Retarded Functional Differential Equations Research Articles

Legendre–Tau Approximations for Functional-Differential Equations

In this paper we consider the numerical approximation of solutions to linear retarded functional differential equations using the so-called Legendre–tau method. The functional differential equation is first eformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

A sharp version of Henry's theorem on small solutions

A small solution of a linear autonomous retarded functional differential equation (rfde) is a solution that goes to zero faster than any exponential. Henry's theorem on small solutions states that there exists a time T—depending on the dimension and the delay of the equation—such that all small solutions vanish a.e. for t ⩾ T. In this paper we shall give an explicit characterisation for the smallest possible time T, in terms of properties of the specific kernel. This characterisation helps to establish new results concerning completeness and F-completeness of the generalized eigenfunctions of the infinitesimal generator of the C 0-semigroup associated with the linear autonomous rfde.

On the existence of periodic solutions for autonomous retarded functional differential equations on R2

SynopsisLet Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.

Structure and Stability of Finite Dimensional Approximations for Functional Differential Equations

This paper deals with the structural and stability properties of the averaging approximation scheme for linear retarded functional differential equations. Both in the discrete- and in the continuous-time case the structure of the approximating systems is shown to be analogous to the structure of the underlying retarded equation. Moreover, it is shown that the approximating systems are exponentially stable in a uniform sense if the original system is asymptotically stable.

On asymptotic stability of linear time-varying infinite-dimensional systems

It is shown that every asymptotically equistable linear time-varying infinite-dimensional discrete-time system x k+1 = A k x k is uniformly asymptotically equistable, if A k is a collectively compact sequence of bounded linear operators. Next, this result is used to prove that for a broad class of linear retarded functional differential equations, the notions of asymptotic equistability and uniform asymptotic equistability coincide.

The Linear-Quadratic Control Problem for Retarded Systems with Delays in Control and Observation

THE OBJECT of this paper is to solve the linear-quadratic control problem (LQCP) for retarded functional differential equations (RFDE) with delays in the input and output variables. This will be done within the general semigroup-theoretic framework which has been developed in [28]. For retarded systems with undelayed input and output variables the LQCP has been studied by various authors for about twenty years. We mention the work of Krasovskii [20], Kushner & Barnea [21], Alekal, Brunovskii, Chyung & Lee [1], Delfour & Mitter [14], Curtain [6], Manitius [25], Delfour, McCalla & Mitter [13], Delfour, Lee & Manitius [11], Delfour [8], Banks & Burns [2]. First results on systems with a single-point delay in the state and control variables can be found in Koivo & Lee [19] and Kwong [22]. Ichikawa [16] has developed a comprehensive evolution equation approach for the treatment of the LQCP for RFDEs with input delays. His idea was to include a past segment of the input function in the state of the system. A completely different approach to this problem has been developed by Vinter & Kwong [30] for RFDEs with distributed input delays. Their approach has been generalized to RFDEs with general delay, in the state and control variables by Delfour [9, 10] and to neutral systems by Karrakchu [18] in her recent thesis. The LQCP for neutral systems with output delays has been studied by Datko [7] and Ito & Tarn [17].

The asymptotics of solutions of singularly perturbed functional differential equations: Distributed and concentrated delays are different

This paper illustrates the differences between systems with distributed delays and systems having only concentrated delays in what concerns the asymptotic rates of solutions of singularly perturbed linear retarded functional differential equations. An example of a system with distributed delays shows that the introduction of a “slow” variable coupled with the “fast” variable may decrease the asymptotic rates of solutions observed when the perturbation parameter is close to zero. Such a situation cannot happen for ordinary differential equations, or even for differential-difference equations.

Null controllability in function space of nonlinear retarded systems with limited control

The question of controlling both linear and nonlinear retarded functional differential equations from an initial function in a Sobolev space W 2 (1) to zero in W 2 (1) is considered when the controls are square integrable with values in a compact subset U of m-dimensional Euclidean space with zero in its interior. Sufficient computable criteria for null controllability are given.

Exponential estimates for singularly perturbed linear fuctional differential equations

Exponential estimates on the fundamental matrix, uniform on the perturbation parameter, are obtained for singularly perturbed systems of linear retarded functional differential equations, under the assumption that the eigenvalues of a certain coefficient matrix in the system have negative real parts. The exponential rates in the estimates are computable from upper bounds on the real parts of the characteristic values of the system or of associated simpler equations. Differences between differential-difference equations and equations with distributed delays are emphasized.

Convergence and boundary layers in singularly perturbed linear functional differential equations

The convergence of solutions of singularly perturbed systems of linear retarded functional differential equations to solutions of the associated degenerate problem is discussed in connection with the geometry of the flow in phase spaces defined in terms of square-integrable functions.

Invariant manifolds for singularly perturbed linear functional differential equations

The existence of “slow” and “fast” manifolds, and of invariant manifolds approaching the manifold of orbits of the degenerate system, is discussed for singularly perturbed systems of linear retarded functional differential equations (FDE). It is shown that these manifolds exist only in very degenerate situations and, consequently, the geometry of the flow of singularly perturbed ordinary differential equations does not generalize to FDEs.

Stability conditions for linear retarded functional differential equations

The purpose of this note is to study the exponential stability for the linear retarded functional differential equation x ̇ (t) = ∫ −1 0 [dη(θ)] x(t − r(θ)) , where the delay function r( θ) ⩾ 0 is continuous and η( θ) is of bounded variation on the interval [−1, 0]. It is shown that the spectral limit function for the equation above has a continuous dependence on the pair ( η, r). The set of all functions of bounded variation η for which the equation above is exponentially stable for every delay function r, the so-called region of stability globally in the delays, is a cone. Therefore for a fixed r, the set of all η which make our equation exponentially stable, that is, the region of stability for the delay function r, contains a cone. A discussion of the characterization of these regions of stability, as well as of the largest cone contained in each region of stability for a fixed delay function r, is given. Some remarks are made with respect to a similar question for the equation x ̇ (t) = Ax(t) + ∫ − 1 0 [dμ(θ)] x(t−r(θ)) , where A is a real n by n matrix, μ(θ) is bounded variation on [−1, 0] and r( θ) as before. Several examples illustrate the results obtained.

F-controllability and observability of linear retarded systems

The paper gives new results onF-controllability of linear retarded functional differential equations. The concept ofF-controllability, weaker than the approximate controllability in the state space, has been introduced by this author in 1976. In the present paper new conditions are presented which, in the case of systems with one delay, become easily verifiable criteria ofF-controllability. A duality betweenF-controllability and observability is discussed, and several examples are given.

Existence of periodic orbits of autonomous retarded functional differential equations

For proving the existence of periodic orbits of autonomous ordinary differential equations, three different methods are available, namely, the Hopf bifurcation theorem, the torus principle and the Poincaré–Bendixson theorem. Until recently, the Poincaré–Bendixson theorem was applicable only to equations of the second order. However, it was extended in (13, 14) to certain equations of higher order by means of a plane projection technique. In §§ 2, 3 of the present paper this technique is adapted to prove analogues of the Poincaré–Bendixson theorem for a class of autonomous retarded functional differential equations. For such equations the orbits lie in a Banach space studied by Hale ((7), p. 43). The main hypothesis of this theorem is that the equation has a bounded semi-orbit whose ω-limit set contains no critical points. The problem of finding such a semi-orbit is solved in § 4 for a class of dissipative equations. The practical application of these results requires the construction of certain functionals which are similar to the Liapunov functionals of stability theory. In § 5, these functionals are constructed for a large class of retarded feedback control equations and explicit conditions for the existence of periodic orbits are deduced. Some practical details are illustrated in § 6 by applying the theory to certain scalar delay-differential equations.

Ważewski's Principle for retarded functional differential equations

Ważewski Principle is an important tool in the study of the asymptotic behavior of solutions of ordinary differential equations. A direct extension of this principle to retarded functional differential equations (RFDEs) can be obtained by noticing that solutions of RFDEs generate processes on C = C([− r, 0], R n ) and by using the general version of Ważewski Principle for flows on topological spaces. The resulting method is of little use in applications, due to the infinite-dimensionality of the space C. This paper presents a “Razumikhin-type” extension of Ważewski's Principle, which is widely applicable to concrete examples. The main results are Corollaries 3.1 and 3.2. Also, an extension of the method to RFDEs with a merely continuous right-hand side is given, and a few examples illustrate the use of the method. Throughout the paper, a standard notation is used.

Completeness and F-completeness of eigenfunctions associated with retarded functional differential equations

The paper gives general necessary and sufficient conditions for completeness of generalized eigenfunctions associated with systems of linear autonomous retarded functional differential equations (FDE), in the Hilbert space R n × L 2([− h, 0], R n ), and also in the space C([− h, 0], R n ). The generalized eigenfunctions are elements of the nullspaces of ( Iλ − A) m , where A is the infinitesimal generator of a C 0-semigroup of bounded linear operators on R n × L 2([− h, 0], R n ) (resp. C([− h, 0], R n )) corresponding to the FDE in question. In addition to the usual notion of completeness, a new concept of F-completeness is introduced and its significance is explained. In particular, it is shown that the F-completeness is related to the absence of solutions of the transposed equation that vanish in finite time. The results are obtained entirely via the C 0-semigroup theory, which results in simplicity of the proofs. As a by-product, some new results on the adjoint semigroup are obtained. The main results are expressed in an operator form. These are translated into conditions expressed in terms of the original system matrices. For systems with one delay, the F-completeness criterion is translated into a verifiable matrix type criterion, in which the concepts of maximal controllability and invariant subspaces of two matrices play a prominent role.

Capture in linear functional differential games of pursuit

The problem of capture in a pursuit game which is described by a linear retarded functional differential equation is considered. The initial function belongs to the Sobolev space W 2 (1). The target is either a subset of W 2 (1) a point in W 2 (1), a subset of the Euclidean space E n or a point of E n . There is capture if the initial function can be forced to the target by the pursuer no matter what the quarry does. The concept of capture therefore formalizes the concepts of controllability under unpredictable disturbances. This is proved to be equivalent to the controllability of an associated linear retarded functional differential equation. There is nothing in (2) (6) or (7) below which restricts the control sets to be of the same dimension as the phase space. Our results can be applied in (2) for example, if the constraint sets Q′, P′ are subsets of E m and E i respectively with q( t) = C( t) q′( t), − p( t) = B( t) p′( t), q′( t) ϵ E m p′( t) ϵ E r and B( t) is an n × r′-matrices and C( t) an n × m-matrix.

A finite difference technique for solving optimization problems governed by linear functional differential equations

Aspects of the approximation and optimal control of systems governed by linear retarded nonautonomous functional differential equations (FDE) are considered. First, certain FDE are shown to be equivalent to corresponding abstract ordinary differential equations (ODE). Next, it is demonstrated that these abstract ODE may be approximated by difference equations in finite dimensional spaces. The optimal control problem for systems governed by FDE is then reduced to a sequence of mathematical programming problems. Finally, numerical results for two examples are presented and discussed.

A star-shaped condition for stability of linear retarded functional differential equations†

SYNOPSISSufficient conditions are developed for asymptotic stability of the autonomous linear functional differential equation of retarded type. If the asymptotic stability ofimplies the asymptotic stability ofthen these conditions are also necessary. Necessary and sufficient conditions are developed for the largest cone in the region of stability. These results are illustrated with the example

Methods for approximating solutions to linear hereditary quadratic optimal control problems

A certain approximation scheme based on approximations of L 2 spaces is employed to formulate a numerical method for solving quadratic optimal control problems governed by linear retarded functional differential equations. This piecewise linear method is an extension of the so-called averaging technique. It is shown that the Riccati equation for the linear approximation is solved by simple transformation of the averaging solution. In fact, the numerical procedures are identical, except for the computation of an initial condition. Numerical examples are included.