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Abstract

The study of problems and structural properties of regular and extended state space theory may be reduced to a study of first—order linear differential equations.

Matrix pencil theory is the key tool for the study of S(F,G) differential systems. The development of a general theory for S(F,G) systems, and thus the development of a unifying theory for linear systems, necessitates the further enrichment of the classical theory of strict equivalence for matrix pencils. To cover the needs of linear systems, a matrix pencil theory has to be general enough and should have a geometric, dynamic, topological, invariant theory and computational dimension. This thesis aspires to contribute in the development of the matrix pencil theory along the above lines and thus con-tribute in the foundations of a matrix pencil based unifying theory for li-near systems.

The theory of strict equivalence is detached from its algebraic context and it is presented as a theory of ordered pairs (F,G). The strict equivalence invariants are defined in a number theoretic way, by the properties of appropriate Peicewise Arithmetic progression sequences defined on a pair (F,G). This new characterisation of strict equivalence invariants allows the derivation of new procedures for computing the Kronecker canonical form, for constructing minimal bases, and provides the means for the establishment of the geometric theory of strict equivalence invariants. The subspaces of the domain of (F,G) are classified in terms of the invariants of the restriction pencil. A variety of notions of invariant subspaces emerges, such as (F,G)- , (G,F)-, complete-(F,G)-invariant subspaces and extended-(F,G)-, (G,F)-, complete- (F,G)-invariant subspaces. These notions of invariant subspaces are the counterparts of the standard notions of invariant subspaces of the geo-metric theory. The properties of the solution space of S(F,G) differential systems are studied and the different notions of invariant subspaces are characterised dynamically in terms of the properties of C distributional-holdability and C -, distributional reachability. Once more, these dynamic properties are generalisations of the fundamental notions of geometric theory. The theory of invatiants of matrix pencils, or ordered pairs, is enriched by the study of invariants under Bilinear strict equivalence; a complete set of invariants is defined under this equivalence. This study leads to a ’’space frequency” relativistic classification of the dynamic and geometric proper-ties of S(F,G) systems and their invariant subspaces; furthermore, it provides the means for a systematic study of dual systems and problems in linear systems. The further development of the theory of invariant forced realisations, allows the translation of results and properties derived on S(F,G) back to linear systems theory. Finally, the problem of defining appropriate topological settings for the study of properties of pencils under under uncertainty in their description is examined. New metric topologies are introduced and their links to. known results of the perturbation theory of the generalised eigenvalue-eigenvector problem are established.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics T Technology > TK Electrical engineering. Electronics Nuclear engineering
Departments:	School of Science & Technology > Department of Engineering School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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