J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... #(#PP) 2 [# 1 P A 1 A 0 P 1 1 P # 2 P A 1 A 1 P 1 A 1 P (# 1 1 # 2 1 )P] #PP) # = #P In this case the required observer gain is given by: L = P 1 (C 0 C 1 ) Note that the proofs of Lemma 1 and Theorem 1 given in [2] are based on some results of [3, 4, 5, 6]. In [2] an algorithm is proposed in order to minimize # and the following sufficient condition is provide. Proposition 1 [2] For a given time delay system of the form (1) if the pair [ C 0 C 1 ) A 0 A 1 ) is detectable, then there exists an H# observer of the form (2) with certain ....

J. Hale, Theory of functional differential equations. Springer-Verlag, 1977.

....in stochastic systems. Also, continuous time deterministic systems may behave differently from their discrete counterparts. The feedback delay has been much more systematically studied in the deterministic systems (see e:g: Fendick et al. 3] Yin and Hluchyj [11] for network applications and Hale [4] for a general theory) than for the stochastic systems (see however Fendick and Rodrigues [2] Pazhyannur and Agrawal [6] Our immediate motivation to study the general question of the effect of delay in the feedback is the congestion control and scheduling in communication networks. In long ....

J. K. Hale, "Theory of functional differential equations " , NY 1997.

....for DDEs are the books by Bellman Cooke [21] and Elsgol ts Norkin [60] These are rich sources for analytical techniques and many interesting examples. Kolmanovskii et al. 105, 106] gave a rigorous treatment of a wide class of problems. Starting from the first edition, the monograph of Hale [83], subsequently Hale Verduyn Lunel [85] is a standard source on the theory of delay equations. Another substantial monograph is by Diekmann et al. 53] Kuang [107] and R. Banks [18] pay particular attention to problems in population dynamics; the former also looked at neutral equations. ....

J.K. Hale, Theory of Functional Differential Equations (Springer, New York, 1977).

....in figure 4. To emphasise the speed of the method, only a third order Runge Kutta scheme is used to estimate the stepsize. This choice could be revised in a working code. For reasons of simplicity, and because of the interest of the author in the solution of functional differential equations [7, 12], the above examples all describe order and stepsize decreases. This is, however, not necessary. An application involving potential stepsize and order increases is given in section 11. Further, numerical experience indicates that the estimators proposed in this paper may help to recover an ....

Hale, J. "Theory of functional differential equations." Applied Math. Sciences 3, Springer Verlag (1977). 16

....area equation) is shown in Figure 1. 3 Analytical Stability Analysis of Combustion Model The purpose of this section is to carry out the stability and bifurcation analysis of the model derived in the previous section. Note that this model falls into a category of neutral differential equations [10]. We rewrite the model for one mode and with obvious changes in notation in the following form j 2ff j 2 j = N d dt [H ( j(t Gamma ) 1 A) 12) A = b 1 (oe j Gamma A) 13) We assume that nonlinearity H( Delta) as well as some other model coefficients implicitly ....

J. K. Hale. Theory of Functional Differential Equations. Springer Verlag, New York, 1977.

....on [0; 1] satisfies (1.2) and (0) Gamma P m j=1 p j ( Gammaoe j ) f(0) x(s) s) for s 2 [ GammaT ; 0] As we can see from Chapter 12 in [17, pp. 273 274] neutral differential equations defined there include difference equations. Thus, some basic properties of (1. 2) can be found in [17]. Let C 1 ( Gammar; 0] R) denote the set of all continuously differentiable functions mapping [ Gammar; 0] into R. The function x is said to be a solution of (1.3) with initial function 2 C 1 ( GammaT ; 0] R) if x(t) P m j=1 p j x(t Gamma oe j ) is continuously differentiable for ....

....function x is said to be a solution of (1.3) with initial function 2 C 1 ( GammaT ; 0] R) if x(t) P m j=1 p j x(t Gamma oe j ) is continuously differentiable for t 0 and x satisfies (1.3) for t 0, and x(s) s) for s 2 [ GammaT ; 0] The fundamental theory of (1. 3) is studied in [4, 17]. The following definitions of oscillation are used in this paper [19, 20] A function x is said to be oscillatory if for any t 1 0, we have inf [t 1 ; 1) x(t) 0 sup [t 1 ; 1) x(t) A function x is said to be strongly oscillatory if we have lim inf t 1 x(t) 0 lim sup t 1 x(t) ....

Hale J., Theory of Functional Differential Equations, Springer--Verlag, New York, 1977.

.... 3t 2 2 l max A t R 1 A t Hence, for f q c F 0 , q 4 2t 0 , one gets x t Px t 0 V x t 0 V x t 0 0 g 1 ( t L t 0 Therefore, for any initial condition in F 0 , the system (5) verifies the conditions of the Krasovskii Theorem [11] and V x t is a local strictly decreasing Lyapunov function. Thus, the asymptotic stability of system (5) is ensured. P 3.3 Disturbance case (w : 0) A solution to Problem 1 in the case w : 0 is presented now. Proposition 2 For a given t 0, if there exist matrices Y m ....

.... relations (32) and (33) one gets x t Px t 0 V x t w 0 V x t 0 w 0 g 1 O) t L t 0 For any initial condition in F 0 defined in (34) and any admissible disturbance satisfying assumption A2, system (5) verifies the conditions of the Krasovskii Theorem [11] and V x t w is a local strictly decreasing Lyapunov function. Thus, the stability of system (5) is assured. P The results derived from Proposition 2 are potentially conservative. This source of conservatism comes in particular from: 1. The polytopic representation used for the ....

J.K. Hale. Theory of functional differential equations. Springer-Verlag, New York, 1977.

....following canonical form: A = 2 6 6 6 6 6 4 0 1 0 1 . 0 1 Gammap 0 Gammap 1 Delta Delta Delta Gammap n Gamma2 Gammap n Gamma1 3 7 7 7 7 7 5 ; b = 2 6 6 6 6 6 4 0 0 . 0 1 3 7 7 7 7 7 5 (1. 4) Then, from the known results on retarded functional difference equations (see [4]) we know that the problem of stabilizability of [A; b] is equivalent to the problem of finding a vector K = Gammaq 0 ; Gammaq 1 ; Delta Delta Delta ; Gammaq n Gamma1 ) T 2 IR n ; such that all the roots of the following polynomial det(zI Gamma A Gamma bK T e Gamma z ) ....

J. K. Hale, Theory of Functional Differential Equations. Springer - Verlag, New York, Berlin, 1977.

....function H(z) is said to be stable if N(H(z) ae C Gamma . The author is now at the Department of Mathematics, University of Notre Dame, IN 46556, USA. AMS Subject Classification : 34D20. This research was supported in part by the Chinese Education Commission Science Foundation. 1 From [8] or [13] we know, generally, the problem of asymptotic stability of differential difference equations is equvialent to the problem of stability of transcendental polynomials of the kind h(z; e z ) For H(z) h(z; e z ) in the early 1940 s, L.S. Pontryagin [16] gave several theorems for ....

J.K. Hale, Theory of Functional Differential Equations, Springer- Verlag, New York, 1977.

.... Kolmogorov Sinai entropy, such that the typical minimal inter point distance which can be reached by reasonably large data sets (say, N = 10 6 ) is still too large for a detection of the deterministic structure [6] A special class of dynamical systems is that of the delayed feedback systems [7]. Despite a small number of physical variables, their phase space is infinite dimensional, namely the space of all differentiable functions on the time interval [0; 0 ] where 0 is the delay time of the feedback. They can thus possess chaotic attractors of arbitrarily high dimension. The direct ....

J. Hale, Theory of Functional Differential Equations (Springer, Berlin, Heidelberg, 1977).

....t n ) V n ) be the unique maximal solution of the system hu n t ; vi ha g(jru n j 2 )ru n ; rvi = hf n ; vi (2.10) for all v 2 V n , with initial data defined by hu n (0) Gamma u 0 ; vi = 0 for all v 2 V n . It follows from standard results on functional differential equations ([10]) that u n exists and is unique for each n; possibly, t n T 0 , in which case ku n (t)k 1 as t t n : It is also easy to see (by differentiating the equation) that in fact u n 2 C 2 ( 0; t n ) V n ) A Priori Estimate I. Let d 2 W 1;1 ( 0; T 0 ] R) be the resolvent kernel ....

J. K. Hale, Theory of Functional Differential Equations, Applied Math. Sciences Vol. 3, Springer, Berlin, Heidelberg, New York, 1977.

....assume the reader is familiar with the notion of a delay differential equation and with the basic concepts of bifurcation analysis for ordinary differential equations. The theory on delay differential equations and a large number of examples are described in several books. Most notably the early [4, 8, 7, 15, 22] and the more recent [2, 20, 16, 5, 21] Several excellent books contain introductions to dynamical systems and bifurcation theory of ordinary differential equations, see, e.g. 27, 14, 1, 28, 23] A large number of packages exist for bifurcation analysis of systems of ordinary differential ....

J. K. Hale. Theory of Functional Differential Equations, volume 3 of Applied Mathematical Sciences. Springer-Verlag, 1977.

....In section 6 we describe the test models which we use, and we present our actual numerical results in section 7. Section 8 contains conclusions. 2 Two Kinds of Boundary Value Problems A general two point boundary value problem for delay differential equations (1) may be formulated as (cf. e.g. [17, 23]) ae dx(t) dt = f(x(t) x(t Gamma ) t 2 [0; T ] b(x 0 ; x T ) 0; 3) where b : C Theta C C and T 0 are given. However, in the current literature on the numerical solution of BVPs for DDEs (or more general functional differential equations) see, e.g. 4, 7, 3, 8, 29, 24] and [2, ....

....solutions bifurcates from the zero solution and that stable periodic solutions of period T = 4 exist whenever ff =2. Here we consider such a stable periodic solution at ff = 2. 11 Model 3 This is the well known delayed logistic equation dx dt = Gamma x(t Gamma 1) x(t) 29) cf. e.g. [17, 21]. We computed a stable periodic solution at = 1:7 with T 4:0964. Model 4 The following system models recurrent neural feedback [28, 9] ae dv dt = h(v(t) Gamma w(t) I(v(t Gamma ) dw dt = ae (v(t) a Gamma bw(t) 30) with 8 : h(v) v Gamma v 3 =3; I(v) v Gamma v 0 ) ....

J. K. Hale. Theory of Functional Differential Equations, volume 3 of Applied Mathematical Sciences. Springer-Verlag, 1977.

.... OE) 2 Z : OE 2 H 1 ( Gammar; 0) OE(0) jg, and A(j; OE) aj bOE( Gammar) d d OE) If the state at time t 0 is defined to be z(t) x(t) x t ) where x t ( x(t ) for Gammar 0, then (3) can be reformulated as the abstract Cauchy problem (1) It is well known (see [1] [8]) that A is the infinitesimal generator of a strongly continuous semigroup T (t) of bounded linear operators on Z. Let us now describe some spline based Galerkin approximation schemes for (3) and for simplicity consider only linear splines. Following [2] let N j = Gammaj r=N , j = 0; 1; ....

Hale, J.K., Theory of Functional Differential Equations, Springer, Berlin, New York, 1977.

.... as the abstract Cauchy problem z(t) Az(t) z(0) z 0 : 2:1) It is well known that A is the infinitesimal generator of a strongly continuous semigroup T (t) of bounded linear operators on Z and that the spectrum of A consists of those complex which satisfy the characteristic equation (see [8]) Delta( j a Gamma be Gammar = 0: 2:2) It is also known (see [8] 9] that there are constants M 1 and ff such that the spectrum of A lies in the left half plane Re ff, and that for every ffl 0, k T (t) k Me (ff ffl)t : 2:3) For the rest of this paper we will assume that b 0. ....

.... known that A is the infinitesimal generator of a strongly continuous semigroup T (t) of bounded linear operators on Z and that the spectrum of A consists of those complex which satisfy the characteristic equation (see [8] Delta( j a Gamma be Gammar = 0: 2:2) It is also known (see [8], 9] that there are constants M 1 and ff such that the spectrum of A lies in the left half plane Re ff, and that for every ffl 0, k T (t) k Me (ff ffl)t : 2:3) For the rest of this paper we will assume that b 0. The reason for this assumption is that the main goal of the paper is to ....

[Article contains additional citation context not shown here]

Hale, J.K., Theory of Functional Differential Equations, Springer, Berlin, New York, 1977.

....for DDEs are the books by Bellman Cooke [21] and Elsgol ts Norkin [60] These are rich sources for analytical techniques and many interesting examples. Kolmanovskii et al. 105, 106] gave a rigorous treatment of a wide class of problems. Starting from the first edition, the monograph of Hale [83], subsequently Hale Verduyn Lunel [85] is a standard source on the theory of delay equations. Another substantial monograph is by Diekmann et al. 53] Kuang [107] and R. Banks [18] pay particular attention to problems in population dynamics; the former also looked at neutral equations. ....

J.K. Hale, Theory of Functional Differential Equations (Springer, New York, 1977).

No context found.

J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.

No context found.

J. K. Hale, "Theory of Functional Differential Equations," Springer-Verlag, New York, 1977.

No context found.

J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, 1977.

No context found.

J.K. Hale, Theory of Functional Differential Equations, Springer, 1977.

No context found.

J.K. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York, NY, 1977). 23

No context found.

J. Hale, "Theory of Functional Differential Equations", Springer, New York (1977).

No context found.

J. Hale, Theory of Functional Differential Equations. Springer-Verlag, New York, 1977.

No context found.

J. Hale, Theory of functional differential equations, Springer-Verlag, New York-Heidelberg-Berlin, (1977).

No context found.

J. K. Hale, Theory of Functional Differential Equations, Springer, New York 1977.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC