  Continuous linear dynamical systems are used extensively in mathematics,
computer science, physics, and engineering to model the evolution of a system
over time. A central technique for certifying safety properties of such systems
is by synthesising inductive invariants. This is the task of finding a set of
states that is closed under the dynamics of the system and is disjoint from a
given set of error states. In this paper we study the problem of synthesising
inductive invariants that are definable in o-minimal expansions of the ordered
field of real numbers. In particular, assuming Schanuel's conjecture in
transcendental number theory, we establish effective synthesis of o-minimal
invariants in the case of semi-algebraic error sets. Without using Schanuel's
conjecture, we give a procedure for synthesizing o-minimal invariants that
contain all but a bounded initial segment of the orbit and are disjoint from a
given semi-algebraic error set. We further prove that effective synthesis of
semi-algebraic invariants that contain the whole orbit, is at least as hard as
a certain open problem in transcendental number theory.
