On the semi-global stabilizability of the Korteweg-de Vries Equationviamodel predictive control

Abstract

Stabilization of the nonlinear Korteweg-de Vries (KdV) equation on a bounded interval by model predictive control (MPC) is investigated. This MPC strategy does not need any terminal cost or terminal constraint to guarantee the stability. The semi-global stabilizability is the key condition. Based on this condition, the suboptimality and exponential stability of the model predictive control are investigated. Finally, numerical experiment is presented which validates the theoretical results.

A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying Parabolic Equations

“…In the light of our recent investigations on analysis of RHC for infinite-dimensional systems in [3,6,4], the novelty of the present paper lies in the following facts: 1. Here we deal with time-varying systems.…”

“…The last term in (3) is the L 1 -penalization of the switching constraint u i (t)u j (t) = 0 for i = j and t > 0. See e.g., [21,22].…”

Analysis and Performance of the Barzilai-Borwein Step-Size Rules for Optimization Problems in Hilbert Spaces

“…Due to Algorithm 1, for every j ∈ {r, • • • , r + Θ} we have one of the cases α k+j = α BB1 k+j or α k+j = α BB2 k+j . Further, using (9), the fact that A is self-adjoint, and the spectral property (8), we have for every k ≥ 1 and q = 0, 1, 2, that…”

“…By putting m = r n u η , (44) holds. Moreover, the equivalence of ( 44) with ( 45) is justified due the fact that, similarly to (9) for G k , it can easily be shown that…”

“…For these problems, the corresponding reduced formulations lead to infinite-dimensional quadratic and near quadratic unconstrained optimization problems. In this respect, we mention [7,8,9] in which the BB-method was efficiently employed in the context of the model predictive control for PDEs. In view of the above discussion, we are motivated to study the BB-method for a more general class of problems, namely, unconstrained problems posed in infinite-dimensional Hilbert spaces.…”