Treatment Of Linear Systems Research Articles

Minimality and observability of group systems

Group systems are a generalization of Willems-type linear systems that are useful in error control coding. It is shown that the basic ideas of Willems's treatment of linear systems are easily generalized to linear systems over arbitrary rings and to group systems. The interplay between systems (behaviors) and trellises (evolution laws) is discussed with respect to completeness, minimality, controllability, and observability. It is pointed out that, for trellises of group systems and Willems-type linear systems, minimality is essentially the same as observability. The development is universal-algebraic in nature and holds unconditionally for linear systems over the real numbers.

A Variational Principle for Nonlinear Systems

The equations describing a reactor system are sometimes nonlinear and do not admit a solution for the neutron density that is separable into a function of time only and a function of the remaining variables. An appropriate variational principle is given by demanding that the calculation of the observable nature of the reactor is insensitive to the value employed for the density, thus obtaining an equation for the optimum distribution of detectors to measure the observable behavior. This optimum weighting function is not identical with the conventional adjoint function or importance in the nonlinear range but the conventional treatment of linear systems is found to be a special case of the general principle. It is shown that the approximate treatment of nonlinear systems as eigenvalue systems is fundamentally unsound. (auth)

Jacobi's Problem on the Order of a System of Differential Equations

where j1, * *, j. is a permutation of 1, * * *, n. He arrives at the conclusion that the number of arbitrary constants in the solution of (1) does not exceed the greates sum (2). Now the number of arbitrary constants in the solution of a system is not a notion which is definite in advance. A system may have many families of solutions, each with its own arbitrary constants. Not every unknown need be determined by the system to within arbitrary constants. It seems safe to assume that before one can speak with completeness on the question of arbitrary constants, one will have to possess a theory for the decomposition of a system into systems of some normal type. Apparently no such theory exists at present for systems other than algebraic systems. It is thus not surprising that the argument on which Jacobi bases his conclusion, in the first of the above mentioned papers, should be whimsical in aspect. Without justification, Jacobi replaces the given equations by their equations of variation, which are linear. He then asserts without proof that, for his purpose, a linear system may be imagined to have constant coefficients. His treatment of linear systems with constant coefficients was completed by Chrystal.2 There are indications in Jacobi's second paper that his conclusion was actually based on a speculation as to the number of times which the given equations would have to be differentiated to permit the elimination of all unknowns except one. Investigations of the problem from this point of view were given by Nan-