Set Invariance Research Articles

The invariance of limit sets for retarded differential equations

On démontre la semi-invariance ou la quasi-invariance des ensembles limites pour une équation différentielle à retard, sans supposer l'unicité ni la prolongeabilité des solutions, et sans supposer non plus que la solution engendrant l'ensemble limite soit contenue dans un ensemble compact ou fermé.

The Invariance of Limit Sets for Autonomous Functional–Differential Equations

The Invariance of Limit Sets for Autonomous Functional–Differential Equations

Asymptotic behavior of a class of abstract dynamical systems

In recent years the extension of the second or direct method of Liapunov to distributed parameter systems has gained much attention; specifically Movchan [I] and Zubov [2] have given results paralleling Liapunov’s classical theorems. However, in dealing with the question of asymptotic stability of equilibria, these theorems are not always applicable because of their rather stringent hypotheses on the negative definiteness of the time derivative of Liapunov type functionals. In [3] Barbashin and Krasovskii did weaken somewhat the hypotheses of Liapunov’s theorems for autonomous and periodic systems but it was LaSalle [46] who exploited the invariance of limit sets to obtain a much more general theory, his principal theorem being stated in the form of an invariance principle. This was then extended by Hale [7] for the infinite dimensional space of autonomous functional differential equations and by Miller in [8,9] f or almost periodic ordinary and functional differential equations. These developments suggest the desirability of formulating invariance principles for infinite dimensional spaces suitable for partial differential equations. This paper is in that direction. The direct generalization of autonomous systems of ordinary differential equations to infinite dimensions is the abstract dynamical system. Thus, one would like to state an invariance principle similar to the theorem of LaSalle for autonomous ordinary differential equations which is valid for dynamical systems in general. Brayton and Miranker [lo] stated a type of invariance principle for abstract dynamical systems but their work was limited to a particular class of systems

Closed systems of functions and predicates

In this paper we show that there is a one to one correspondence between systems of functions defined on a finite set A and systems of predicates defined on A. This result implies that a complete set of invariants for a universal algebra on A is given by predicates defined on A. Conversely functions on A provide a complete system of invariants for sets of predicates closed under conjunction, change of variable and application of the existential quantifier. We begin in § 2 by giving a definition of closure for systems of functions and predicates. This is followed by a definition of commutivity of a function and a predicate which gives a correspondence between the two types of systems. In Theorems 1 and 2 of § 3 we show that the correspondence is a Galois connection. In Theorem 3 we consider sets of predicates closed under the existential quantifier and show that the corresponding systems are determined by functions defined for all values of the arguments. In Theorems 4 and 5 we include disjunction and then negation in the definition of closure of a set of predicates. We also require that equality be among the predicates. The corresponding systems consist of essentially first order functions and essentially first order permutations respectively. We conclude in § 4 with some comments on the infinite case and some general comments on these results.