A broad class of evolution equations are approximately controllable, but never exactly controllable

Controllability of the semilinear Benjamin-Bona-Mahony equation

“…In the particular case that a = 0 and b = 1 the operator H define by (4.4) is compact and Rang(H) is compact set (see [3]) , and as a consequence we obtain the interior approximate controllability of the semilinear heat equation (see [12]). …”

Approximate Controllability of a Semilinear Heat Equation

“…In fact, since is smooth and satisfies (12) and the semigroup { ( )} ≥0 given by (19) is compact (see [19,20]), then using the result from [1], we obtain that the operator is compact, which implies that the operator is compact. Moreover,…”

“…It is clear that exact controllability of the system (1) implies approximate controllability, null controllability, and controllability to trajectories of the system. But it is well known (see [1]) that due to the diffusion effect or the compactness of the semigroup generated by −Δ, the heat equation can never be exactly controllable. We observe also that in the linear case, controllability to trajectories and null controllability are equivalent.…”

“…Recently, the interior controllability of the semilinear heat equation (1) has been proved in [5][6][7] under the following condition: there are constants , ∈ R, with ̸ = − 1, such that…”

On partialS-controllability of semilinear partially observable systems

“…But the distinction is significant if the dimension is infinite. Indeed, there is a broad class of infinite-dimensional systems that are approximately controllable, but not exactly controllable (Bárcenas, Leiva, & Sívoli, 2005 …”