Boundary sliding mode control of a system of linear hyperbolic equations: A Lyapunov approach

Super-Twisting Sliding Mode Control for the Stabilization of a Linear Hyperbolic System

“…Proof: Note that it is enough to prove σ(t) = γ(t) to be able to conclude W (t) = η(t). For this proof, we refer to [14,Section 4.1]. This concludes the proof of Lemma 4.…”

“…Theorem 2 (Global asymptotic stability): Assume that (13) holds. Then, for any (R 0 1 , R 0 2 , W 0 ) ∈ X × IR, 0 is globally asymptotically stable for (14). In other words, there exists a KL-function τ such that for any (R 0 1 , R 0 2 , W 0 ) ∈ X × R and for any t ≥ 0:…”

“…This section provides a proof of Theorem 1. More precisely, the aim consists in proving the well-posedness of the closed−loop system (14) and the regularity of the function S defined by (7). Let (R 0 1 , R 0 2 , W 0 ) ∈ X × IR and consider the following ODE…”

“…The aim is now to prove that, for any Filippov solution (γ, η) of ( 22) with initial condition (S 0 , W 0 ), the solution 16)-( 26)-( 22) is a mild solution of (14).…”

Boundary stabilization of 1D hyperbolic systems

“…• Could other methods like the sliding mode be applied in this case (see for instance the promising work of [113])?…”