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On operators of transition in Krein spaces

Grod A., Kuzhel S., Sudilovskaya V.

Opuscula Mathematica

2011

Vol. 31, no. 1

49-59

The paper is devoted to investigation of operators of transition and the corresponding decompositions of Krein spaces. The obtained results are applied to the study of relationship between solutions of operator Riccati equations and properties of the associated operator matrix L. In this way, we complete the known result (see Theorem 5.2 in the paper of S. Albeverio, A. Motovilov, A. Skhalikov, Integral Equ. Oper. Theory 64 (2004), 455-486) and show the equivalence between the existence of a strong solution K (//K// < 1) of the Riccati equation and similarity of the J-self-adjoint operator L to a self-adjoint one.

Feedback control for time-varying descriptor systems.

Kurina GA.

Systems Science

2000

Vol. 26, nr 3

47-59

The control in the feedback form and the optimal cost are obtained for different linear-quadratic optimal control problems by time-varying descriptor systems in Hilbert space. For that purpose the operator which is the solution of the operator Riccati equation is used. This operator acts in all state space, unlike the operator in the subspace in the case of a singular operator before the derivative, which determines an optimal control with the help of part of the state variable. In contrast to previous works of other authors the regularity of the pencil of operators from the state equation is not required. At the end of the paper the feedback control for linear-quadratic optimal control problems with the degenerate Legendre condition is considered.