Define 0 < M ≤ 2 N as number of reachable system states, determined as soon as a state is repeated during the signal propagation.
The true sequence of reachable system states is represented in an unknown matrix Γ* = {γ* itr : i ∈ 1,…, N, t ∈ 1,…, T, r ∈ 1,…, R}.
In this way, all reachable system states are computed and stored in a matrix Γ={ γ i k ∈{0,1}: i∈1,…, N, k∈1,…, M}, holding column-wise the activation states of all proteins at transition step k.
The result: a one-piece, no-wires, nondislodgeable, easily reachable personal music system.
But it must obtain all the system reachable states first, which is not realistic in medium- or large-scale systems.
end{aligned} Then the reachable sets of system (1) are bounded by a ball (B 0,r)={ zin R^{n}||z|leq r}) with r=frac{1}{sqrt{lambda_{min}(P }}.
end{aligned} Then the reachable sets of system (2) are bounded in a ball (B 0,r)={zin R^{n}|Vert zVert leq r}) with r=frac{1}{sqrt{lambda_{min}(P }}.
The set K b ( f ) = { x ( b ) ∈ X : u ∈ L 2 ( [ 0, b ], U ), x is the mild solution of (1.1) with control u }. is called the reachable set of system (1.1).
end{aligned} Then the reachable sets of system (1) are bounded in a ball (B 0,r)={zin R^{n}|Vert zVert leq r}) with r=frac{1}{sqrt{lambda_{min}(P_{i})}}, quad i=1,2,ldots,n.
The reachable set of system (23) is plotted in Figure 1 with (P=left [ {scriptsizebegin{matrix} 1.2692 & -0.4984 cr -0.4984 & 0.7255 end{matrix}} right ] ). Figure 1 The reachable set of Example 4. In the reference [29], Domoshnitsky discussed the stability of more complicated linear neutral systems with uncertain coefficients and uncertain delays.
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 ).
