Richard Datko

An example on the effect of time delays in boundary feedback stabilization of wave equations

This note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations. The question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays.

Extending a theorem of A. M. Liapunov to Hilbert space

The purpose of this paper is to extend a well known theorem of A.M. Liapunov concerning Hurwitzian matrices (n x n matrices with eigenvalues in the half plane Re z < 0) to strongly continuous semi-groups of operators on a complex Hilbert space. Liapunov’s result may be phrased as follows: Let A be a complex n x n matrix and A* its adjoint. Then A has all its characteristic roots lying in the half plane Re z < 0 if and only if the solution in B of the matrix equation A*B + BA = -I (the identity matrix) is a unique positive definite Hermitian matrix (see e.g. ([I] p. 245)). A partial extension of Liapunov’s theorem has been given in [2]. This paper is a sequel to that paper in the sense that much sharper results than those in [2] are obtained if we restrict ourselves to Hilbert spaces instead of general Banach spaces.

Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks

We present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls. We show that in a general case, when the controls are distributed, stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks. We also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems.

Two examples of ill-posedness with respect to time delays revisited

We revisit two examples of elastic control systems which are stabilized by boundary feedback and then destabilized when time delays occur in the feedback. In the first example, a wave equation with a feedback gain greater than or equal to one, arbitrary divergence rates are obtained for small time delays. In the second example, a Euler-Bernoulli beam, any time delay results in an arbitrary divergence rate.

Linear autonomous neutral differential equations in a Banach space

Abstract The existence of solutions for a class of linear functional differential equations defined on a general Banach space is established; the solutions are shown to generate a semigroup of class C0; a representation of the solutions in terms of a particular family of linear transformations is developed.

Two questions concerning the boundary control of certain elastic systems

Abstract We present two questions connected with the stabilization of certain hyperbolic partial differential equations which are controlled on their boundaries. The uncontrolled versions of these systems are conservative and therefore represent only approximate realizations of the physical phenomena. The first question concerns the qualitative behavior of a given stabilizing feedback for such a system vis a vis that of an appropriately damped realization using the same feedback. The second question concerns the robustness of both the original system and a damped version when time delays occur in a given feedback stabilization scheme.

Representation of solutions and stability of linear differential-difference equations in a Banach space

Abstract In this paper the theory of linear delay differential equations is extended in three directions. One, the underlying phase space is allowed to be a Banach space so that equations with unbounded operators may be considered. Two, the delay is permitted to be effective over an infinite interval and a connection is made between this type of system and neutral systems whose delay is effective over a finite interval. Three, a theory of uniform asymptotic stability for linear delay differential equations in a Hilbert space is developed.

Remarks concerning the asymptotic stability and stabilization of linear delay differential equations

Abstract This note concerns the asymptotic stability for all values of the delays of controlled and uncontrolled linear delay differential equations. In the case of uncontrolled systems it is shown by example that structural considerations must be accounted for. In the controlled case, although arbitrary pole placement may not be possible, there is an algorithm which sometimes reduces the spectrum of the feedback system to one which is finite and is contained in the left half plane.

AN ALGORITHM FOR COMPUTING LIAPUNOV FUNCTIONALS FOR SOME DIFFERENTIAL-DIFFERENCE EQUATIONS

Publisher Summary This chapter discusses an algorithm for computing Liapunov functionals for some differential-difference equations. In this note, an algorithm is developed for determining the stability behavior of linear autonomous differential-difference equations with a single lag. The technique is to treat the solutions to the differential-difference equation as the range of a strongly continuous semi-group of the operators of class C0 and to attempt to find Hermitian forms H and R such that a particular algebraic relationship holds between the forms H and R and the infinitesimal generator, A, of the semi-group. It was assumed that R has a particular representation in terms of unknown matrices and using knowledge of H and A one hope to determine these matrices. This process leads to the solution of a two-point boundary value problem for a system of linear ordinary differential equations.

Lyapunov functionals for certain linear delay differential equations in a Hilbert space

Abstract A theory of stability via Lyapunov functionals is developed for a general class of autonomous delay differential equation whose values lie in a Hilbert space.