In this paper we propose a novel Lyapunov based approach for the gain adaptation of super-twisting algorithm ensuring global finite time convergence to the desired sliding surface for linear time invariant systems with absolutely continuous matched uncertainties/disturbances bounded together with their gradients by known functions which could be not globally Lipshitz.
Proof Since q is of bounded variation for t ≥ 0, all solutions of (3) are bounded together with their derivatives, see e.g. [[18], Theorem 2].
Theorem 3.1 If any solution of the linear differential equation (19) with coefficients bounded together with their derivatives up to order p satisfies (3) on ℝ, then the linear system of differential equations (4) has the integral manifold given by (6), where K ( t ) is a p m × q m matrix.
we look for solutions y 1 ( t ), y 2 ( t ) : R → R 2 of F ( y 1 ) + ε H 1 ( y 1, y 2, η, ε ) = 0, F ( y 2 ) + ε H 2 ( y 1, y 2, η, ε ) = 0 (35). in the Banach space of C 1 -functions on ℝ, bounded together with their derivatives and with small norms.
In this work we use the method of lower and upper solutions to generate a sequence of modified nonlinear problems, each having a unique solution; in this way, we obtain a sequence of functions, which is uniformly bounded together with their first and second order derivatives.
Now we apply Theorem 4.1 to (28) with F ( y ), H 1 ( y 1, y 2, η, ε ), H 2 ( y 1, y 2, η, ε ) as in (33), (34) and Y = C b 1 ( R, R 2 ), X = C b 0 ( R, R 2 ), η = ∈ R × R m, where C b k ( R, R 2 ) is the Banach space of C k -functions bounded together with their derivatives with the usual sup-norm.
Since is bounded together with being demicompact, there exists a subsequence of which converges strongly to some.
An infinitely differentiable function φ ( x ) of the continuous argument x ∈ R n that is continuous and bounded together with all its derivatives is said to be smooth.
An infinitely differentiable function of the continuous argument (yin mathbb{R}^{n-1}) that is continuous and bounded together with all its derivatives is said to be smooth.
We shall show that the sequence of modified problems (17) is such that each problem has a unique solution, which is uniformly bounded, together with its first and second order derivatives.
System (3.2) is said to be reducible (or reducible to zero), if there exists which is bounded together with its inverse on such that is a constant (or zero) matrix on.
