Sentence examples for bounded and the solution from inspiring English sources

When is bounded and, the solution of of (1.1) obeys as ; moreover, Miller [1] has shown when in addition, the solution of (1.2) obeys as.

In the case that such a matrix has some pair of complex conjugate eigenvalues then the solution is bounded and the solution trajectory may oscillate, and if there is some eigenvalue with positive real part (i.e., within the instability region), then the state trajectory solution is unbounded.

If 0<α i <1, then the recursive relation in (29) is stable since the input, H i, is bounded and the solution to (29) is given by C_{i} tau+1) = sum_{j=0}^{infty} left 1-alpha_{i}right)^{j} H_{i}(tau-j), (30).

Remarkably, by using the function scaling gain strategy and homogeneous domination method, two kinds of explicit universal adaptive controllers are designed such that all the states of the closed-loop system are globally bounded and the solutions of the original system converge to zero.

Therefore, N 1 + N 2 is ultimately bounded and all the solutions of system (1) globally exists on the interval [ 0, ∞ ). The aforementioned discussions can be summarized into the following results.

Therefore, the feasible region is bounded, and the problem has an efficient solution.

The function is bounded because solutions are bounded, and the functions have mean value zero.

It is proved that the numerical solutions are bounded and the numerical methods can achieve a convergence rate of (mathcal{O} tau^{2} + h^{4})) in the maximum norm.

The positivity preserving variational formulation is designed to maintain the bounded and stable solution of the Allen Cahn equation.

It is easy to see that is the desired bounded and nonoscillatory solution of  (2.9) (b) Sufficiency.

Finally, Figure 8 shows the upper dual bound, the lower bound and the solution given by weight adjustment methods as a function of the number of RT users.