# zbMATH — the first resource for mathematics

Remarks on the theory of implicit linear continuous-time systems. (English) Zbl 0833.93031
Implicit linear continuous-time systems described by differential equations of the form: $$Ex' (t)= Fx (t)+ Gu(t)$$ which may fail to have solutions in the classical sense are considered. The distributional theory of such equations is sketched and the space of admissible initial conditions as well as the controllable spaces are determined.
Reviewer: L. Goras (Iaşi)

##### MSC:
 93C35 Multivariable systems, multidimensional control systems 93C05 Linear systems in control theory 93C99 Model systems in control theory
Full Text:
##### References:
 [1] A. Banaszuk M. Kociecki, K. M. Przyluski: On almost invariant subspaces for implicit linear discrete-time systems. Systems Control Lett. 11 (1988), 289-297. · Zbl 0666.93052 [2] A. Banaszuk M. Kociecki, K.M. Przyluski: Implicit linear discrete-time systems. Math. Control Signals Systems 3 (1990), 271-297. · Zbl 0715.93038 [3] A. Banaszuk M. Kociecki, K.M. Przyluski: Kalman-type decomposition for implicit linear discrete-time systems and its applications. Internat. J. Control 52 (1990), 1263-1271. · Zbl 0707.93007 [4] D. Cobb: On the solutions of linear differential equations with singular coefficients. J. Differential Equations 46 (1982), 310-323. · Zbl 0489.34006 [5] D. Cobb: A further interpretation of inconsistent initial conditions in descriptor variable systems. IEEE Trans. Automat. Control AC-28 (1983), 920-922. · Zbl 0522.93037 [6] D. Cobb: Controllability, observability and duality in singular systems. IEEE Trans. Automat. Control AC-29 (1984), 920-922. · Zbl 0522.93037 [7] V. Dolezal: Dynamics of Linear Systems. Academia, Prague 1967. · Zbl 0173.30303 [8] H. Frankowska: On controllability and observability of implicit systems. Systems Control Lett. 14 (1990), 219-223. · Zbl 0699.93003 [9] F. R. Gantmacher: The Theory of Matrices. Chelsea, New York 1959. · Zbl 0085.01001 [10] T. Geerts: Free end-point linear-quadratic control subject to implicit continuous-time systems: necessary and sufficient conditions for solvability. Preprints of the Second IFAC Workshop on System Structure and Control, Prague 1992, pp. 28-31 [11] T. Geerts: Invariant subspaces and invertibility properties of singular systems: the general case. Res. Memo. 557, Dept. of Economics, Tilburg University 1992. · Zbl 0774.93038 [12] T. Geerts: Solvability conditions, consistency and weak consistency for linear differential-algebraic equations and time-invariant linear systems: the general case. Res. Memo. 558, Dept. of Economics, Tilburg University 1992. · Zbl 0772.34002 [13] T. Geerts, V. Mehrmann: Linear differential equations with constant coefficients: A distributional approach. Preprint 09-073, Universität Bielefeld 1990. [14] M. L. J. Hautus, L. M. Silverman: System structure and singular control. Linear Algebra Appl. 50 (1983), 369-402. · Zbl 0522.93021 [15] L. Hormander: The Analysis of Linear Partial Differential Operators: Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin 1983. [16] K. Ozcaldiran, L. Haliloglu: Structural properties of singular systems. Part 1: Controllability. Preprints of the 2 IFAC Workshop on System Structure and Control, Prague 1992, pp. 344-347 [17] V.-P. Peltola: Singular systems of ordinary linear differential equations with constant coefficients. Report-MAT-A221, Helsinki University of Technology 1984. [18] K. M. Przyluski, A. Sosnowski: Remarks on implicit linear continuous-time systems. Preprints of the 2 IFAC Workshop on System Structure and Control, Prague 1992, pp. 328-331 [19] V. S. Vladimirov: Generalized Functions in Mathematical Physics. Mir Publishers, Moscow 1979. · Zbl 0515.46034 [20] K.-T. Wong: The eigenvalue problem $$\lambda Tx + Sx$$. J. Differential Equations 16 (1974), 270-280. · Zbl 0327.15015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.