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We consider the one-dimensional wave equation, with a damping term that convolies the past history with an infmite sum of exponentials. The system is reformulated, as an abstract Cauchy problem. By means of semigroup theory it is shown that the systems with truncated sum of exponentials approximate the original one in L 2 sense. On the other hand it is demonstrated that the system and the approximations have different spectral propertie Wir untersuchen die eindimensionale Wellengleichung mit einem Reibungsterm, der die vergangene Entwicklung mit einer unendlichen Samme ton Exponentiolfunktionen faltet. Das System wird als abstraktes Cauchy-Problem reformu liert. Mit Hilje der Theorie der C 0 -Halbgruppen wird gezeigt, dass die Systeme mit abgebrochenen Summen das Ausgangssystem im L d -Sinn approximieren. Es wird bemiesen. dass das Ausgangssystem und dic Approximationen unterschiedliche spektrale Eigenschaften besitzen.

A computational method for a system of partial integrodifferential equations of Volterra type is developed. These equations describe the damping of pressure waves in tubes of circular form, taking into account the nonstationary increase of friction. The analytical solution of the model equations is given. A rapidly convergent expansion as well as sharp error estimates for the approximate solutions are obtained. Finally an efficient algorithm using recursion formulas is presented and the algorithm is compared numerically with the method of characteristics. © 1995 John Wiley & Sons, Inc.

The linear quadratic control problem for systems governed by evolution equations of parabolic type is considered. The optimal feedback control uses a Riccati operator which is approximated by a Galerkin scheme. Optimal convergence rates (except for a logarithmic factor) are derived for the approximation of the Riccati operator for both the finite and infinite time interval. This is done in an abstract Hilbert space framework assuming a certain rate of convergence for the uncontrolled system. In the numerical part, the Galerkin method is compared with the collocation method with special emphasis on the feedback gains.