Exact Controllability of C0-groups with One-Dimensional Input Operators

Assume that A generates a C0-group T(t) on the complex separable Hilbert space H and that b ∈ D(A*)’ is an admissible control operator for the semigroup T(t). It is known that exact controllability of the system defined by A and b implies that every element of the spectrum of A is an eigenvalue. We develop equivalent conditions for exact controllability of the system defined by A and b. These conditions are given in terms of the eigenvalues and eigenvectors of A and the control operator b. Also necessary conditions are included. A necessary condition is that one over the eigenvalues is a sequence in l 1+ɛ,ɛ > 0. If additionally A is a diagonal operator then we prove that the conjecture of Russell and Weiss[12] holds.

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