Self-conjugate differential and difference operators arising in the optimal control of descriptor systems

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Author(s) : Volker Mehrmann , Lena Scholz

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 26-2012

MSC 2000

93C05 Linear systems
93C55 Discrete-time systems
93C15 Systems governed by ordinary differential equations
65L80 Methods for differential-algebraic equations
49K15 Problems involving ordinary differential equations
34H05 Control problems

Abstract :
We analyze the structure of the linear differential and difference operators associated with the necessary optimality conditions of optimal control problems for descriptor systems in continuous- and discrete-time. It has previously been shown that in continuous-time the associated optimality system is a self-conjugate operator associated with a self-adjoint pair of coefficient matrices and we show that the same is true in the discrete-time setting. We also extend these results to the case of higher order systems. Finally, we discuss how to turn higher order systems with this structure into first order systems with the same structure.

Keywords : Differential-algebraic equation, self-conjugate difference operator, self-adjoint pair, discrete-time optimal control, necessary optimality condition, congruence transformation, higher order systems.