Canonical Dual Control for Nonconvex Distributed-Parameter Systems: Theory and Method1

This paper presents a potentially powerful canonical dual transformation method and associated duality theory for solving fully nonlinear distributed-parameter control problems. The extended Lagrange duality and the interesting triality theory proposed recently in finite deformation theory are generalized into nonconvex dynamical systems. A bifurcation criterion is proposed, which leads to an effective dual feedback control against the chaotic vibration in Duffing system. 1 Problems and Motivations We shall study a duality approach for solving the following very general abstract distributed parameter problem ((P) for short), (P) : ρu,tt +A(u, μ) = 0 ∀u ∈ Uk, (1) where the feasible space Uk is a convex, non-empty subset of a reflexive Banach space U over an open space-time domain Ωt = Ω×(0, tc) ⊂ Rn×R+, in which, certain essential boundaryinitial conditions are prescribed. We assume that for a given distributed parameter control field μ(x, t) over Ωt, the mapping A(u, μ) is a potential operator from Uk into its dual space U∗, i.e., there exists a Gâteaux differentiable potential functional Pμ(u) = P (u;μ), such that the directional derivative of P at ū ∈ Uk in the direction δu can be written as δPμ(ū; δu) = 〈DPμ(ū), δu〉 ∀δu ∈ Uk, where the operator DPμ(ū) = A(ū, μ) is the Gâteaux derivative of Pμ at the point ū; the bilinear form 〈·, ·〉 : U × U∗ → R places U and U∗ in duality. By nonlinear operator theory we know that the mapping A : Uk → U∗ is monotone if and only if P is convex on Uk. Published in Control of Nonlinear Distributed Parameter Systems, Goong Chen, Irena Lasiecka and Jianxin Zhou (eds). Marcel Dekker, 2001, 85-112.