Exact null controllability of degenerate evolution equations with scalar control

On control of degenerate distributed systems

“…It was proven in [23,Thm. 5] that operator is strongly ( , 0)-radial, where , correspond to a class of boundary value problems involving problem (21), (22). In accordance with this result, if > 0, − 2 ̸ = 0, operator is strongly ( , 0)-radial, where , are defined in the present section.…”

“…We choose operators , as in Section 7. By Theorem 12 and in view of Remark 3, in the case of -controllability in a free time of system (32), (22), the identity dom 0 = span { −1 0 0 (0), = 1, 2, . .…”

“…(22) iscontrollable in a free time if and only if (0) ̸ = 0. If (0) ̸ = 0, ( ) ̸ = 0, then system (21), (22) is -controllable in time > 0.…”

“…It is to make sure that operator ( ) in this problem is so that for 0 we have im ( ) = if and only if 11 ( ) 22 (23), (24) is -controllable in a free time. (23), (24) in time .…”

“…We also mention work [22] dealing with distributed control system (1) involving a strongly ( , )-radial operator . Here the criterion of complete (or exact) controllability system in a Hilbert space was formulated by means of notion of strongly minimal sequence of generalized exponentials.…”