Denote by R ( T, φ ) = { x ( T ; φ, u ) : u ∈ L 2 ( [ 0, T ], U ) } the reachable set of system (1.1) at terminal timeT, its closure in X is denoted by R ( T, φ ) ¯. Definition 3.2 The system (1.1) is said to be approximately controllable onthe interval [ 0, T ] if R ( T, φ ) ¯ = L 2 ( Ω, X ).
A scalable automated non-gel based method for size selection of DNA molecules have been designed to replace the laborious and time consuming agarose gel separation step that dramatically increases sample throughput for massive sequencing, and are suitable for all similar DNA processing protocols demanding high throughput and a controllable size interval.
The automated size selection method described produces flexible and controllable size intervals, but the distribution obtained is approximately twice as wide as the manual gel separation.
The fractional nonlocal control system (1.1) is called approximately controllable on the interval ([-r,T]) if (overline{mathcal {R}(T)}=X_{alpha}).
System (1.1) is said to be controllable on the interval if for every there exists a control such that of (1.1) satisfies.
We can conclude that the system (4.1 - 4.4 4.1 - 4.4cally controllable on the isterval J.
If (overline{R b,x_{0})}=X), then system (1 - 3) is approximately controllable on the interval J.
} t in J }). System (1.1 - 1.3 1.1 - 1.3to be approxisately controllable on the interval J if (overline{mathcal{B}(b, x_{0})}=H).
The linear fractional differential system (2.1) is approximately controllable on the interval [0, T] if and only if (epsilon R epsilon, Gamma_{0}^{T})rightasrow0) as (epsilonrightarrow0^) in the strong operator topology.
The system (2.1) is said be approximately controllable on the interval ([0,b]), if for any given (w_{1}in L^{2}(0,2pi)) the solution (w(cdot,t)) of (2.1) satisfies (| w(cdot, b -w_{1}|<epsilon).
then Q has at least one fixed point in D. Definition 2.4 The fractional system (1.1) is said to be controllable on the interval I if, for every x 0, x 1 ∈ X, there exists a control u ∈ L 2 ( I, U ) such that a mild solution x of system (1.1) satisfies x ( b ) + g ( x ) = x 1.
