Computability, noncomputability and undecidability of maximal intervals of IVPs

Abstract

Let ( α , β ) ⊆ R (\alpha ,\beta )\subseteq \mathbb {R} denote the maximal interval of existence of solutions for the initial-value problem \[ { d x d t = f ( t , x ) , x ( t 0 ) = x 0 , \left \{ \begin {array} [c]{l}\frac {dx}{dt}=f(t,x), x(t_{0})=x_{0}, \end {array} \right . \] where E E is an open subset of R m + 1 \mathbb {R}^{m+1} , f f is continuous in E E and ( t 0 , x 0 ) ∈ E (t_{0},x_{0})\in E . We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable f f and ( t 0 , x 0 ) (t_{0},x_{0}) whose maximal interval of existence ( α , β ) (\alpha ,\beta ) is noncomputable. The fact that f f may be taken to be analytic shows that this is not a lack of the regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that ( α , β ) (\alpha ,\beta ) is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.