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Preprint 13-2011

Self-adjoint differential-algebraic equations

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Author(s) : Peter Kunkel , Volker Mehrmann , Lena Scholz

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 13-2011

MSC 2000

93C10 Nonlinear systems
93C15 Systems governed by ordinary differential equations

Abstract :
Motivated from linear-quadratic optimal control problems for differential-algebraic equations (DAEs), we study the functional analytic properties of the operator associated with the necessary optimality boundary value problem and show that it is associated with a self-conjugate operator and a self-adjoint pair of matrix functions. We then study general self-adjoint pairs of matrix valued functions and derive condensed forms under orthogonal congruence transformations that preserve the self-adjointness. We analyze the relationship between self-adjoint DAEs and Hamiltonian systems with symplectic flows. We also show how to extract self-adjoint and Hamiltonian reduced systems from derivative arrays.

Keywords : Differential-algebraic equation, self-conjugate operator, self-adjoint pair, optimal control, necessary optimality condition, strangeness index, condensed form, congruence transformation, Hamiltonian system, symplectic flow

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