Stabilization of infinite-dimensional semilinear second-order systems by position feedback

In this paper stabilization of infinite-dimensional undampedsemilinear second-order systems is considered in the case whereany velocity feedback is not available. For such a case parallelcompensators are effective. For the systems with collocatedinput and output operators the stabilizer is constructed bya P-controller for the augmented system which consists of thecontrolled system and a parallel compensator. The asymptoticstability of the closed-loop system is proved by LaSalle's invarianceprinciple under compactness of the resolvent.     

Low-gain adaptive stabilization of infinite-dimensional second-order systems

In this paper, low-gain adaptive stabilization of infinite-dimensional undamped second-order systems is considered in the case where the input and output operators are collocated. The systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low-gain adaptive PI controller (proportional plus integral controller). An energy-like function and a multiplier function are introduced and low-gain adaptive stabilization of the linear second-order systems is analyzed.     

Low‐gain adaptive stabilization of semilinear second‐order hyperbolic systems

In this paper low‐gain adaptive stabilization of undamped semilinear second‐order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low‐gain adaptive velocity feedback. The closed‐loop system is governed by a non‐linear evolution equation. First, the well‐posedness of the closed‐loop system is shown. Next, an energy‐like function and a multiplier function are introduced and the exponential stability of the closed‐loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd.     

Stabilization of infinite-dimensional undamped second order systems by using a parallel compensator

In this paper stabilization of infinite-dimensional undamped second-order systems is considered in the case where the input and output operators are collocated. The systems have an infinitenumber of poles and zeros on the imaginary axis. In the casewhere only position feedback is available, a parallel compensatoris effective. The stabilizer is constructed by a P-controllerfor the augmented system which consists of the controlled systemand a parallel compensator. The asymptotic stability of theaugmented system is proved by LaSalle's invariance principleunder compactness of the resolvent.     

Exponential stabilization of distributed semilinear systems by optimal control

This paper studies exponential stabilization of distributed semilinear systems. The paper (i) gives a constrained feedback control that ensure the exponential stabilizability and (ii) shows that this control is the unique solution of an appropriate minimization problem. Examples of hyperbolic partial equations are provided.     

Some second-order vibrating systems cannot tolerate small time delays in their damping

We show that certain infinite-dimensional damped second-order systems of linear differential equations become unstable when arbitrarily small time delays occur in the damping.     

Controllability of semilinear systems

The most general results known about controllability of semilinear systems are found in Refs. 1 and 2. The authors of these papers used different approaches, and no connection between them has ever been pointed out. The purpose of this paper is to present a general theorem under which these results easily follow.This paper is taken from a chapter of the author's doctoral thesis written at University of California, Los Angeles, California.     

On controllability of semilinear stochastic systems in Hilbert spaces

The Nussbaum fixed-point theorem together with conditions forapproximate controllability of linear systems are used to obtainsufficient conditions for approximate controllability of associatedsemilinear stochastic systems in Hilbert spaces.     

Internal stabilization of semilinear parabolic systems

This work is concerned with the internal stabilization of the steady-state solutions to semilinear parabolic systems via finite-dimensional feedback controllers. The internal controller is active on a nonempty open subset and in one equation only.     

Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function

Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchinger's control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.     

Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations

In this article, we investigate the application of pseudo‐transient‐continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, use the PTC‐methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction‐type PTC‐method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC ‐Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2005–2022, 2017     

Finite-time stabilization of state dependent impulsive dynamical linear systems

The problem of finite-time stabilizing control design for state-dependent impulsive dynamical linear systems (SD-IDLS) is tackled in this paper. Such systems are characterized by continuous-time, linear, possibly time-varying, dynamics coupled with discrete-time, linear, possibly time-varying, dynamics. The continuous-time part determines the system evolution in any time interval between two consecutive resetting events, while the discrete-time part governs its instantaneous state jump whenever the system trajectory intersects a resetting set, i.e. a region of the state space assumed to be time-independent. By making use of a quadratic control Lyapunov function, the finite-time stabilization of SD-IDLS through a static output feedback control design is specifically discussed in this paper. A sufficient and constructive result is provided based on the conical hulls of the resetting set subregions and on some cone copositivity properties of the chosen control Lyapunov function. Such a result is based on the solution of a feasibility problem that involves a set of coupled Difference/Differential Linear Matrix Inequalities (D/DLMI), which is shown to be less conservative and more numerically amenable with respect to other results available in the literature. An example illustrates the effectiveness of the proposed approach.     

Adaptive control of infinite-dimensional systems without parameter estimation: an overview

This paper contains an overview of adaptive control of infinite-dimensionalsystems without parameter estimation or identification. We describethe main problems and present the most comprehensive results.High-gain, low-gain and switching controllers are considered.The literature is discussed and a number of open problems areposed.     

On the linearization of infinite-dimensional random dynamical systems

We present a new version of the Grobman–Hartman's linearization theorem for random dynamics. Our result holds for infinite-dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on the nonlinear perturbations, we do not require for the linear part to be nonuniformly hyperbolic in the sense of Pesin but rather (besides requiring the existence of stable and unstable directions) allow for the existence of a third (central) direction on which we do not prescribe any behavior for the dynamics. Moreover, under some additional nonuniform growth condition, we prove that the conjugacies given by the linearization procedure are Hölder continuous when restricted to bounded subsets of the space.     

Adaptive stabilization of uncertain unified chaotic systems with nonlinear input

This paper addresses the problem of adaptive stabilization of uncertain unified chaotic systems with nonlinear input in the sector form. A novel representation of nonlinear input function, that is, a linear input with bounded time-varying coefficient, is firstly established. Then, an adaptive control scheme is proposed based on the new nonlinear input model. By using Barbalat’s lemma, the asymptotic stability of the closed-loop system is proved in spite of system uncertainties, external disturbance and input nonlinearity. One of the advantages of the proposed design method is that the prior knowledge on the plant parameter, the bound parameters of the uncertainties and the slope parameters inside the sector nonlinearity is not required. Finally, numerical simulations are performed to verify the analytical results.     

Robust stabilization for a class of uncertain dynamical systems with time delay

In this paper, a new and simple approach whereby we derive several sufficient conditions on robust stabilizability for a class of uncertain dynamical systems with time delay is presented. Some analytical methods and the Bellman-Gronwall inequality are employed to investigate these sufficient conditions. The notable features of the results obtained are their simplicity in testing the stability of uncertain dynamical systems with time delay and their clarity in giving insight into system analysis. Finally, several numerical examples are given to demonstrate the utilization of the results.The authors would like to acknowledge the many helpful comments provided by the reviewer. Particularly, in the light of these comments, the proof of Theorem 3.1 has been considerably shortened.     

Decentralized stabilization of a class of interconnected systems

Abstract. This paper is concerned with the decentralized stabilization of continuous and discretelinear interconnected systems with the structural constraints about the interconnection matri-ces. For the continuous case,the main improvement in the paper as compared with the corre-sponding results in the literature is to extend the considered class of systems from S to S“ (bothwill be defined in the paper) without resulting in high decentralized gain and difficult numericalcomputation. The algorithm for obtaining decentralized state feedback control to stable theoverall system is presented. The discrete case and some very useful results are discussed aswell.     

Uniqueness results for semilinear elliptic systems on

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form As an application we consider the nonlinear Schrödinger system for and exponents q which satisfy in case and in case . Generalizing the results of Wei and Yao for we find new sufficient conditions and necessary conditions on such that precisely one positive solution exists. Our results dealing with the special case are optimal. Finally, an application to a multi‐component nonlinear Schrödinger system is given.     

Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x∈H, a spectral decomposition is introduced, and for any function f:R→R, we define a corresponding vector-valued function f^H(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x∈H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.     

Robust stabilization of uncertain linear dynamical systems with time-varying delay

In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered. By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties. In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh ^–. Some analytical methods are employed to investigate these stability conditions. Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay. Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper.