System (3.9) is said to be approximately controllable at time whenever the set is densely embedded in ; that is, (4.44).
The system (SE) is said to be approximately controllable at time T if Cl{x(T; g, u): u ∈ L 2 0, T; U } = V* where Cl denotes the closure in V*.
Therefore, the system (3.6) is approximately controllable at time T. In order to investigate the controllability of the nonlinear control system, we need to impose the following condition.
System (1.1) is said to be approximately controllable on I if for every (x_{0}, x_{1}inmathbb{X}), there is some control (uin L^{2}(I,U)), the closure of the reachable set (R(b)) is dense in (mathbb{X}), i.e., (overline{R(b)}=mathbb{X}), where (R(b)={x(b;u):u(cdot)in L^{2}(I,U }).
The system (2.8) is said to be approximately controllable on if for every, there exists such that the solution of (3.2) corresponding to verifies: (3.3).
System (1.2) is said to be approximately controllable on if the space is dense in and approximately controllable in finite time if the space is dense in.
} 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 system (NCE) is said to be approximately controllable in the time interval, if for every given final state,, and there is a control function such that.
Definition 2.8 The fractional system is said to be approximately controllable on [ 0, b ] if K b ( f ) ¯ = X, where K b ( f ) ¯ denotes the closure of K b ( f ).
The system (NCE) is said to be approximately controllable in the time interval if for every desired final state and, there exists a control function such that the solution of (NCE) satisfies, that is, if where is the closure of in, then the system (NCE) is called approximately controllable at time.
