2 Ordinary differential equations
2.1 Gronwall–Bellman inequality
A function obeying an integral inequality is bounded by the solution of the corresponding integral equation. The two specializations are what the rest of the development actually calls: constant \(\Lambda \) gives exponential growth, and constant \(\Lambda \) and \(\mu \) gives the \(C e^{\mu (t-a)}\) bound behind every continuous-dependence estimate.
Khalil, Lemma A.1. Let \(\Lambda , \mu : [a,b] \to \mathbb {R}\) be continuous with \(\mu \ge 0\), and let \(y : [a,b] \to \mathbb {R}\) be continuous satisfying
Then
An integrating-factor argument. Write \(z(t) = \int _a^t \mu y\), \(M(t) = \int _a^t \mu \), and \(v = z + \Lambda - y\), so that \(v \ge 0\) is exactly the hypothesis. The transform \(w = e^{-M}z\) has, by the product rule and lemma 22,
the inequality because \(e^{-M}\), \(\mu \) and \(v\) are all nonnegative — this is the only place \(\mu \ge 0\) is used, and it is essential: a sign change would reverse it. Integrating from \(a\), where \(w(a) = 0\), gives \(w(t) \le \int _a^t e^{-M(s)}\mu (s)\Lambda (s)\, ds\). Multiplying by \(e^{M(t)}\) and folding \(e^{M(t)}e^{-M(s)} = e^{\int _s^t \mu }\) turns this into the stated bound on \(z(t)\), and \(y \le \Lambda + z\) finishes.
Khalil, Lemma A.1 (the case \(\Lambda (t) \equiv \Lambda \) constant). Let \(\mu \) be continuous and nonnegative on \([a,b]\), and let \(y\) be continuous with
Then \(y(t) \le C\, e^{\int _{a}^{t}\mu (\tau )\, \mathrm{d}\tau }\) on \([a,b]\).
theorem 28 at \(\Lambda \equiv C\) gives \(y(t) \le C + \int _a^t C\mu (s)e^{\int _s^t \mu }\, ds\), and the integral evaluates in closed form: \(s \mapsto -C\, e^{M(t)-M(s)}\) is an antiderivative of the integrand, so the integral is \(C e^{M(t)} - C\). The two \(C\)’s cancel.
Khalil, Lemma A.1 (the case \(\Lambda \) and \(\mu \ge 0\) both constant). Let \(\mu \ge 0\) be constant and let \(y\) be continuous with
Then \(y(t) \le C\, e^{\mu (t-a)}\) on \([a,b]\). This is the form the continuous-dependence estimates use, and the reason their bounds carry a factor \(e^{L(t-t_0)}\).
lemma 29 with \(\int _a^t \mu = \mu (t-a)\).
2.2 Existence, uniqueness and continuous dependence
A function \(x\), defined on the segment between \(t_0\) and \(t_1\), is an integral solution of \(\dot{x} = F(t,x)\) with initial value \(x_0\) when
for every \(t\) between \(t_0\) and \(t_1\).
Let \(x\) be a continuous integral solution of \(\dot x = F(t,x)\) with initial value \(x_0\) on the segment between \(t_0\) and \(t_1\), with \(F\) jointly continuous. Then for any \(s, r\) on that segment, \(x\) is an integral solution on the segment between \(s\) and \(r\), with initial value \(x(s)\):
Leaving both endpoints free means the domain need not shrink — taking \(s = t_1\) and \(r = t_0\) re-anchors at one end of the original segment while retaining all of it.
Subtract the defining equations at \(t\) and at \(s\) and split the integral at \(s\), which is legitimate because \(w \mapsto F(w, x(w))\) is interval-integrable on each piece.
Let \(x\) be an integral solution of \(\dot x = F(t,x)\) with initial value \(x_0\) on the segment between \(t_0\) and \(t_1\). Then \(\sigma \mapsto x(t_0 + t_1 - \sigma )\) is an integral solution, with the same initial value \(x_0\), of the reflected field
on the segment between \(t_1\) and \(t_0\).
The map \(\sigma \mapsto t_0 + t_1 - \sigma \) is an involution exchanging the two endpoints, so reflection turns a statement anchored at \(t_0\) into one anchored at \(t_1\) and nothing more. Its purpose is to let an argument that is inherently forward-marching — a Grönwall bootstrap, say — be applied unchanged in backward time.
Substituting \(r \mapsto t_0 + t_1 - r\) in the defining integral reverses the orientation of the interval, contributing one sign, and the negation of the field contributes another; the two cancel.
2.2.1 The integral and differential formulations
An ODE can be written as a differential equation, \(\dot x = F(t,x)\), or as the equivalent integral equation. The continuous-dependence and Gronwall arguments want the integral form, which presupposes no differentiability and carries its initial condition; the Lyapunov arguments want the differential form, so that \(V \circ \varphi \) can be differentiated. The results below are the two directions of the fundamental theorem of calculus relating them.
They are provision rather than current practice: nothing in the stability chapters has yet been routed through this equivalence, so it is the intended path from a trajectory to the results of this chapter rather than an exercised one.
Let \(x\) be an integral curve of \(F\) on the segment between \(t_0\) and \(t_1\), with \(s \mapsto F(s, x(s))\) continuous there. Then \(x\) is an integral solution on that segment, anchored at its own value \(x(t_0)\):
Continuity of \(x\) itself is not assumed — it follows from differentiability.
The fundamental theorem of calculus. The derivative hypothesis holds on an open neighbourhood of each interior point, so it yields the right-derivative form the theorem needs, and continuity of \(F\) along \(x\) gives the integrability side condition.
Let \(x\) be an integral solution of \(\dot x = F(t,x)\) on the segment between \(t_0\) and \(t_1\), with \(s \mapsto F(s, x(s))\) continuous there. Then \(x\) is an integral curve of \(F\) on that segment: it is differentiable, with \(\dot x(t) = F(t, x(t))\).
Differentiating the primitive: the integral term has derivative \(F(t, x(t))\) by the fundamental theorem of calculus, and the constant \(x_0\) contributes nothing. Transport that along the defining equation, which says \(x\) agrees with the primitive.
Given continuity of \(s \mapsto F(s, x(s))\) on the segment between \(t_0\) and \(t_1\), the two formulations agree: \(x\) is an integral solution anchored at \(x(t_0)\) if and only if it is an integral curve of \(F\) there. Continuity of \(F\) along \(x\) is exactly the price of the equivalence — without it the integral form is the weaker notion.
lemma 36 in forward time: for \(t_0 \le t_1\), the equivalence holds over \([t_0, t_1]\). This is the interval the finite-forward stability predicates are phrased over, so this is the form the stability track would use to reach the results of this chapter.
Let \(g\) be a continuous autonomous field and let \(x\) be an integral curve of \(g\) on \([t_0,t_1]\), \(t_0 \le t_1\). Then \(x\) is an integral solution there, anchored at \(x(t_0)\).
This is the autonomous, forward-time specialization of lemma 37, and so the shape a trajectory of an autonomous system has: continuity of the field is the only hypothesis, the continuity of \(x\) along which \(F\) must be continuous being supplied by the curve itself.
2.2.2 Time invariance and local existence
A time-invariant field has no preferred time origin: a solution can be shifted to start at time \(0\).
If \(x\) is an integral curve of an autonomous field \(g\) on a set \(s\), then \(t \mapsto x(t + \Delta t)\) is an integral curve of the same \(g\) on \(s - \Delta t\), for every \(\Delta t\).
An autonomous field has no preferred time origin, so translating a solution translates its interval of definition and changes nothing else. For a time-varying field the translated solution would solve the translated equation instead.
lemma 39 in the form the finite-segment predicates need: a solution segment of an autonomous field on \([t_0, t_1]\) becomes one on \([0, t_1 - t_0]\) under \(s \mapsto x(s + t_0)\).
This is why the autonomous stability predicates may fix the initial time at \(0\) without loss: any segment can be shifted there.
Let \(g\) be an autonomous field that is \(C^1\) at \(x_0\). Then there is a \(T {\gt} 0\) and a curve \(\varphi \) with \(\varphi (0) = x_0\) that is an integral curve of \(g\) on \([0, T]\): a nontrivial solution segment exists from every point at which the field is \(C^1\).
The anchor \(0\) is a construction choice rather than a restriction — by lemma 39 a segment from any other anchor follows.
Reference: the Picard–Lindelöf local existence theorem.
Picard–Lindelöf local existence: being \(C^1\) at \(x_0\) gives a closed ball on which \(g\) is Lipschitz, and hence a solution on some \((-\varepsilon , \varepsilon )\); take \(T := \varepsilon /2\).
Let \(f : \mathbb {R} \to E \to E\) be jointly continuous and \(K\)-Lipschitz in the state variable, uniformly in time, on a Banach space \(E\). Then for every \(t_0 \le t_1\) and every \(x_0 \in E\) there is an \(\alpha \) with \(\alpha (t_0) = x_0\) solving \(\dot\alpha = f(t, \alpha )\) on all of \([t_0, t_1]\).
Global — rather than merely local — Lipschitz continuity is what makes the solution reach \(t_1\): it forces linear growth in the state, ruling out finite-time blowup.
Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 3.2 (Global Existence and Uniqueness), proved in his Appendix C.1. Difference: Khalil assumes \(f\) piecewise continuous in \(t\); here it is jointly continuous.
Mathlib’s Picard–Lindelöf theorem supplies a solution on a time interval of length \(\Delta t\) only when \(\| f\| \le L\) on a ball of radius \(a\) about \(x_0\) with \(L\, \Delta t \le a\). For a \(K\)-Lipschitz field the sharpest available bound on that ball is \(L = M + K a\), where \(M\) bounds \(\| f(t, x_0)\| \), so the condition becomes \(M\Delta t \le a(1 - K \Delta t)\) — solvable in \(a\) exactly when \(K \Delta t {\lt} 1\). One application therefore never covers a long interval.
Take instead a step \(h\) with \(Kh \le 1/2\); then \(a = 2Mh\) and \(L = 2M\) satisfy the hypotheses, since \(\| f(t,x)\| \le M + Ka = M(1 + 2Kh) \le 2M\) and \(Lh = 2Mh = a\). This gives a solution on \([s, s+h]\) from any initial state. Chain \(n\) such steps, splicing consecutive segments at their shared endpoint: away from a junction the glued curve agrees with one piece on a neighbourhood, and at a junction the two one-sided derivatives combine. Choosing \(n\) with \(nh \ge t_1 - t_0\) covers \([t_0, t_1]\).
If \(\alpha \) is an integral curve of \(f\) on \([t_0, t_1]\), then for every \(s \in [t_0, t_1)\) it has right derivative \(f(s, \alpha (s))\) at \(s\), i.e. \(\alpha \) has derivative \(f(s,\alpha (s))\) within \([s,\infty )\) at \(s\).
Let \(f\) be \(K\)-Lipschitz in the state variable, uniformly in time, and let \(\alpha , \beta \) be integral curves of \(f\) on \([t_0, t_1]\). Then
Uniqueness (lemma 45) is the case where the initial distance is zero; continuous dependence on the initial state is the statement read as a bound on the whole interval.
Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 3.4, in the unperturbed case: two solutions of the same field, so the perturbation bound is zero.
Let \(f\) be \(K\)-Lipschitz in the state variable, uniformly in time, and let \(\alpha , \beta \) be integral curves of \(f\) on \([t_0, t_1]\) with \(\alpha (t_0) = \beta (t_0)\). Then \(\alpha = \beta \) on \([t_0, t_1]\).
Reference: Khalil, Nonlinear Systems (3rd ed.), the uniqueness half of Theorem 3.2.
The initial distance in lemma 44 is zero, so the bound forces the distance to vanish throughout.
Let \(f\) be jointly continuous and \(K\)-Lipschitz in the state variable, uniformly in time. Then for every \(t_0\) and \(x_0\) there is an \(\alpha \) with \(\alpha (t_0) = x_0\) solving \(\dot\alpha = f(t,\alpha )\) on all of \([t_0,\infty )\), and any other such solution agrees with it on \([t_0,\infty )\).
Uniqueness is agreement on \([t_0,\infty )\), not equality of functions: the defining condition says nothing about \(t {\lt} t_0\), so solutions may differ there.
Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 3.2, whose \(t_1\) is arbitrary, extended to the forward ray.
For each \(n\) take a solution \(\alpha _n\) on \([t_0, t_0+n]\) (theorem 42). By uniqueness (lemma 45) these agree wherever two of them are both defined, so \(\Phi (t) := \alpha _{\lceil t - t_0\rceil }(t)\) is well defined and agrees with \(\alpha _n\) on all of \([t_0, t_0+n]\). Near any \(t \ge t_0\) choose \(n {\gt} t - t_0\); then \([t_0, t_0+n]\) is a neighbourhood of \(t\) within \([t_0,\infty )\) on which \(\Phi = \alpha _n\), so \(\Phi \) inherits the derivative there. Uniqueness on the ray follows by applying the compact-interval uniqueness at each \(n\).
Let \(g : \mathbb {R} \to \mathbb {R} \to \mathbb {R}\) be jointly continuous and \(L\)-Lipschitz in its state argument, uniformly in time, and let \(t_0 \le t_1\). Then for every \(x_0\) there is a \(z\), continuous on \([t_0, t_1]\), that is an integral solution of \(\dot z = g(t, z)\) with \(z(t_0) = x_0\) and has right derivative \(g(s, z(s))\) at each \(s \in [t_0, t_1)\).
This is the scalar specialization of theorem 42; it discharges the perturbed-solution hypothesis that theorem 51 takes for generality.
Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 3.2, scalar case.
Apply theorem 42 with \(E = \mathbb {R}\). Continuity on \([t_0,t_1]\) is automatic for an integral curve, which converts the differential form into the integral one (lemma 37); the right derivatives are lemma 43.
2.2.3 Continuous dependence
Two solutions of the same equation, started at different points or subject to different perturbations, separate no faster than exponentially — at a rate fixed by the Lipschitz constant of the field. Everything downstream that needs uniqueness gets it from here, by taking the two solutions to agree initially and the perturbation to vanish, which collapses the bound to zero.
Khalil, Theorem 3.4. Let \(f\) be jointly continuous and \(L\)-Lipschitz in its state argument on the segment between \(t_0\) and \(t_1\), let \(y\) be an integral solution of \(\dot y = f(t,y)\) with \(y(t_0) = y_0\), and let \(z\) be an integral solution of the perturbed equation \(\dot z = f(t,z) + g(t,z)\) with \(z(t_0) = z_0\), where \(\| g(t,x)\| \le \mu \). Then for every \(t\) on that segment,
Two solutions may separate, but no faster than exponentially, at a rate set by the Lipschitz constant alone; the two error sources — a different starting point and a perturbed field — contribute additively. The bound is stated on the unordered segment, so it runs in either time direction.
The perturbation \(g\) is assumed only bounded and integrable along \(z\), never continuous, so \(z\) need not be differentiable anywhere. That is why the theorem is phrased over the integral formulation rather than the differential one.
The forward case is a Grönwall bootstrap. Subtracting the two integral equations and applying the triangle inequality gives
the Lipschitz hypothesis bounding the \(f\)-difference and the uniform bound on \(g\) the rest. Adding \(\mu /L\) to both sides absorbs the linear term \(\mu (\tau - t_0)\) into the integral, putting the inequality in the shape lemma 30 expects, and Grönwall delivers the exponential.
That bootstrap is inherently forward-marching, so rather than repeat it for the backward case, reflect: lemma 33 sends \(\sigma \mapsto t_0 + t_1 - \sigma \) and negates the field, turning the backward instance into a forward one on \([t_1, t_0]\) to which the same argument applies unchanged.
Khalil, Theorem 3.5. If \(y\) solves \(\dot{y} = f(t,y)\) and \(z\) solves \(\dot{z} = f(t,z) + g(t,z)\) with \(\| g(t,x)\| \le \alpha \) and \(\| z_0 - y_0\| \le \alpha \), and \(\alpha (1 + 1/L)e^{L(t_1-t_0)} \le \varepsilon \), then \(\| y(t) - z(t)\| \le \varepsilon \) for all \(t \in [t_0, t_1]\).
Specialise theorem 48 with \(\mu = \alpha \) and \(\| y_0 - z_0\| \le \alpha \): for \(t \in [t_0, t_1]\) it gives \(\| y(t) - z(t)\| \le \alpha e^{L(t-t_0)} + \frac{\alpha }{L}(e^{L(t-t_0)} - 1) = \alpha (1 + 1/L)e^{L(t-t_0)} - \alpha /L\). Drop the \(-\alpha /L\) and bound \(e^{L(t-t_0)}\) by \(e^{L(t_1-t_0)}\) to get at most \(\varepsilon \).
2.3 Comparison lemma
The comparison lemma bounds a function satisfying a differential inequality by the solution of the corresponding differential equation. The inequality is stated with the Dini derivative, so \(v\) need only be continuous — which is what makes the lemma applicable to \(\| x(t)\| \), not differentiable where a trajectory crosses the origin.
Let \(u\) be continuous on \([t_0, t_1]\) with \(u(t_0) = u_0\) and \(\dot u(t) = f(t, u(t))\) at every interior point, and let \(f\) be jointly continuous. Then \(u\) is an integral solution:
.
The fundamental theorem of calculus on \([t_0, s]\); the integrand is continuous there, being a composition of continuous maps.
Khalil, Lemma 3.4. Let \(f\) be jointly continuous and \(L\)-Lipschitz in its state argument on \([t_0, t_1]\), and suppose
\(u\) is continuous on \([t_0,t_1]\) and solves \(\dot u = f(t,u)\) with \(u(t_0) = u_0\);
\(v\) is continuous on \([t_0,t_1]\) with \(v(t_0) \le u_0\), satisfies the Dini subsolution inequality \(D^{+}v(t) \le f(t, v(t))\), and has forward difference quotients bounded above at each \(t\);
for every \(\lambda {\gt} 0\) the perturbed equation \(\dot z = f(t,z) + \lambda \), \(z(t_0) = u_0\), has a solution on \([t_0,t_1]\).
Then \(v(t) \le u(t)\) for all \(t \in [t_0, t_1]\).
The perturbed solutions in (3) are taken as a hypothesis for generality, so that callers with solutions already in hand need not re-derive them. Under the hypotheses already listed they always exist: \(f + \lambda \) is Lipschitz exactly when \(f\) is, so theorem 47 discharges (3).
Fix \(\lambda {\gt} 0\) and let \(z_\lambda \) be as in hypothesis (3).
Claim: \(v \le z_\lambda \) on \([t_0,t_1]\). Otherwise \(v(t_{\mathrm{bad}}) {\gt} z_\lambda (t_{\mathrm{bad}})\) for some \(t_{\mathrm{bad}}\), and since \(v(t_0) \le z_\lambda (t_0)\) the continuous difference \(v - z_\lambda \) has a last zero \(a {\lt} t_{\mathrm{bad}}\), with \(v {\gt} z_\lambda \) on \((a, t_{\mathrm{bad}}]\). Being below \(v\) immediately to the right of \(a\) forces \(\dot z_\lambda (a) \le D^{+}v(a)\), so
contradicting \(\lambda {\gt} 0\).
Now let \(\varepsilon {\gt} 0\) and choose \(\lambda \) small enough that theorem 49 — which applies because \(f\) is Lipschitz in its state argument, and which treats \(\lambda \) as a perturbation of size \(\lambda \) — gives \(\| u - z_\lambda \| \le \varepsilon /2\) on \([t_0,t_1]\). Then \(v \le z_\lambda \le u + \varepsilon \). As \(\varepsilon \) was arbitrary, \(v \le u\).
2.4 Lie derivatives
The rate of change of a function \(V(t, x)\) along the integral curves of a time-dependent vector field, with no differential-equation content beyond the chain rule: what every Lyapunov, Chetaev and invariance argument differentiates.
For \(V : \mathbb {R} \times E \to \mathbb {R}\) and a time-dependent vector field \(v : \mathbb {R} \times E \to E\), the Lie derivative of \(V\) along \(v\) is
the derivative of \(V\) at \((t, x)\) in the direction \((1, v(t, x))\): the Lie derivative of \(V\), as a function on \(\mathbb {R} \times E\), along the autonomous field \((t, x) \mapsto (1, v(t, x))\).
If \(V\) does not depend on \(t\) and is differentiable at \(x\), then \(L_v V(t, x) = DV(x)\, v(t, x)\).
If \(V\) does not depend on \(t\) and is differentiable at \(x\), then the Lie derivative of \((t, x) \mapsto V(x) + ct\) along \(v\) is \(DV(x)\, v(t, x) + c\).
Let \(V\) be differentiable and \(\gamma \) an integral curve of \(v\) on \(I\). Then for \(t \in I\), within \(I\),
Let \(V\) be differentiable and \(\gamma \) an integral curve of \(v\) on an interval \(s\) with \(L_v V(t, \gamma (t)) \le 0\) at every interior point \(t\) of \(s\). Then \(t \mapsto V(t, \gamma (t))\) is nonincreasing on \(s\).
The mean value theorem, with the derivative given by lemma 55.
Let \(V\) be differentiable, \(\gamma \) an integral curve of \(v\) on \([t_0, t_1]\), and \(c \in \mathbb {R}\) with \(L_v V(t, \gamma (t)) \le -c\, V(t, \gamma (t))\) for \(t \in (t_0, t_1)\). Then \(V(t, \gamma (t)) \le e^{-c(t - t_0)}\, V(t_0, \gamma (t_0))\) on \([t_0, t_1]\).
The derivative of \(e^{c t} V(t, \gamma (t))\) is \(e^{c t}\bigl(c V + L_v V\bigr) \le 0\) (lemma 55), so it does not increase.
If \(V\) is \(C^1\) and \(v\) is continuous, then \((t, x) \mapsto L_v V(t, x)\) is continuous.
2.5 \(\omega \)-limit sets
The points a curve or sequence returns arbitrarily close to, arbitrarily late: its cluster points at infinity, which is the \(\omega \)-limit of topological dynamics for a single orbit. These are the topological facts behind LaSalle’s invariance principle, in continuous and in discrete time alike: a continuous function converging along the orbit is constant on its \(\omega \)-limit set, an orbit trapped in a closed set has its \(\omega \)-limit set there, and an orbit trapped in a compact set converges to its \(\omega \)-limit set.
The \(\omega \)-limit set of \(\varphi : \tau \to X\), for \(\tau \) a preordered index set such as \(\mathbb {R}\) or \(\mathbb {N}\), is the set of cluster points of \(\varphi \) as \(t \to \infty \):
the points \(y\) such that \(\varphi (t)\) is frequently in every neighbourhood of \(y\) as \(t \to \infty \).
\(y \in \omega (\varphi )\) (definition 59) if and only if for every neighbourhood \(U\) of \(y\) there are arbitrarily large \(t\) with \(\varphi (t) \in U\).
definition 59 is the \(\omega \)-limit, in the sense of topological dynamics, of the one-member family \(\{ \varphi \} \).
Let \(g : X \to Y\) be continuous, \(Y\) Hausdorff, and \(g(\varphi (t)) \to L\) as \(t \to \infty \). Then \(g(y) = L\) for every \(y \in \omega (\varphi )\) (definition 59).
\(g(y)\) is a cluster point of \(g \circ \varphi \), which converges to \(L\); in a Hausdorff space the two coincide.
If \(\Omega \) is closed and \(\varphi (t) \in \Omega \) for all large \(t\), then \(\omega (\varphi ) \subseteq \Omega \).
A point outside \(\Omega \) has the open neighbourhood \(X \setminus \Omega \), which \(\varphi \) eventually avoids.
If \(\varphi (t)\) lies in a compact set for all large \(t\) and \(\omega (\varphi ) \subseteq M\), then \(\varphi (t) \to M\): for every open \(W \supseteq M\), \(\varphi (t) \in W\) for all large \(t\).
Mathlib’s eventually_mapsTo_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset (lemma 61): a family eventually absorbed by a compact set is eventually inside every open set containing its \(\omega \)-limit set.