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# Perturbation theory, stability and control for classes of delay differential-algebraic equations

Abstract:

A neutral delay differential equation can be written as a system of a retarded delay differential equation coupled with a difference equation and such a system can be considered as a delay differential-algebraic equation. In contrast to the case for retarded delay equations, there is not yet an effective perturbation theory for difference equations and more general for delay differential-algebraic equations available.

For a given neutral delay differential equation, one defines a semigroup of operators by shifting along the solution. The need to develop a perturbation theory for such solution semigroups arises in the context of stability and control problems. The answer employs the variation-of-constants formula, which involves integration in a Banach space. The reason that there does not yet exist a perturbation theory for delay differential-algebraic equation is related to the fact that in general solutions of such equations lack smoothness properties, see [3].

A first key idea is to describe a perturbation at the generator level by a cumulative output map [1] with finite dimensional range and to construct the semigroup by solving a linear finite dimensional convolution equation. Once this is done, all that remains is to explicate the integral in the variation-of-constants formula in the nonsmooth case. Already in 1953 Feller [2] emphasized that one might use the Lebesgue integral for scalar valued functions of time obtained by pairing a linear semigroup acting on an element with an element of the dual space and that there is no need to require strong continuity. More recently this point of view was elaborated by Kunze [4] who defined a Pettis type integral in the framework of a norming dual pair of spaces.

The aim of the lecture, which is based on joint work with O. Diekmann, is to derive in this manner a powerful version of the variation-of-constants formula for neutral delay equations.

We illustrate our perturbation with some results about stability and control of difference equations with applications to boundary control of partial differential equations.

[1] O. Diekmann, M. Gyllenberg and H.R. Thieme , Perturbing semigroups by solving Stieltjes Renewal Equations, Diff. Int. Equa. 6 (1993) 155-181.

[2] W. Feller , Semi-groups of transformations in general weak topologies, Ann. of Math. 57 (1953) 287-308.

[3] J.K. Hale and S.M. Verduyn Lunel, Effects of small delays on stability and control, In: Operator Theory and Analysis, The M.A. Kaashoek Anniversary Volume (eds. H. Bart, I. Gohberg and A.C.M. Ran), Operator Theory: Advances and Applications 122, Birkhäuser, 2001

[4] M. Kunze , A Pettis-type integral and applications to transition semigroups, Czech. Math. J. 61 (2011) 437-459.