Strong stabilizability and the steady state Riccati equation

We explore strong stabilizability (as opposed to exponential stabilizability) with the aid of the steady state Riccati equation. We show that the latter can have at most one strongly stable solution and obtain some sufficient conditions for existence. We also indicate an application to steady state Kalman filtering where the observation operator is compact so that we may not have exponential stability.Research supported in part under Grant No. 78-3550, AFOSR, USAF, Applied Math Division.     

Robust exponential stability and stabilizability of linear parameter dependent systems with delays

The robust exponential stability and stabilizability problems are addressed in this paper for a class of linear parameter dependent systems with interval time-varying and constant delays. In this paper, restrictions on the derivative of the time-varying delay is not required which allows the time-delay to be a fast time-varying function. Based on the Lyapunov-Krasovskii theory, we derive delay-dependent exponential stability and stabilizability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. Numerical examples are given to illustrate the effectiveness of our theoretical results.     

Stabilizability of a class of stochastic bilinear hybrid systems

The problem of the stabilizability of stochastic nonlinear hybrid systems with a Markovian or any switching rule is considered. Using the Lyapunov technique sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found. In particular, the asymptotic stabilizability in probability problem of stochastic bilinear hybrid systems with a Markovian or any switching rule is discussed and a closed-loop controller is found. Also the sufficient conditions for the exponential mean-square stabilizability for bilinear hybrid systems with any switching based on the Lie algebra approach are formulated and an open-loop controller is designed. The obtained results are illustrated by examples and simulations.     

Boundary Observability and Stabilization for Westervelt Type Wave Equations without Interior Damping

In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear acoustics. We prove local existence for undamped generalized Westervelt equations with homogeneous Dirichlet boundary conditions as well as global existence and exponential decay with absorbing type boundary conditions.     

Feedback stabilization of 2D Navier–Stokes equations with Navier slip boundary conditions

We study the local exponential stabilizability with internally distributed feedback controllers for the incompressible 2D-Navier–Stokes equations with Navier slip boundary conditions. These controllers are localized in a subdomain and take values in a finite-dimensional space.     

Exponential stabilizability and robustness analysis for discrete-time nonlinear systems

In this paper, we derive some results on the exponential stabilizability and robustness analysis for discrete-time nonlinear control systems. Using the discrete Gronwall’s inequality, we also derive an important absolute estimate for the robustness index of the controlled discrete-time nonlinear system.     

Qualitative properties and stabilizability of a model for blood thrombin formation

We perform a qualitative analysis of a differential equation that was originally introduced by Stortelder, Hemker and Hemker to model the formation of thrombin, and discuss issues of controllability and stabilizability. Results include a general proof of convergence to equilibrium, and of local exponential stabilizability.     

基于开关控制的线性奇异系统的二次状态反馈镇定问题

主要讨论基于开关控制的线性奇异系统的二次状态反馈镇定问题.利用二次反馈镇定的概念,给出了线性奇异系统基于异步开关控制的二次状态反馈镇定问题可解的两个充分条件.进一步,对于带有范数有界的不确定项的奇异线性系统,给出了其可以基于异步开关控制的二次状态反馈鲁棒镇定的可解性条件.     

On practical asymptotic stabilizability of switched affine systems

In this paper, we report some new results on practical asymptotic stabilizability of switched systems consisting of affine subsystems. We first briefly review some practical asymptotic stabilizability notions and some results from our previous papers. Then we propose a new approach to estimate the region of attraction for switched affine systems. Based on this new approach, we present several new sufficient conditions for the practical asymptotic stabilizability and global practical asymptotic stabilizability of such systems. Finally, a computational approach to check the new sufficient conditions is proposed and it is applied to several numerical examples.     

Boundary feedback stabilization of the Boussinesq system with mixed boundary conditions

We study the feedback stabilization of the Boussinesq system in a two dimensional domain, with mixed boundary conditions. After ascertaining the precise loss of regularity of the solution in such models, we prove first Green's formulas for functions belonging to weighted Sobolev spaces and then correctly define the underlying control system. This provides a rigorous mathematical framework for models studied in the engineering literature. We prove the stabilizability by extending to the linearized Boussinesq system a local Carleman estimate already established for the Oseen system. Then we determine a feedback control law able to stabilize the linearized system around the stationary solution, with any prescribed exponential decay rate, and able to stabilize locally the nonlinear system.     

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.     

Exponential Behaviour of Stochastic 2D Navier-Stokes Equations Driven by Levy Noise

In this paper, some results on the pathwise exponential stability are established for the weak solutions of stochastic 2D Navier-Stokes equation driven by noise. Also, some results and comments concerning the stabilizability and stabilization of these equations are stated.     

Hybrid discrete-continuous systems: I. stability and stabilizability

For hybrid discrete-continuous linear stationary systems, we consider the basic problems of qualitative control theory, namely, stability and stabilization. For such systems, we obtain parametric criteria for asymptotic and exponential stability ensuring a prescribed stability exponent and a rank criterion for stabilizability. We consider the problem of finding the minimum number of inputs for which the considered system is stabilizable. We suggest an effective algorithm for constructing the matrix describing the structure of the input device of a minimum-input stabilizable system. An example illustrating the results is given.     

The Exponential Behaviour and Stabilizability of Stochastic 2D-Navier-Stokes Equations

Some results on the pathwise exponential stability of the weak solutions to a stochastic 2D-Navier-Stokes equation are established. The first ones are proved as a consequence of the exponential mean square stability of the solutions. However, some of them are improved by avoiding the previous mean square stability in some more particular and restrictive situations. Also, some results and comments concerning the stabilizability and stabilization of these equations are stated.     

Stabilization of Euler- Bernoulli Beam by A Boundary Control

A flexible beam is considered. One end of the beam is free and another end is clamped under changeable angle, with angular velocity taken as control variable. A suitable stability criterion is introduced and the uniform exponential stabilizability is established for a linear feedback control applied at the clamped end.     

Discrete Ingham type inequalities and simultaneous observability of strings or beams

Harmonic analysis has been an efficient tool in control theory for a long time, see, e.g., Russell [D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev. 20 (1978) 639-739] and its numerous references. Here we establish discrete Ingham type and Haraux type inequalities for exponential sums satisfying a weakened gap condition. They enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams.     

Quadratic control for linear periodic systems

We propose a new quadratic control problem for linear periodic systems which can be finite or infinite dimensional. We consider both deterministic and stochastic cases. It is a generalization of average cost criterion, which is usually considered for time-invariant systems. We give sufficient conditions for the existence of periodic solutions.Under stabilizability and detectability conditions we show that the optimal control is given by a periodic feedback which involves the periodic solution of a Riccati equation. The optimal closed-loop system has a unique periodic solution which is globally exponentially asymptotically stable. In the stochastic case we also consider the quadratic problem under partial observation. Under two sets of stabilizability and detectability conditions we obtain the separation principle. The filter equation is not periodic, but we show that it can be effectively replaced by a periodic filter. The theory is illustrated by simple examples.This work was done while this author was a visiting professor at the Scuola Normale Superiore, Pisa.     

Feedback stabilization for perturbed linear systems *

In this paper, we consider the questions of stabilization of perturbed (or uncertain) linear systems in Hilbert space. Perturbations of the system operator represent the uncertainty in the modelling process and could be bounded or even unbounded. Sufficient conditions are presented that guarantee stabilizability of the perturbed system given that the nominal (unperturbed) system is stabilizable. In particular it is shown that for certain class of perturbations weak and strong stabilizability properties are preserved for the same state feedback control. Two numerical examples are given to illustrate our theory.     

Internal stabilization of standard sampled‐data systems

The paper presents necessary and sufficient conditions under which for a standard sampled‐data system there exists a digital controller, that ensures the internal stability of the system. For the case of stabilizability a general expression for the set of all stabilizing controllers is given. The stabilizability conditions are formulated as divisibility of certain polynomials, controllability and observability concepts are not used.     

Controllability and stabilization for linear systems of algebraic and differential equations

In this paper, we study several reachability and controllability definitions for linear, autonomous systems of degenerate ordinary differential equations. We try to identify the most interesting among them; we prove the connections with stabilizability and full stabilizability properties of the systems, which are also studied. In particular, we show the connection of the stabilizability property with the existence of special solutions of a suitable version of the Riccati equation.This paper was written under the auspices of GNAFA-CNR (Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche), Rome Italy.