(\(\Rightarrow \)) Apply stability at \(\varepsilon = 1\) to get \(\delta (1, t_{0})\). Every response from the ball of radius \(\delta \) stays inside the unit ball, and scaling turns that into the bound with \(C = 2/\delta \).

(\(\Leftarrow \)) Given \(\varepsilon \), take \(\delta = \varepsilon /(C+1)\). A trajectory is its own response (theorem 180), so \(\| \varphi (t)\| \le C\| \varphi (t_{0})\| {\lt} \varepsilon \).

The state-transition form is lemma 356. A bound on \(\Phi \) gives the response bound \(C\| x_{0}\| \); conversely a family of operators bounded at every point of a Banach space is norm-bounded, by the uniform boundedness principle.

Identical to lemma 356, with the margin \(\delta \) and the bound \(C\) both carried outside the quantifier on \(t_{0}\).

The stability conjunct is theorem 357. For attractivity: an arbitrary \(x_{0}\) is rescaled into the attractivity ball, and the limit scales back, since the response is linear in the initial state; conversely every trajectory is a response (theorem 180), so convergence of the responses is convergence of the trajectories, from any radius.

Both directions are the trajectory characterization plus homogeneity: the exponential estimate along every trajectory is the estimate applied to the initial state, and conversely the estimate on any ball scales to the estimate everywhere.

As in the global case; the radius \(c\) of the local definition is absorbed by the scaling argument, and \(c = 1\) serves in the converse.

Both are equivalent to the same estimate, theorem 361 and theorem 360.

The negation of theorem 357.

A complex eigenvalue \(\mu \) of \(A\) becomes \(\mu + \alpha \) after the shift, and \(\operatorname {Re}(\mu ) {\lt} -\alpha \) is equivalent to \(\operatorname {Re}(\mu + \alpha ) {\lt} 0\).

The spectrum is compact and nonempty, so \(\operatorname {Re}\) attains a maximum on it at some \(\mu _0\), which is an eigenvalue and hence has \(\operatorname {Re}(\mu _0) {\lt} 0\). Take \(c = -\operatorname {Re}(\mu _0)/2\).

Gelfand’s formula gives \(\| a^m\| ^{1/m} \to r(a)\), so \(\| a^m\| ^{1/m} {\lt} 1\) for all large \(m\); pick any such \(m\) with \(m \neq 0\). Were \(\| a^m\| \ge 1\), raising to the positive power \(1/m\) would give \(\| a^m\| ^{1/m} \ge 1\), a contradiction.

Both unfold to a condition on \(\operatorname {Re}\mu \), the first to \(\operatorname {Re}\mu {\lt} 0\) and the second to \(\operatorname {Re}\mu {\lt} -0\).

Each eigenvalue satisfies \(\operatorname {Re}\mu {\lt} -\alpha \le -\beta \).

There is no nonzero vector, so the hypothesis of the predicate is never met.

The only eigenvalue is \(-\gamma \), and \(\operatorname {Re}(-\gamma ) = -\gamma {\lt} -\alpha \).

The affine map \(t \mapsto -\gamma (t - t_{0})\) tends to \(-\infty \) because \(-\gamma {\lt} 0\); compose with \(\exp \) and scale by \(K\).

The squeeze theorem: \(\| f\| \) is bounded below by \(0\) and above by an envelope that tends to \(0\) (lemma 373, after reassociating \(k \cdot E \cdot c = (kc) \cdot E\)).

A contracting block. The spectral radius of \(e^{A_{\mathbb C}}\) is below \(1\) (lemma 201), so by Gelfand’s formula (lemma 368) some integer time \(p {\gt} 0\) has \(\| e^{Ap}\| {\lt} 1\). Fix \(c\) with \(\| e^{Ap}\| \le c {\lt} 1\), kept away from \(0\) so that \(\log c\) exists.

One block. \(\| e^{Ar}\| \) is continuous, hence bounded by some \(M \ge 1\) on the compact interval \([0, p]\).

Spreading. Write \(t = qp + r\) with \(q \in \mathbb {N}\) and \(r \in [0, p)\). The exponents commute, so \(e^{At} = e^{Ar}(e^{Ap})^{q}\), and submultiplicativity gives \(\| e^{At}\| \le M c^{q}\) by induction on \(q\). Setting \(\gamma = -\log c / p\) turns \(c^{q}\) into \(e^{-\gamma q p}\), and \(qp \ge t - p\) costs one further factor \(c^{-1}\). The bound is \(\| e^{At}\| \le (M/c)\, e^{-\gamma t}\).

The contraction is taken in the \(L^{\infty }\) operator norm, which is what makes it usable against \(\| Ax\| _{\infty } \le \| A\| _{\infty }\| x\| _{\infty }\) in theorem 376.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.1.

theorem 375 bounds the matrix; one application of \(\| Ax\| _{\infty } \le \| A\| _{\infty }\| x\| _{\infty }\) transfers it to the state. This is the step at which the choice of the \(L^{\infty }\) operator norm earns its keep.

Suppose \(\operatorname {Re}\mu \ge 0\) for some eigenpair \((\mu , v)\) with \(v \ne 0\). Complexification is continuous, so \(e^{At}_{\mathbb C}v \to 0\) as well. But \(v\) is an eigenvector of \(tA_{\mathbb C}\) with eigenvalue \(t\mu \), hence of its exponential (lemma 196), so

for every \(t \ge 0\), contradicting convergence to \(0\).

Feeding the hypothesis the standard basis vectors returns the columns of \(e^{At}\), and convergence in the matrix space is entrywise, so \(e^{At} \to 0\); apply theorem 377.

Forwards, theorem 375 supplies an exponentially decaying envelope and lemma 374 squeezes; backwards is theorem 377.

Forwards is theorem 375. Backwards, the envelope forces \(e^{At} \to 0\) (lemma 374), so theorem 379 applies.

For constant \(A\) the state transition matrix is \(e^{A(t-t_0)}\) (theorem 186), so theorem 359 turns the statement into one about \(e^{At}x_0\).

If the response converges then \(A\) is Hurwitz (theorem 378). Conversely a Hurwitz \(A\) gives an exponentially decaying envelope for \(\| e^{At}x_0\| \) (theorem 376); the envelope is bounded by \(k\| x_0\| \), which supplies the stability half, and tends to zero (lemma 373), which supplies the attractivity half.

As for theorem 381, but against theorem 361, which already asks for an exponential envelope. The forward direction squeezes \(\| e^{At}x\| \) to zero under that envelope and applies theorem 378; the converse is theorem 376 re-anchored from \(0\) to \(t_0\).

Complexify the equation and pair it against an eigenvector \(v\) of \(M\) with eigenvalue \(\mu \). The two terms contribute \(\mu \, v^{*}Pv\) and \(\bar\mu \, v^{*}Pv\), so the left-hand side is \(2\operatorname {Re}(\mu )\, v^{*}Pv\) while the right-hand side is \(-v^{*}Qv\). Both \(v^{*}Pv\) and \(v^{*}Qv\) are strictly positive (lemma 193), so \(\operatorname {Re}(\mu ) {\lt} 0\).

Take the forcing to be minus the residual and apply theorem 385.

Transposition and the quadratic form \(P \mapsto x^{\mathsf T}Px\) are continuous and linear, so they commute with the sum: the sum is symmetric, and its quadratic form is a sum of nonnegative terms.

Write \(L(P) = PA + A^{\mathsf T}P\) for the Lyapunov operator, \(C = e^{Am}\), and \(T(W) = C^{\mathsf T}WC\) for the congruence (Stein) operator.

One block. Let \(R = \int _{0}^{m} e^{tA^{\mathsf T}}Qe^{tA}\, \mathrm{d}t\). Differentiating the integrand two ways and equating the derivatives by uniqueness gives the key identity \(L(R) = T(Q) - Q\). \(R\) is positive definite because the integrand is, pointwise, and positive definiteness survives the integral.

Summing the blocks. \(\| C\| {\lt} 1\) makes \(\| T\| {\lt} 1\), so \(\sum _{k} T^{k}R\) converges; call it \(P\). It is positive definite, being a convergent sum of positive definite terms. Since \(A\) and \(C\) commute, \(L\) commutes with \(T\), so \(L(T^{k}R) = T^{k+1}Q - T^{k}Q\) and the series for \(L(P)\) telescopes to \(-Q\).

Uniqueness. A contraction has only the trivial fixed point, so \(L\) is injective; two solutions with the same forcing therefore coincide.

The contraction is taken in the Frobenius norm, the only place in the construction where \(\| C^{\mathsf T}\| = \| C\| \) is needed.

A contracting integer-time matrix exponential is obtained from the Hurwitz spectrum. Integrating over one time block and summing the resulting discrete Lyapunov series constructs \(P\); contraction proves convergence and uniqueness.

Instantiate the forcing at \(Q = I\); the residual is then \(-I\), whose negation is the identity. The edge carries no mathematics beyond the instantiation.

Split \(-(PA + A^{\mathsf T}P)\) as \(\bigl(-(PA + A^{\mathsf T}P) - 2cP\bigr) + 2cP\): positive semidefinite plus positive definite is positive definite, so theorem 386 applies.

Solve the Lyapunov equation for the shifted matrix. A Hurwitz \(A\) has a positive spectral margin \(c\) (theorem 367), so \(A + cI\) is still Hurwitz (theorem 366). Solving \(P(A + cI) + (A + cI)^{\mathsf T}P = -I\) (theorem 389) and expanding gives \(PA + A^{\mathsf T}P = -I - 2cP\), so the residual \(-(PA + A^{\mathsf T}P) - 2cP\) is exactly the identity.

No domination bound between two positive definite matrices is needed: the residual is not merely bounded below, it is \(I\).

Forwards is theorem 389. Backwards, instantiate the forcing at \(Q = I\) and apply theorem 385.

Forwards, theorem 389 produces a solution for each forcing and lemma 390 extracts one certificate from them; backwards is theorem 386.

theorem 392 and theorem 391.

Every clause is tied to clause 3 and the cycle is closed by transitivity: \(1 \Leftrightarrow 3\) is theorem 381, \(2 \Leftrightarrow 3\) is theorem 382, \(3 \Leftrightarrow 4\) is theorem 393, \(3 \Leftrightarrow 5\) is theorem 394, and \(3 \Leftrightarrow 6\) is theorem 395.

The derivative of the quadratic form in the direction \(Ay\) is \(\langle y, Py'\rangle + \langle y', Py\rangle \) with \(y' = Ay\), which is the form represented by \(PA + A^{\mathsf T}P\). Transporting the equation through \(\mathtt{toEuclideanCLM}\) — under which transpose becomes adjoint — rewrites that as \(-Q\).

theorem 397 at \(Q = I\), where the quadratic form is the squared Euclidean norm.

Strong induction on \(k + l\). The base cases are the zero generalized eigenspaces, where the vector is \(0\). Otherwise write \(Tx = N_{x} + \xi x\) and \(Ty = N_{y} + \nu y\) with \(N_{x}, N_{y}\) one order lower; the induction hypothesis kills \(B(N_{x}, y)\) and \(B(x, N_{y})\), leaving \((\xi + \nu - c)B(x,y) = 0\), and the coefficient is nonzero by nonresonance.

Membership gives \(x \in \ker (T - \xi )^{k}\) for some \(k\), necessarily \(k \ne 0\) since \(x \ne 0\); so that generalized eigenspace is nontrivial, and a nontrivial generalized eigenspace forces an eigenvalue.

Over \(\mathbb {C}\) the generalized eigenspaces of \(T\) span \(V\), so it suffices to check \(B(x,y) = 0\) for \(x, y\) in generalized eigenspaces. A nonzero such vector witnesses its eigenvalue (lemma 400), so nonresonance applies and lemma 399 gives \(B(x,y) = 0\).

Both sides unfold to \(x \cdot (HAy)\) by associativity of matrix-vector multiplication.

Move the transpose across the dot product: \(x \cdot (A^{\mathsf T}Hy) = (Ax) \cdot (Hy)\).

Move the transpose across the dot product and apply \(H^{\mathsf T} = H\).

Choose a shift avoiding the finitely many pairwise spectral resonances, solve the resulting Lyapunov–Sylvester equation by finite-dimensional injectivity, and extract \(w\) from the real or imaginary part of the supplied eigenvector.

(\(\Rightarrow \)) Stability at \(\varepsilon = 1\) gives a margin \(\delta \); every response from the \(\delta \)-ball stays in the unit ball, and homogeneity scales that to the bound with \(C = 2/\delta \). (\(\Leftarrow \)) Given \(\varepsilon \), take \(\delta = \varepsilon /(C+1)\): a trajectory is its own homogeneous response (theorem 224).

The state-transition form is lemma 406. A bound on \(\Phi \) bounds each response by \(C\| x_0\| \); conversely a family of operators bounded at every point of a Banach space is norm-bounded, by the uniform boundedness principle.

As lemma 406, with the margin and the bound both carried outside the quantifier on \(t_0\).

The stability conjunct is theorem 407. For attractivity an arbitrary \(x_0\) is rescaled into the attractivity ball and the limit scales back; conversely every trajectory is a response (theorem 224).

The estimate along every trajectory is the estimate applied to its initial state, since a trajectory is its own homogeneous response; conversely the response from \(x\) is a trajectory.

As in the global case; the radius of the local definition is absorbed by the scaling argument, and radius \(1\) serves in the converse.

Both are equivalent to the same estimate: theorem 411 and theorem 410.

The negation of theorem 407.

The spectral radius of \(A_{\mathbb C}\) is below \(1\), so by Gelfand’s formula (lemma 368) some power has \(\| A^{m}\| {\lt} 1\); fix \(\| A^{m}\| \le c {\lt} 1\) with \(c {\gt} 0\), and let \(M\) bound \(\| A^{r}\| \) for \(r {\lt} m\). Writing \(k = r + mq\) with \(r {\lt} m\), submultiplicativity gives \(\| A^{k}\| \le M c^{q}\), and with \(\rho = c^{1/m}\), \(c^{q} = \rho ^{mq} \le \rho ^{k}/c\).

theorem 415 and \(\| Ax\| _{\infty } \le \| A\| _{\infty }\| x\| _{\infty }\).

Suppose \(|\mu | \ge 1\) for an eigenpair \((\mu , v)\) of \(A_{\mathbb C}\) with \(v \ne 0\). Complexification is continuous, so \(A^{k}_{\mathbb C}v \to 0\); but \(A^{k}_{\mathbb C}v = \mu ^{k}v\) (lemma 198) has norm \(|\mu |^{k}\| v\| \ge \| v\| {\gt} 0\).

The standard basis vectors return the columns of \(A^{k}\), so \(A^{k} \to 0\); apply theorem 417.

Forwards, theorem 415 supplies a geometric envelope; backwards is theorem 417.

Forwards is theorem 415; backwards, the envelope forces \(A^{k} \to 0\) and theorem 419 applies.

For constant \(A\) the state transition matrix is \(A^{t - t_0}\) (theorem 220), so theorem 409 turns the statement into one about \(A^{k}x_0\). If the responses converge then \(A\) is Schur (theorem 418); conversely a Schur \(A\) has a geometric envelope \(\| A^{k}x\| \le C\rho ^{k}\| x\| \) (theorem 416), which bounds the responses and makes them vanish.

Against theorem 411, which already asks for a geometric envelope \(\| A^{t - t_0}x\| \le c\rho ^{t - t_0}\| x\| \). Forwards, at \(t_0 = 0\) the envelope makes \(A^{k}x \to 0\), so \(A\) is Schur (theorem 418); backwards is theorem 416.

\((Ax)^{\mathsf T}P(Ax) = x^{\mathsf T}(A^{\mathsf T}PA)x\), and \(A^{\mathsf T}PA = P - Q\).

\(A\) and \(A^{\mathsf T}\) have the same characteristic polynomial, hence the same spectrum over \(\mathbb {C}\), and the spectrum is the set of eigenvalues (lemma 194).

Take \(P = \sum _{k} (A^{\mathsf T})^{k} Q A^{k}\). It converges because both \(A\) and \(A^{\mathsf T}\) are Schur (lemma 425), so their powers decay geometrically (theorem 415). Shifting the index, \(A^{\mathsf T}PA = P - Q\). The first term is \(Q\) and the others are positive semidefinite, so \(P\) is positive definite (lemma 387). For uniqueness, a difference \(W\) of two solutions satisfies \(W = A^{\mathsf T}WA\), hence \(W = (A^{\mathsf T})^{k}WA^{k} \to 0\).

Complexify the equation and pair it against an eigenvector \(v\) with eigenvalue \(\mu \): the left-hand side is \((|\mu |^{2} - 1)\, v^{*}Pv\) and the right-hand side is \(-v^{*}Qv\). Both quadratic forms are strictly positive (lemma 193), so \(|\mu |^{2} {\lt} 1\).

Take the forcing to be minus the residual and apply theorem 427.

Forwards is theorem 426. Backwards, instantiate the forcing at \(Q = I\) and apply theorem 427.

Forwards, solve the equation with \(Q = I\) (theorem 426); the residual is then \(-I\). Backwards is theorem 428.

Every clause is tied to clause 3: \(1 \Leftrightarrow 3\) is theorem 421, \(2 \Leftrightarrow 3\) is theorem 422, \(3 \Leftrightarrow 4\) is theorem 429, and \(3 \Leftrightarrow 5\) is theorem 430.