Structure of nonasymptotic stability sets of families of linear differential systems with a multiplying parameter

摘要

We consider families of linear differential systems depending on a real parameter that occurs only as a factor multiplying the matrix of the system. The stability (respectively, asymptotic stability) set of such a family is defined as the set of parameter values for which the corresponding systems in the family are stable (respectively, asymptotically stable). We show that a pair (P,Q) of sets on the real line is a pair consisting of the stability set P and the asymptotic stability set Q of some family if and only if the following conditions are satisfied: 0 ∈ P; if Q ≠ ∅, then P and Q are F σ - and F σδ -sets, respectively, lie on one of the closed rays issuing from zero, and satisfy Q ⊂ P{0}; if Q = ∅, then P is an F σ -set on the real line. In addition, for any pair (P,Q) of sets with these properties, the coefficient matrix of a family with multiplying parameter whose stability and asymptotic stability sets coincide with P and Q, respectively, can be chosen to be infinitely differentiable and uniformly bounded on the time half-line. The same problem of complete description of pairs consisting of stability and asymptotic stability sets is also solved for general one-parameter families of linear differential systems whose solutions continuously depend on a parameter.