Abstract
Boundary control for a nonlinear system actuated by a first-order time-varying hyperbolic partial differential equation is investigated. Two-step backstepping transformation is applied during the design. The first-step backstepping transformation is given to transfer the first-order hyperbolic PDE to a transport PDE, and the second-step backstepping transformation is used to design the compensator for the closed-loop system. Globally asymptotical stability of the closed-loop system is proved by constructing a Lyapunov functional. The validity of the proposed design is illustrated by an example.
Highlights
Boundary feedback control is investigated for the first-order hyperbolic PDE
The first step backstepping transformation is given to transfer the hyperbolic PDE to a transport PDE, and the second backstepping transformation is used to design the compensator for the closed-loop system
The cascaded system of nonlinear ODE and the first-order hyperbolic PDE with time-varying parameter is investigated in this paper
Summary
In [1] where t ≥ 0, 0 ≤ x ≤ 1, v ∈ R, U ∈ R and functions λ, g, hare continuous. Using the backstepping transformation and boundary feedback the unstable PDE is changed to a transport PDE, which converges to zero in finite time [1]. Infinite dimension backstepping method is applied in stabilization n+1 coupled first-order hyperbolic linear PDEs with a single boundary input in [16]. Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS where t ≥ 0, 0 ≤ x ≤ D and X ∈ Rn, v ∈ R, U ∈ R are ODE state, PDE state, and control input, respectively, and f : Rn × R → Rn is locally Lipschitz with f (0, 0) = 0, and λ, g : [0, D] × R+ → R, h : [0, D] × [0, D] × R+ → R are continuous. For a scalar function u ∈ L∞[0, D], | · | denotes the Euclidean norm, and u(t) ∞ denotes its supremum norm
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