If (d^{m} f cdot)) is continuous at (x_{0}) then f is said to be mth-order continuously Fréchet differentiable at (x_{0}).
Virtually all modern financial markets match orders continuously – that is, as orders arrive to the exchange.
Then, we prove the existence of smooth kernels with second order continuously derivative for the forward and inverse transformations in the backstepping feedback control law design.
Furthermore, a refined SS-TVD flux-limiter, referred to henceforth as TCDF (Third-order Continuously Differentiable Function), is proposed for steady-state calculations based on the review.
Fourth, the main technical novelty is the construction of three new nth-order continuously differentiable switching functions, which are used to design the desired controller.
Thirdly, the main technical novelty is to construct three new nth-order continuously differentiable switching functions such that multiswitching-based adaptive neural backstepping controllers are designed successfully.
Third, the main technical novelty is to construct three new nth-order continuously differentiable functions which are used to design the actual controller, the virtual control variables and the adaptive laws.
Theorem 2.1 Suppose that f : J → R is second-order continuously differentiable.
For the simplicity of our proof in Theorem 3.1, f ( t, x ) is a first-order continuously differentiable function satisfying Lipschitz condition.
For ensuring that x ( t ) in Eq. (5) makes sense, σ has to be at least first order continuously differentiable, i.e., σ ∈ C 1 [ 0, π ].
Theorem 3.1 Let f : [ 0, π ] × X → X be a first-order continuously differentiable function satisfying the condition ∥ f ( t, x ) − f ( t, y ) ∥ ≤ L ∥ x − y ∥, ∀ t ∈ [ 0, π ], x, y ∈ X.
