singular linear systems, particularly when the necessary and sufficient condition to obtain a Krylov solution is not satisfied. Thanks to some new results which may be applied in exact arithmetic or in finite precision, we analyze the convergence of GMRES and restarted GMRES. These formulas can also be used in the case when the systems are nonsingular. In particular, it allows us to understand what is often referred to as stagnation of the residual norm of GMRES. [ABSTRACT FROM AUTHOR]

Recently, a general numerical procedure has been developed for solvable systems of singular differential equations E(t)x′(t)+F(t)x(t)=f(t). This paper shows how to exploit the structure present in many control problems to reduce the computational effort substantially. An example is worked which shows that additional reductions are possible in some cases. [ABSTRACT FROM AUTHOR]

This paper investigates the event-triggered sliding mode control (SMC) problem for singular systems with disturbance. Firstly, an event-triggered sliding mode control law is designed to guarantee the reachability of sliding surface. Different from the related methods, in order to deal with the difficulty caused by event-triggered SMC strategy, a novel Lemma is proposed in this paper. Secondly, the admissibility of sliding motion is presented, which is used to solve the controller gain. Then, a positive lower bound of the inter execution time can be guaranteed and the Zeno behavior is avoided. Finally, two simulation examples are presented to show the effectiveness of derived theoretical results. [ABSTRACT FROM AUTHOR]

This paper mathematically explains how state derivative space (SDS) system form with state derivative related feedback can supplement standard state space system with state related feedback in control designs. Practically, inverse optimal control is attractive because it can construct a stable closed-loop system while optimal control may not have exact solution. Unlike the previous algorithms which mainly applied state feedback, in this paper inverse optimal control are carried out utilizing state derivative alone in SDS system. The effectiveness of proposed algorithms are verified by design examples of DC motor tracking control without tachometer and very challenging control problem of singular system with impulse mode. Feedback of direct measurement of state derivatives without integrations can simplify implementation and reduce cost. In addition, the proposed design methods in SDS system with state derivative feedback are analogous to those in state space system with state feedback. Furthermore, with state derivative feedback control in SDS system, wider range of problems such as singular system control can be handled effectively. These are main advantages of carrying out control designs in SDS system. [ABSTRACT FROM AUTHOR]

Synthesis of variable structure controller for discrete uncertain singular system with multiple time-delays was studied. Firstly, the system was decomposed into two low dimensional uncertain subsystems. Secondly, switching, function with difference compensator was designed to guarantee the asymptotically stable movement of the system in quasi-switching manifold. Thirdly, variable structure controller was designed, by which the solution trajectory of system was forced to arrive at the quasi-switching manifold in finite steps and ultimately vanished to zero, or entered into a small neighborhood of the quasi-switching manifold in finite steps and converged to a small chattering in the neighborhood. Finally, a numerical example was given to demonstrate effectiveness and simplicity of the design method. [ABSTRACT FROM AUTHOR]

This paper considers the problem of reachable set estimation for linear singular systems. The contribution of this paper lies in the following three aspects: • A new inequality scaling method is used to deal with Lyapunov functional with e index and the conservatism of the result is reduced. • Different from the Lyapunov functional construction in the comparison literatures, the candidate Lyapunov functional in this paper contains more time-delay information based on time-delay segmentation method. • A practical example shows the effectiveness of the proposed method. This paper investigates the problem of reachable set estimation for a general class of continuous-time singular linear systems with time-varying delay. By employing a new Lyapunov functional candidate, a criterion with the help of linear matrix inequalities (LMIs) is developed utilizing the new inequality scaling method, free-weighting matrix and the time-delay segmentation method. The reachable set of the underlying system is bounded by the intersection of ellipsoids. The effectiveness of our results are demonstrated by the examples. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of at least one smooth positive solution for a singular quasilinear elliptic system involving gradient terms. The approach combines the sub-supersolutions method and Schauder's fixed point theorem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of sliding mode observer design and observer-based sliding mode controller synthesis for polynomial fuzzy singular systems with time-delay. The polynomial fuzzy model is constructed for singular nonlinear system to eliminate the upper and lower bounds restriction of states in the system and reduce the calculation quantity. An integral-type sliding surface function, which has a stabilizing effect on the sliding variable, is proposed for the observer system. In terms of sum-of-squares approach, observer-based sliding mode passive controller is designed for the system with unknown membership functions, which drives the system trajectories onto the predefined sliding surface in a finite time, meanwhile, the stability and robust passivity of the closed-loop system are guaranteed. The information of membership functions is included in the stability conditions, which can reduce the conservatism. Finally, an example is performed to verify the merits of the method proposed. [ABSTRACT FROM AUTHOR]

This paper investigates the H ∞ control problems of singular fractional-order interval systems with commensurate order 0<α<1. Firstly, novel conditions for the bounded real lemma in the sense of H ∞ norm for nominal singular fractional-order systems are proposed, based on which the sufficient conditions for checking the admissibility and H ∞ performance of singular fractional-order interval systems are derived. Secondly, the state feedback H ∞ controller synthesis for singular fractional-order interval systems is addressed and the state feedback controller is designed. Thirdly, the dynamic output feedback H ∞ controller synthesis for singular fractional-order interval systems is addressed and the dynamic output feedback controller is designed. Finally, three illustrative examples are given to show the effectiveness of the proposed results., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2020 ISA. Published by Elsevier Ltd. All rights reserved.)