khalil2002nonlinear

Nonlinear Systems (3rd edition)

Authors: Hassan K. Khalil · Year: 2002 · Venue: Prentice Hall, Upper Saddle River, NJ (3rd ed.) Raw: md (R3 block, pp. 110–205 only) · pdf

Partial corpus coverage

Only the R3 block, pp. 110–205 is OCR’d into the corpus — Chapter 3 (Fundamental Properties) through Chapter 4 (Lyapunov Stability). The full 767-page book is not filed; the PDF is complete but the markdown is this block only. Sections outside pp. 110–205 (e.g. Barbalat in §8.3, input–output stability, singular perturbations) are not in the corpus markdown. Edition confirmed from the PDF title/copyright page (“Third Edition”; © 2002, 1996 Prentice Hall). Cite by edition — chapter numbering differs from the 2nd ed. (khalil1996nonlinear).

Summary

The canonical graduate text on nonlinear control and Lyapunov stability. The filed block covers the analytical backbone the stability rungs of this project lean on: existence, uniqueness and continuity of solutions (Ch. 3), and Lyapunov’s direct method for autonomous and non-autonomous systems, LaSalle’s invariance principle, class-𝒦/𝒦ℒ comparison functions, and input-to-state stability (Ch. 4).

Key Claims

• Existence & uniqueness (Ch. 3). Local Lipschitz continuity gives a unique local solution (Thm 3.1); global Lipschitz gives a unique global solution (Thm 3.2); Thm 3.3 gives global existence for merely locally-Lipschitz $f$ whose solutions stay in a compact set. Thm 3.4 bounds the separation of two solutions under a perturbed vector field (the Gronwall-type continuous-dependence estimate).
• Lyapunov’s direct method (§4.1). An equilibrium is stable if a positive-definite $V$ has $\dot{V}\le 0$, and asymptotically stable if $\dot{V}<0$ (Thms 4.1–4.2).
• LaSalle’s invariance principle (§4.2, Thm 4.4). Solutions in a compact positively-invariant set approach the largest invariant set inside $\{\dot{V}=0\}$; $V$ need not be positive definite. Corollaries 4.1–4.2 (Barbashin–Krasovskii) specialize it to global asymptotic stability.
• Comparison functions (§4.4, Def. 4.2–4.3). Class 𝒦, 𝒦∞ and 𝒦ℒ functions — the substrate for uniform (non-autonomous) stability definitions used “in the rest of the chapter, and indeed the rest of the book.”
• Input-to-state stability (Ch. 4). ISS via an ISS-Lyapunov function, with gain $\gamma ={\alpha }_1^{-1}\circ {\alpha }_2\circ \rho$.

Method

Textbook exposition: definitions, theorems with full proofs, and worked examples. Autonomous theory (§4.1–4.4) is extended to non-autonomous systems (§4.5–4.6) through the comparison-function machinery.

Regime note. A general nonlinear-systems reference — no spacecraft, manipulator, or base-actuation model — so the free-flying vs free-floating distinction does not arise. It is a tool text.

Relevance to thesis

This is the section-and-theorem authority for the Lean stability library. §4.4 (Def. 4.2–4.3) is the canonical cite pinned by the comparison-function module ([comparison_functions in the Lean wiki]); the Lyapunov and LaSalle theorems back the block-Lyapunov and comparison rungs. Barbalat’s lemma, the one workhorse not in this OCR block (it sits in §8.3, outside pp. 110–205), is cited instead from slotine1991applied.

Connections

Topics: Lyapunov Stability · Input to State Stability · Cascaded Systems Sources: khalil1996nonlinear (2nd ed. — distinct book) · slotine1991applied (Barbalat’s lemma)

Key Equations / Quotes

Comparison-function motivation (§4.4):

“In Section 4.4, we introduce class $\mathcal{K}$ and class $\mathcal{K}\mathcal{L}$ functions, which are used extensively in the rest of the chapter, and indeed the rest of the book.”

Continuous-dependence estimate (Thm 3.4):

$$\Vert y(t)-z(t)\Vert \le \Vert y_0-z_0\Vert \exp [L(t-t_0)]+\frac{\mu }{L}\{\exp [L(t-t_0)]-1\}.$$

Open Questions

• Full book is not in the corpus markdown — only pp. 110–205. §8.3 (Barbalat), Ch. 5 (input–output stability) and later chapters remain to be OCR’d if cited.