Decidability and complexity of event detection problems for ODEs

Abstract

The ability of ordinary differential equations (ODEs) to simulate discrete machines with a universal computing power indicates a new source of difficulties for event detection problems. Indeed, nearly any kind of event detection is algorithmically undecidable for infinite or finite half-open time intervals, and explicitly given “well-behaved” ODEs (see [18]). Practical event detection, however, usually takes place on finite closed time intervals. In this article, the undecidability of general event detection is extended to such intervals. On the other hand, on finite closed time intervals, event detection in a certain approximate sense is quite generally decidable, which partly saves the case for practicable event detection. The capability of simulating universal Turing machines is still there, and is used to give complexity lower bounds in terms of accuracy of event detection. The ODEs used here are, of course, quite complicated, but not artificial, in that even from the point of view of practical event detection, it would appear difficult to find criteria to exclude them. © 1997 John Wiley & Sons, Inc.