Constrained Controllability in Non Reflexive Banach Spaces

The dual quasisemigroup and controllability of evolution equations

“…Bárcenas and Leiva [3] generalize simultaneously the findings of Peichl and Schappecher [12], Papageorgiou [11], and Bárcenas and Diestel [2] by showing that the same approaches of null controllability are valid for systems governed by strongly continuous evolution operators with strongly continuous dual evolution operators in (0, +∞).…”

“…Bárcenas and Diestel [2], generalize the results of Peichl and Schappecher [12] to the case where the dual semigroup is strongly continuous in general Banach space; which allow them to apply this result to important partial differential equations in nonreflexive Banach spaces.…”

A note on the controllability of linear first order systems with holomorphic initial functions in elliptic complex numbers

“…Bárcenas and Diestel proved in [2] the following useful controllability criterion: Let X and U be Banach spaces, let B : U → X be a bounded linear operator, and A : X → X be the infinitesimal generator of a c 0 -semigroup {S(t)} t≥0 on X whose dual semigroup is strongly continuous on (0, ∞). Suppose Ω is a non-empty separable weakly compact convex subset of U containing 0.…”

“…is solvable provided the coefficients of (1) and (2) and the initial functions possess power-series representations. In [6] there are given conditions on the coefficients of (1) and (2) under which each initial value problem (3) and 4is solvable by assuming that the initial functions φ and ψ satisfy the Cauchy-Riemann conditions.Those results are generalized in [1], with the assumption that the initial value functions are holomorphic in the sense of an algebra with the structure polynomial X 2 +βX +α, where α and β are real numbers. In these algebras complex numbers are written as z = x+iy where i 2 = −βi−α and the two real functions u(x, y) and v(x, y) satisfy the Cauchy-Riemann equations if…”