Exact boundary controllability of the linear advection equation

The advection equation is used to model the transport of material whose mass is conserved. If the vector field governing the advection is divergence-free, then the advection equation reduces to the usual transport equation which is used to model linear diffusion processes such as transport of neutrons, scattering of light and propagation of -rays in a scattering medium [1]. The uncontrolled advection and transport equations are solved using the Perron-Frobenius and Koopman semigroup evolution operators which are adjoints of each other. For more on the relationship between these two operators and ergodic theory, we refer the reader to [2]. The Perron-Frobenius (P-F) operator was recently used to develop a weaker notion of stability (almost everywhere stability) for ordinary differential equations in [3,4]. In [4], almost everywhere stability of invariant sets in dynamics governed by (3) was characterized using the Lyapunov density as a stability certificate. Roughly speaking, an invariant set is said to be almost everywhere stable if every initial condition except for a set of zero Lebesgue measure flows asymptotically into the invariant set. This allows for dynamics with unstable invariant sets (equilibrium points in one dimension and periodic orbits in higher dimensions) to be classified as stable provided such sets have zero Lebesgue measure. Keeping in line with this new development, we have chosen LðXÞ as the state space for exact controllability for two reasons. First, we want to allow for the existence of unstable invariant sets but of Lebesgue measure zero in the interior of X and hence need a state space that can consider states differing on a zero Lebesgue measure