DOCSLIB.ORG

Adding Integral Action for Open-Loop Exponentially Stable Semigroups and Application to Boundary Control of PDE Systems A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Adding Integral Action for Open-Loop Exponentially Stable Semigroups and Application to Boundary Control of PDE Systems A 1 Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems A. Terrand-Jeanne, V. Andrieu, V. Dos Santos Martins, C.-Z. Xu Abstract—The paper deals with output feedback stabilization of boundary regulation is considered for a general class of of exponentially stable systems by an integral controller. We hyperbolic PDE systems in Section III. The proof of the propose appropriate Lyapunov functionals to prove exponential theorem obtained in the context of hyperbolic systems is given stability of the closed-loop system. An example of parabolic PDE (partial differential equation) systems and an example of in Section IV. hyperbolic systems are worked out to show how exponentially This paper is an extended version of the paper presented in stabilizing integral controllers are designed. The proof is based [16]. Compare to this preliminary version all proofs are given on a novel Lyapunov functional construction which employs the and moreover more general classes of hyperbolic systems are forwarding techniques. considered. Notation: subscripts t, s, tt, ::: denote the first or second derivative w.r.t. the variable t or s. For an integer n, I I. INTRODUCTION dn is the identity matrix in Rn×n. Given an operator A over a ∗ The use of integral action to achieve output regulation and Hilbert space, A denotes the adjoint operator. Dn is the set cancel constant disturbances for infinite dimensional systems of diagonal matrix in Rn×n. has been initiated by S. Pojohlainen in [12]. It has been extended in a series of papers by the same author (see [13] II. GENERAL ABSTRACT CAUCHY PROBLEMS for instance) and some other (see [23]) always considering A. Problem description bounded control operator and following a spectral approach Let X be a Hilbert space with scalar product h; i and A : (see also [11]). X D(A) ⊂ X ! X be the infinitesimal generator of a C - In the last two decades, Lyapunov approaches have allowed 0 semigroup denoted t 7! eAt. Let B and C be linear operators, to consider a large class of boundary control problems (see for B from m to X and C from D(C) ⊆ X to m. instance [2]). In this work our aim is to follow a Lyapunov R R In this section, we consider the controlled Cauchy problem approach to solve an output regulation problem. The results with output Σ(A; B; C) in Kalman form, as follows are separated into two parts. In a first part, abstract Cauchy problems are considered. 't = A' + Bu + w ; y = C'; (1) It is shown how a Lyapunov functional can be constructed where w 2 X is an unknown constant vector and u : for a linear system in closed loop with an integral controller 7! m is the controlled input. We consider the following when some bounds are assumed on the control operator and for R+ R exponential stability property for the operator A. an admissible measurement operator. This gives an alternative Assumption 1 (Exponential Stability): The operator A gen- proof to the results of S. Pojohlainen in [12] (and [23]). It erates a C -semigroup which is exponentially stable. In other allows also to give explicit value to the integral gain that solves 0 words, there exist ν and k both positive constants such that, arXiv:1901.02208v1 [math.AP] 8 Jan 2019 the output regulation problem. 8' 2 X and t 2 In a second part, following the same Lyapunov functional 0 R+ At design procedure, we consider a boundary regulation problem ke '0kX 6 k exp(−νt)k'0kX : (2) for a class of hyperbolic PDE systems. This result generalizes many others which have been obtained so far in the regulation We are interested in the regulation problem. More precisely of PDE hyperbolic systems (see for instance [8], [24], [3], [20], we are concerned with the problem of regulation of the output [21], [18]). y via the integral control The paper is organized as follows. Section II is devoted u = kiKiz ; zt = y − yref ; (3) to the regulation of the measured output for stable abstract m m Cauchy problems. It is given a general procedure, for an where yref 2 R is a prescribed reference, z 2 R , Ki 2 m×m exponentially stable semigroup in open-loop, to construct a R is a full rank matrix and ki a positive real number. Lyapunov functional for the closed loop system obtained with The control law being dynamical, the state space has been an integral controller. Inspired by this procedure, the case extended. Considering the system Σ(A; B; C) in closed loop, with integral control law given in (3) the state space is now m All authors are with LAGEP-CNRS, Universite´ Claude Bernard Lyon1, Xe = X × R which is a Hilbert space with inner product Universite´ de Lyon, Domaine Universitaire de la Doua, 43 bd du 11 Novembre > 1918, 69622 Villeurbanne Cedex, France. h'ea;'ebiXe = h'a;'biX + za zb ; 2 'a 'b where 'ea = and 'eb = . The associated norm is On another hand, if one wants to address nonlinear abstract za zb Cauchy problems or unbounded operators, we may need to denoted k · k . Let A :(D(A) \ D(C)) × m !X be the Xe e R e follow a Lyapunov approach. For instance in the context of extended operator defined as boundary control, a Lyapunov functional approach has allowed ABK k to tackle feedback stabilization of a large class of PDEs (see A = i i : (4) e C 0 for instance [2] or [5]). It is well known (see for instance [9, Theorem 8.1.3]) The regulation problem to solve can be rephrased as the that exponential stability of the operator A is equivalent to following. existence of a bounded positive and self adjoint operator P in Regulation problem: We wish to find a positive real number L(X) such that m ki and a full rank matrix Ki such that 8(w; yref ) 2 X × R : 1) The system (1)-(3) is well-posed. In other words, for hA'; P'iX + hP'; A'iX 6 −µk'kX ; 8 ' 2 D(A); (8) all ' = (' ; z ) 2 X there exists a unique (weak) e0 0 0 e where µ is a positive real number. We assume that this '(t) 0 Lyapunov operator P is given. The first question, we intend solution denoted 'e(t) = 2 C (R+; Xe) defined z(t) to solve is the following: Knowing the Lyapunov operator P, 8t 0 and initial condition 'e(0) = 'e0. > is it possible to construct a Lyapunov operator Pe associated 2) There exists an equilibrium point denoted 'e1 = to the extended operator Ae? '1 2 Xe, depending on w and yref , which is To answer this question, we first give a construction based z1 exponentially stable for the system (1)-(3). In other on a well-known technique in the nonlinear finite dimensional the forwarding words, there exist positive real numbers ν and k such control community named (see for instance e e [10], [15] or more recently [4], or [1]). that for all t > 0 −νet k'e(t) − 'e1kXe 6 ke exp k'e0 − 'e1kXe : B. A Lyapunov approach for regulation 3) The output y is regulated toward the reference yref . Inspired by the forwarding techniques, the following result More precisely, can be obtained. Theorem 2 (Forwarding Lyapunov functional): Assume that 8'e0; lim jC'(t) − yref j = 0: (5) t!+1 all assumptions of Theorem 1 are satisfied and let P in L(X) be a positive self adjoint operator such that (8) holds. Then We know with the work of S. Pohjolainen in [13] that when there exist a bounded operator M : X! Rm and positive ∗ ∗ A generates an exponentially stable analytic semi-group, when real numbers p and ki , such that for all 0 < ki < ki , there B is bounded and when C is A-bounded, with a rank condition, exists µe > 0 such that the operator the regulation may be achieved. This result has been extended P + pM∗M −pM∗ to more general exponentially stable semi-groups in [23]. Pe = (9) −pM p Id Theorem 1 ([23]): Assume that X is separable and that A m satisfies Assumption 1. Assume moreover that : is positive and satisfies 8 'e = ('; z) 2 D(A) × R 1) the operator B is bounded; 2 2 hAe'e; Pe'eiXe + hPe'e; Ae'eiXe 6 −µe(k'kX + jzj ): 2) the operator C is A-admissible (see [22]), i.e. (10) • it is A-bounded : Proof: The operator A satisfying Assumption 1, 0 is in its jC'j 6 c(k'kX + kA'kX ) ; 8 ' 2 D(A); (6) resolvent set and consequently A−1 : X 7! D(A) is well for some positive real number c; defined and bounded. Let M : X! Rm be defined by M = −1 • there exist T > 0 and cT > 0 such that CA which is well defined due to the fact that D(A) ⊆ D(C) Z T since C is A-bounded. Moreover, with (6) 8' 2 X At 2 2 2 jCe 'j dt 6 cT k'kX ; 8 ' 2 D(A); −1 −1 0 jM'j = jCA 'j 6 c kA 'kX + k'kX 6 c~k'kX ; 3) the rank condition holds. In other words operators A, B where c~ is a positive real number. Hence, M is a bounded and C satisfy linear operator. Moreover, M satisfies the following equation rankfCA−1Bg = m; (7) MA' = C'; 8' 2 D(A) : (11) ∗ K = (CA−1B)−1 then there exists a positive real number ki and a m×m matrix Let i which exists due to the third assump- ∗ Ki, such that for all 0 < ki < ki the operator Ae given in tion of Theorem 1.
Load more

Recommended publications
  • Exponential Stability for the Resonant D’Alembert Model of Celestial Mechanics
    DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 12, Number 4, April 2005 pp. 569{594 EXPONENTIAL STABILITY FOR THE RESONANT D'ALEMBERT MODEL OF CELESTIAL MECHANICS Luca Biasco and Luigi Chierchia Dipartimento di Matematica Universit`a\Roma Tre" Largo S. L. Murialdo 1, 00146 Roma (Italy) (Communicated by Carlangelo Liverani) Abstract. We consider the classical D'Alembert Hamiltonian model for a rotationally symmetric planet revolving on Keplerian ellipse around a ¯xed star in an almost exact \day/year" resonance and prove that, notwithstanding proper degeneracies, the system is stable for exponentially long times, provided the oblateness and the eccentricity are suitably small. 1. Introduction. Perturbative techniques are a basic tool for a deep understand- ing of the long time behavior of conservative Dynamical Systems. Such techniques, which include the so{called KAM and Nekhoroshev theories (compare [1] for gen- eralities), have, by now, reached a high degree of sophistication and have been applied to a great corpus of di®erent situations (including in¯nite dimensional sys- tems and PDE's). KAM and Nekhoroshev techniques work under suitable non{ degeneracy assumptions (e.g, invertibility of the frequency map for KAM or steep- ness for Nekhoroshev: see [1]); however such assumptions are strongly violated exactly in those typical examples of Celestial Mechanics, which { as well known { were the main motivation for the Dynamical System investigations of Poincar¶e, Birkho®, Siegel, Kolmogorov, Arnold, Moser,... In this paper we shall consider the day/year (or spin/orbit) resonant planetary D'Alembert model (see [10]) and will address the problem of the long time stability for such a model.
    [Show full text]
  • Exponential Stability of Distributed Parameter Systems Governed by Symmetric Hyperbolic Partial Differential Equations Using Lyapunov’S Second Method ∗
    ESAIM: COCV 15 (2009) 403–425 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv:2008033 www.esaim-cocv.org EXPONENTIAL STABILITY OF DISTRIBUTED PARAMETER SYSTEMS GOVERNED BY SYMMETRIC HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS USING LYAPUNOV’S SECOND METHOD ∗ Abdoua Tchousso1, 2, Thibaut Besson1 and Cheng-Zhong Xu1 Abstract. In this paper we study asymptotic behaviour of distributed parameter systems governed by partial differential equations (abbreviated to PDE). We first review some recently developed results on the stability analysis of PDE systems by Lyapunov’s second method. On constructing Lyapunov functionals we prove next an asymptotic exponential stability result for a class of symmetric hyper- bolic PDE systems. Then we apply the result to establish exponential stability of various chemical engineering processes and, in particular, exponential stability of heat exchangers. Through concrete examples we show how Lyapunov’s second method may be extended to stability analysis of nonlin- ear hyperbolic PDE. Meanwhile we explain how the method is adapted to the framework of Banach spaces Lp,1<p≤∞. Mathematics Subject Classification. 37L15, 37L45, 93C20. Received November 2, 2005. Revised September 4, 2006 and December 7, 2007. Published online May 30, 2008. 1. Introduction Lyapunov’s second method, called also Lyapunov’s direct method, is one of efficient and classical methods for studying asymptotic behavior and stability of the dynamical systems governed by ordinary differential equations. It dates back to Lyapunov’s thesis published at the beginning of the twentieth century [22]. The applications of the method and its theoretical development have been marked by the works of LaSalle and Lefschetz in the past [19,35] and numerous successful achievements later in the control of mechanical, robotic and aerospace systems (cf.
    [Show full text]
  • Stability Analysis for a Class of Partial Differential Equations Via
    1 Stability Analysis for a Class of Partial terms of differential matrix inequalities. The proposed Differential Equations via Semidefinite method is then applied to study the stability of systems q Programming of inhomogeneous PDEs, using weighted norms (on Hilbert spaces) as LF candidates. H Giorgio Valmorbida, Mohamadreza Ahmadi, For the case of integrands that are polynomial in the Antonis Papachristodoulou independent variables as well, the differential matrix in- equality formulated by the proposed method becomes a polynomial matrix inequality. We then exploit the semidef- Abstract—This paper studies scalar integral inequal- ities in one-dimensional bounded domains with poly- inite programming (SDP) formulation [21] of optimiza- nomial integrands. We propose conditions to verify tion problems with linear objective functions and poly- the integral inequalities in terms of differential matrix nomial (sum-of-squares (SOS)) constraints [3] to obtain inequalities. These conditions allow for the verification a numerical solution to the integral inequalities. Several of the inequalities in subspaces defined by boundary analysis and feedback design problems have been studied values of the dependent variables. The results are applied to solve integral inequalities arising from the using polynomial optimization: the stability of time-delay Lyapunov stability analysis of partial differential equa- systems [23], synthesis control laws [24], [29], applied to tions. Examples illustrate the results. optimal control law design [15] and system analysis [11]. Index Terms—Distributed Parameter Systems, In the context of PDEs, preliminary results have been PDEs, Stability Analysis, Sum of Squares. presented in [19], where a stability test was formulated as a SOS programme (SOSP) and in [27] where quadratic integrands were studied.
    [Show full text]
  • Introduction to Lyapunov Stability Analysis
    Tools for Analysis of Dynamic Systems: Lyapunov’s Methods Stanisław H. Żak School of Electrical and Computer Engineering ECE 680 Fall 2013 1 A. M. Lyapunov’s (1857--1918) Thesis 2 Lyapunov’s Thesis 3 Lyapunov’s Thesis Translated 4 Some Details About Translation 5 Outline Notation using simple examples of dynamical system models Objective of analysis of a nonlinear system Equilibrium points Lyapunov functions Stability Barbalat’s lemma 6 A Spring-Mass Mechanical System x---displacement of the mass from the rest position 7 Modeling the Mass-Spring System Assume a linear mass, where k is the linear spring constant Apply Newton’s law to obtain Define state variables: x1=x and x2=dx/dt The model in state-space format: 8 Analysis of the Spring-Mass System Model The spring-mass system model is linear time-invariant (LTI) Representing the LTI system in standard state-space format 9 Modeling of the Simple Pendulum The simple pendulum 10 The Simple Pendulum Model Apply Newton’s second law J mglsin where J is the moment of inertia, 2 J ml Combining gives g sin l 11 State-Space Model of the Simple Pendulum Represent the second-order differential equation as an equivalent system of two first-order differential equations First define state variables, x1=θ and x2=dθ/dt Use the above to obtain state–space model (nonlinear, time invariant) 12 Objectives of Analysis of Nonlinear Systems Similar to the objectives pursued when investigating complex linear systems Not interested in detailed solutions, rather one seeks to characterize the system behavior---equilibrium points and their stability properties A device needed for nonlinear system analysis summarizing the system behavior, suppressing detail 13 Summarizing Function (D.G.
    [Show full text]
  • Small Denominators and Exponential Stability: from Poincare´ to the Present Time
    SMALL DENOMINATORS AND EXPONENTIAL STABILITY: FROM POINCARE´ TO THE PRESENT TIME ANTONIO GIORGILLI Dipartimento di Matematica dell’ Universit`adi Milano Via Saldini 50, 20133 Milano, Italy. Abstract. The classical problem of small denominators is revisited in its historical develop- ment, ending with recent results on exponential stability. Conference given at the Seminario Matematico e Fisico di Milano. Small denominators and exponential stability 1 1. Overview In 1959 J.E. Littlewood published two papers devoted to the stability of the Lagrangian equilateral equilibria of the problem of three bodies.[30][31] Le me quote the incipit of the first of those papers: “ The configuration is one of two point masses S and J, with mJ = µmS, and a body P of zero mass, the law of attraction being the inverse square; the equilateral triangle SJP , of side 1, rotates in equilibrium with unit angular velocity about the centre of gravity of S and J. If µ < µ0, where µ0 is the −2 1 smaller root of µ0(1 + µ0) = 27 , the configuration is stable in the sense that a small disturbance does not double in a few revolutions; there are real periods 2π/λ1, 2π/λ2 of ‘ normal oscillations ’. It is notorious that no non–trivial system whatever is known to be sta- ble (or bounded) over infinite time. It is possible, however, to ask a less far–reaching question: given that the initial disturbance in coordinates and velocities is of order ε, for how long a time, in terms of ε, can we say the disturbance will remain of order ε? We shall find that, for almost all values −1/2 −3/4 of µ < µ0, this time is as long as exp Aε log ε , where A depends only on µ; while not eternity, this is a considerable| | slice of it.
    [Show full text]
  • Existence and Exponential Stability for Neutral Stochastic Integro–Differential Equations with Impulses Driven by a Rosenblatt Process
    DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019251 DYNAMICAL SYSTEMS SERIES B Volume 25, Number 2, February 2020 pp. 507{528 EXISTENCE AND EXPONENTIAL STABILITY FOR NEUTRAL STOCHASTIC INTEGRO{DIFFERENTIAL EQUATIONS WITH IMPULSES DRIVEN BY A ROSENBLATT PROCESS Tomas´ Caraballo Dpto. Ecuaciones Diferenciales y An´alisisNum´erico Facultad de Matem´aticas,Universidad de Sevilla C/ Tarfia s/n, 41012-Sevilla, Spain Carlos Ogouyandjou and Fulbert Kuessi Allognissode Universit´ed'Abomey-Calavi(UAC) Institut de Math´ematiqueset de Sciences Physiques(IMSP) 01 B.P. 613, Porto-Novo, R´epubliquedu B´enin Mamadou Abdoul Diop Universit´eGaston Berger de Saint-Louis, UFR SAT D´epartement de Math´ematiques B.P234, Saint-Louis, S´en´egal Dedicated to Prof. Dr. Juan J. Nieto on the occasion of his 60th birthday Abstract. The existence and uniqueness of mild solution of an impulsive sto- chastic system driven by a Rosenblatt process is analyzed in this work by using the Banach fixed point theorem and the theory of resolvent operator developed by R. Grimmer in [12]. Furthermore, the exponential stability in mean square for the mild solution to neutral stochastic integro–differential equations with Rosenblatt process is obtained by establishing an integral inequality. Finally, an example is exhibited to illustrate the abstract theory. 1. Introduction. Integro-differential equations are of great importance in the mod- eling of several physical phenomena. Their resolution can be done through the theory of resolvent operators (see Grimmer [12]). However, the resolvent operator does not satisfy semigroup properties. The study of the quantitative and qualita- tive properties of solutions to stochastic neutral differential equations like existence, uniqueness and stability, have been widely examined by many researchers by analyz- ing various mathematical models in different areas such as mechanics, electronics, control theory, engineering and economics, etc (see [3,5,7,8, 25]).
    [Show full text]
  • Nonoscillation, Maximum Principles, and Exponential Stability of Second Order Delay Differential Equations Without Damping Term Alexander Domoshnitsky*
    Domoshnitsky Journal of Inequalities and Applications 2014, 2014:361 http://www.journalofinequalitiesandapplications.com/content/2014/1/361 RESEARCH Open Access Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term Alexander Domoshnitsky* *Correspondence: [email protected] Ariel University, Ariel, Israel Abstract Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation x(t)+c(t)x(t) = 0 can be oscillating and asymptotically unstable, the delay equation x(t)+a(t)x(t – h(t)) – b(t)x(t – g(t)) = 0, where c(t)=a(t)–b(t), can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this
    [Show full text]
  • 4 Lyapunov Stability Theory
    4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We present a survey of the results that we shall need in the sequel, with no proofs. The interested reader should consult a standard text, such as Vidyasagar [?] or Khalil [?], for details. 4.1 Basic definitions Consider a dynamical system which satisfies n x˙ = f(x, t) x(t0)=x0 x ∈ R . (4.31) We will assume that f(x, t) satisfies the standard conditions for the exis- tence and uniqueness of solutions. Such conditions are, for instance, that f(x, t) is Lipschitz continuous with respect to x, uniformly in t, and piece- wise continuous in t.Apointx∗ ∈ Rn is an equilibrium point of (4.31) if f(x∗,t) ≡ 0. Intuitively and somewhat crudely speaking, we say an equi- librium point is locally stable if all solutions which start near x∗ (meaning that the initial conditions are in a neighborhood of x∗) remain near x∗ for all time. The equilibrium point x∗ is said to be locally asymptotically stable if x∗ is locally stable and, furthermore, all solutions starting near x∗ tend towards x∗ as t →∞. We say somewhat crude because the time-varying nature of equation (4.31) introduces all kinds of additional subtleties. Nonetheless, it is intuitive that a pendulum has a locally sta- ble equilibrium point when the pendulum is hanging straight down and an unstable equilibrium point when it is pointing straight up.
    [Show full text]
  • Module 06 Stability of Dynamical Systems
    Stability of CT Systems Stability of DT Linear Systems Lyapunov Stability, Nonlinear Systems Module 06 Stability of Dynamical Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: [email protected] Webpage: http://engineering.utsa.edu/ataha October 10, 2017 ©Ahmad F. Taha Module 06 — Stability of Dynamical Systems 1 / 24 Stability of CT Systems Stability of DT Linear Systems Lyapunov Stability, Nonlinear Systems Stability of CT LTV Systems The following CT LTI system without inputs n x˙ (t) = A(t)x(t), x(t) ∈ R has an equilibrium at xe = 0. Asymptotic Stability The above system is asymptotically stable at xe = 0 if its solution x(t) starting from any initial condition x(t0) satisfies x(t) → 0, as t → ∞ Exponential Stability The above system is exponentially stable at xe = 0 if its solution x(t) starting from any initial condition x(t0) satisfies −rt kx(t)k ≤ Ke kx(t0)k, ∀t ≥ t0 for some positive constants K and r. ©Ahmad F. Taha Module 06 — Stability of Dynamical Systems 2 / 24 Stability of CT Systems Stability of DT Linear Systems Lyapunov Stability, Nonlinear Systems Example 1 Consider the following TV LTI system from Homework 4: − 1 0 x˙ (t) = t+1 x(t) − 1 0 t+1 Recall that the solution to this system is 1 0 1 1 x(t) = φ(t, 0)x(0) = t+1 = t+1 − t 1 −1/5 − t − 1/5 t+1 t+1 Is this system asymptotically stable? Solution: it’s not, since the states do not go to zero for any initial conditions ©Ahmad F.
    [Show full text]
  • Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations
    London Mathematical Society ISSN 1461–1570 EXPONENTIAL MEAN-SQUARE STABILITY OF NUMERICAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS DESMOND J. HIGHAM, XUERONG MAO and ANDREW M. STUART Abstract Positive results are proved here about the ability of numerical simula- tions to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial. 1. Introduction Suppose that we are required to find out whether a stochastic differential equation (SDE) is exponentially stable in mean square. In the absence of an appropriate Lyapunov function, we may carry out careful numerical simulations using a numerical method with a ‘small’ step size t. Two key questions then follow. (Q1) If the SDE is exponentially stable in mean square, will the numerical method be exponentially stable in mean square for sufficiently small t? (Q2) If the numerical method is exponentially stable in mean square for small t, can we infer that the underlying SDE is exponentially stable in mean square? These questions deal with an asymptotic (t →∞) property, and hence they cannot be answered directly by applying traditional finite-time convergence results.
    [Show full text]
  • Exponential Stabilization of Driftless Nonlinear Control Systems" Robert Thomas M'closkey I11
    TECHNICAL MEMORANDUM NO. CIT-CDS 95-003 January 1995 "Exponential Stabilization of Driftless Nonlinear Control Systems" Robert Thomas M'Closkey I11 Control and Dynamical Systems California Institute of Technology Pasadena, CA 91125 Exponential Stabilization of Driftless Nonlinear Control Systems Robert Thomas M7Closkey I11 Department of Mechanical Engineering California Institute of Technology Abstract This dissertation lays the foundation for practical exponential stabilization of driftless control systems. Driftless systems have the form, Such systems arise when modeling mechanical systems with nonholonomic con- straints. In engineering applications it is often required to maintain the mechan- ical system around a desired configuration. This task is treated as a stabilization problem where the desired configuration is made an asymptotically stable equi- librium point. The control design is carried out on an approximate system. The approximation process yields a nilpotent set of input vector fields which, in a special coordinate system, are homogeneous with respect to a non-standard dilation. Even though the approximation can be given a coordinate-free in- terpretation, the homogeneous structure is useful to exploit: the feedbacks are required to be homogeneous functions and thus preserve the homogeneous struc- ture in the closed-loop system. The stability achieved is called p-exponential stability. The closed-loop system is stable and the equilibrium point is exponen- tially attractive. This extended notion of exponential stability is required since the feedback, and hence the closed-loop system, is not Lipschitz. However, it is shown that the convergence rate of a Lipschitz closed-loop driftless system cannot be bounded by an exponential envelope. The synthesis methods generate feedbacks which are smooth on Rn \ (0).
    [Show full text]
  • ABSTRACT a General Linear Systems Theory on Time Scales
    ABSTRACT A General Linear Systems Theory on Time Scales: Transforms, Stability, and Control Billy Joe Jackson, Ph.D. Advisor: John M. Davis, Ph.D. In this work, we examine linear systems theory in the arbitrary time scale set- ting by considering Laplace transforms, stability, controllability, observability, and realizability. In particular, we revisit the de¯nition of the Laplace transform given by Bohner and Peterson in [10]. We provide su±cient conditions for a given function to be transformable, as well as an inversion formula for the transform. Su±cient conditions for the inverse transform to exist are provided, and uniqueness of this inverse function is discussed. Convolution under the transform is then considered. In particular, we develop an analogue of the Convolution Theorem for arbitrary time scales and discuss the algebraic properties of the convolution. This naturally leads to an algebraic identity for the convolution operator, which is a time scale analogue of the Dirac delta distribution. Next, we investigate applications of the transform to linear time invariant sys- tems and before discussing linear time varying systems. The focus is on fundamental notions of linear system control such as controllability, observability, and realizabil- ity. Su±cient conditions for a system to possess each of these properties are given in the time varying case, while these same criteria often become necessary and suf- ¯cient in the time invariant case. Finally, several notions of stability are discussed, and linear state feedback is explored. A General Linear Systems Theory on Time Scales: Transforms, Stability, and Control by Billy Joe Jackson, B.S., M.A.
    [Show full text]