An Introductory Course in Functional Analysis

Backstepping PDE Design: A Convex Optimization Approach

“…with continuous Kernel or weakly singular Kernel K, is compact on the Banach space of continuous functions (X = (C(Ω; R), • ∞ )) and on the Hilbert space of square integrable functions (X = (L 2 (Ω; R), •, • L 2 )). Likewise, for square integrable Kernels, A is compact on this Hilbert space [23], [38]. This is the case for the integral operators derived from the Kernel-PDEs in the Backstepping PDE design, as pointed out in [2] (page 19, footnote 2), where the Kernel is bounded and twice continuously differentiable.…”

“…Compact operators resemble the behaviour of operators in finite-dimensional spaces. In most of the traditional Banach spaces and for all Hilbert spaces, every compact operator is a limit of finite rank operators [38]. For continuous Kernels, a simple option for establishing this sequence are polynomials, which are a particular class of degenerate Kernels [23].…”

Backstepping PDE-based adaptive observer for a Single Particle Model of Lithium-Ion Batteries

“…Likewise, for square integrable Kernels, A is compact on this Hilbert space [23], [38]. This is the case for the integral operators derived from the Kernel-PDEs in the Backstepping PDE design, as pointed out in [2] (page 19, footnote 2), where the Kernel is bounded and twice continuously differentiable.…”

“…) and therefore compact [38], [41], [61], [63]. Thus, based on Lemma 2, Lemma 4 and on the property of compactness of F P,Q , according to a particular feature of the Fredholm Alternative Theorem ([23, Corollary 3.5], [37,Corollary 7.27]) 7 , the solution of (67) is unique and the operator T is boundedly invertible in X C and X H .…”

“…In most of the traditional Banach spaces and for all Hilbert spaces, every compact operator is a limit of finite rank operators [38]. For continuous Kernels, a simple option for establishing this sequence are polynomials, which are a particular class of degenerate Kernels [23].For instance if…”