proposition 9.31 Lyapunov's first method: asymptotic stability

Rests on

• depends_on corollary 9.27 Linear stability ¶
  • depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
    • depends_on definition 5.15 Basis ¶
      • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.5 Linear combination ¶
        • depends_on definition 5.7 Vector subspace ¶
        • proves proof ch:03-linear-algebra-representations@proof-3 ¶
      • depends_on definition 5.14 Linear independence ¶
        • depends_on definition 5.5 Linear combination ¶ ↺
    • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      • depends_on definition 5.56 Endomorphism ¶
        • depends_on definition 5.37 Linear transformation ¶
      • depends_on definition 5.39 Kernel, image, nullity, rank ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
        • depends_on definition 5.7 Vector subspace ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-12 ¶
    • depends_on theorem 9.23 Solution of a constant-coefficient system ¶
      • depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
        • depends_on definition 9.2 Linear equation; homogeneity ¶
        • depends_on remark 9.3 Normal form and first-order systems ¶
        • depends_on theorem 9.8 Picard–Lindelöf ¶
        • proves proof ch:07-odes-sturm-liouville@proof-3 ¶
      • depends_on proposition 9.22 The exponential and its derivative ¶
        • depends_on definition 9.21 Matrix exponential ¶
        • depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
        • depends_on theorem 7.51 Termwise differentiation ¶
        • proves proof ch:07-odes-sturm-liouville@proof-11 ¶
      • proves proof ch:07-odes-sturm-liouville@proof-12 ¶
    • proves proof ch:07-odes-sturm-liouville@proof-13 ¶
  • depends_on theorem 9.26 Structure of the solutions ¶
    • depends_on theorem 8.24 Residue theorem ¶
      • depends_on corollary 8.15 Deformation of contours ¶
        • depends_on definition 6.17 Simply connected space ¶
        • depends_on theorem 8.12 Cauchy ¶
        • proves proof ch:06-complex-analysis@proof-9 ¶
      • depends_on definition 8.22 Isolated singularities; residue ¶
        • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
        • depends_on theorem 8.21 Laurent expansion ¶
      • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
      • depends_on theorem 8.21 Laurent expansion ¶ ↺
      • proves proof ch:06-complex-analysis@proof-17 ¶
    • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
      • depends_on definition 5.70 Characteristic polynomial ¶
        • depends_on definition 5.56 Endomorphism ¶ ↺
        • depends_on equation 5.19 eq:lin-leibniz-det ¶
        • depends_on equation 5.74 eq:lin-matrix-rep ¶
        • proves proof ch:03-linear-algebra-representations@proof-25 ¶
      • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
      • depends_on proposition 5.49 Injective, surjective, invertible ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
        • depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
        • depends_on theorem 5.40 Rank–nullity ¶
        • proves proof ch:03-linear-algebra-representations@proof-18 ¶
      • proves proof ch:03-linear-algebra-representations@proof-26 ¶
    • depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
    • proves proof ch:07-odes-sturm-liouville@proof-14 ¶
  • proves proof ch:07-odes-sturm-liouville@proof-15 ¶
• depends_on lemma 9.10 Grönwall's inequality ¶
  • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    • depends_on theorem 7.35 Mean value theorem ¶
      • depends_on proposition 7.29 Linearity ¶
        • depends_on definition 7.11 Sum of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-13 ¶
      • depends_on theorem 7.34 Rolle ¶
        • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • proves proof ch:05-real-analysis@proof-18 ¶
      • proves proof ch:05-real-analysis@proof-19 ¶
    • proves proof ch:05-real-analysis@proof-20 ¶
  • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    • depends_on definition 7.26 Derivative of a function at a point ¶
      • depends_on definition 7.16 Limit ¶
        • depends_on definition 7.2 Absolute value ¶
        • depends_on definition 7.9 Real function ¶
      • depends_on definition 7.9 Real function ¶ ↺
    • depends_on theorem 7.40 Continuous functions are integrable ¶
      • depends_on definition 7.39 Darboux sums and the definite integral ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶
      • depends_on theorem 7.24 Extreme value theorem ¶ ↺
      • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
        • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on theorem 7.7 Bolzano–Weierstrass ¶
        • proves proof ch:05-real-analysis@proof-10 ¶
      • proves proof ch:05-real-analysis@proof-23 ¶
    • depends_on theorem 7.24 Extreme value theorem ¶ ↺
    • proves proof ch:05-real-analysis@proof-24 ¶
  • proves proof ch:07-odes-sturm-liouville@proof-4 ¶
• depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
• proves proof ch:07-odes-sturm-liouville@proof-18 ¶

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