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Algebraic theory of linear systems: a survey. (English) Zbl 1328.93074

Ilchmann, Achim (ed.) et al., Surveys in differential-algebraic equations II. Cham: Springer (ISBN 978-3-319-11049-3/pbk; 978-3-319-11050-9/ebook). Differential-Algebraic Equations Forum, 287-333 (2015).
The introduction contains the key ideas of algebraic approach to the theory of various types linear systems. The presentation via the introduction of a ring \(D\) of operators and a set \(\mathcal{F}\) of functions, where the solutions are sought. In the first three sections these ideas are applied separately to three important classes of linear systems: ordinary differential equations, difference equations and partial differential equations. Existence and uniqueness questions are solved via the construction of formally well-posed initial value problems. In sec. 5 it is shown how the index notion for differential algebraic equation can be reformulated in terms of the used general approach to linear systems. In the last sec., a general approach is presented for analysing abstract linear systems with the aid of algebraic and homological methods with the main emphasis to some theoretical aspects such as autonomy and controllability. The appendix presents the Gröbner bases theory as the fundamental tool in constructive algebra, permitting to perform algorithmically many basic algebraic constructions.
For the entire collection see [Zbl 1305.65006].

MSC:

93B25 Algebraic methods
93B05 Controllability
34A09 Implicit ordinary differential equations, differential-algebraic equations
13N10 Commutative rings of differential operators and their modules
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
35G40 Initial value problems for systems of linear higher-order PDEs
35N10 Overdetermined systems of PDEs with variable coefficients
68W30 Symbolic computation and algebraic computation
93B40 Computational methods in systems theory (MSC2010)
93C05 Linear systems in control theory

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