5 Comparison functions
Comparison functions are the vocabulary of stability estimates: a class \(\mathcal{K}\) function bounds how far a trajectory can go, a class \(\mathcal{L}\) function how fast it decays, and a class \(\mathcal{KL}\) function both at once. This chapter builds them, and the tools that turn a cruder bound into one of them.
5.1 Class \(\mathcal{K}\) and \(\mathcal{K}_{\infty }\) functions
A class \(\mathcal{K}\) function on \([0,a)\) is a continuous strictly increasing map \(\alpha : [0,a) \to [0,b)\) with \(\alpha (0) = 0\). All class K functions have an inverse \(\alpha ^{-1} : [0,b) \to [0,a)\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.2 (class \(\mathcal{K}\)). Differences:
Khalil writes the codomain as \([0,\infty )\); here the bound \(b\) is tracked in the type.
The inverse is bundled as data rather than derived.
A class \(\mathcal{K}_{\infty }\) function is a continuous strictly increasing map \(\alpha : [0,\infty ) \to [0,\infty )\) with \(\alpha (0) = 0\) and \(\alpha (r) \to \infty \) as \(r \to \infty \).
Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.2 (class \(\mathcal{K}_{\infty }\)). The inverse is bundled as data rather than derived.
5.1.1 Building class \(\mathcal{K}\) functions
A recognition criterion and the closure properties. The restatements of the definition itself (strict monotonicity as an iff, the two inverse equations, and so on) are left to the Lean API; they say nothing the definition above does not.
Let \(f : \mathbb {R} \to \mathbb {R}\) satisfy \(f(0) = 0\) and \(f(a) = b\) and be continuous and strictly increasing on \([0, a]\), with \(a, b {\gt} 0\). Then \(f\) is a class \(\mathcal{K}\).
If \(\alpha \) is class \(\mathcal{K}\) on \([0, a) \to [0, b)\) and \(0 {\lt} c {\lt} a\), then \(\alpha \) restricted to \([0, c)\) is class \(\mathcal{K}\) on \([0, c) \to [0, \alpha (c))\).
definition 228 with the codomain bound named: if \(\alpha (c) = e\), the restriction of \(\alpha \) to \([0, c)\) is class \(\mathcal{K}\) on \([0, c) \to [0, e)\).
Class \(\mathcal{K}\) is closed under inversion. The inverse of a class K function is also class K.
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, first bullet. Khalil’s domain \([0, \alpha (a))\) is the codomain bound \(b\) carried by the type.
Class \(\mathcal{K}\) is closed under composition: if \(\alpha \) is class \(\mathcal{K}\) on \([0, a) \to [0, b)\) and \(\beta \) is class \(\mathcal{K}\) on \([0, b) \to [0, c)\), then \(\beta \circ \alpha \) is class \(\mathcal{K}\) on \([0, a) \to [0, c)\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, third bullet.
5.1.2 Building class \(\mathcal{K}_{\infty }\) functions
The same recognition criterion and closure properties on the unbounded domain, together with the closure properties that need the global domain (sum, positive scaling, minimum), the power functions as the standard examples, and the restriction back to class \(\mathcal{K}\).
Let \(f : \mathbb {R} \to \mathbb {R}\) satisfy \(f(0) = 0\), be continuous and strictly increasing on \([0, \infty )\), and have \(f(r) \to \infty \) as \(r \to \infty \). Then \(f\) is a class \(\mathcal{K}_{\infty }\) function, its inverse being the inverse of \(f\) on \([0,\infty )\).
Class \(\mathcal{K}_{\infty }\) is closed under inversion: if \(\alpha \) is class \(\mathcal{K}_{\infty }\), so is \(\alpha ^{-1}\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, second bullet.
Class \(\mathcal{K}_{\infty }\) is closed under composition: if \(\alpha \) and \(\beta \) are class \(\mathcal{K}_{\infty }\), so is \(\beta \circ \alpha \), with inverse \(\alpha ^{-1} \circ \beta ^{-1}\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fourth bullet.
Class \(\mathcal{K}_{\infty }\) is closed under pointwise addition: if \(\alpha \) and \(\beta \) are class \(\mathcal{K}_{\infty }\), so is \(\alpha + \beta \).
Class \(\mathcal{K}_{\infty }\) is closed under multiplication by a positive constant: if \(\alpha \) is class \(\mathcal{K}_{\infty }\) and \(c {\gt} 0\), so is \(c\alpha \).
Class \(\mathcal{K}_{\infty }\) is closed under pointwise minimum: if \(\alpha \) and \(\beta \) are class \(\mathcal{K}_{\infty }\), so is \(\min (\alpha , \beta )\).
For every \(p {\gt} 0\), the map \(r \mapsto r^{p}\) is class \(\mathcal{K}_{\infty }\).
Every class \(\mathcal{K}_{\infty }\) function \(\alpha \) restricts, for each \(a {\gt} 0\), to a class \(\mathcal{K}\) function on \([0, a) \to [0, \alpha (a))\).
Let \(\omega : \mathbb {R} \to \mathbb {R}\) satisfy \(\omega (0) = 0\), be monotone on \([0, \infty )\), and be continuous at \(0\) from the right. Then there is a class \(\mathcal{K}_{\infty }\) function \(\alpha \) with \(\omega (r) \le \alpha (r)\) for all \(r \ge 0\).
lemma 268 supplies a continuous, strictly increasing \(f\) with \(f(0) = 0\), \(f(r) \to \infty \) and \(\omega \le f\) on \([0,\infty )\); by definition 232 that \(f\) is class \(\mathcal{K}_{\infty }\).
5.2 Inverses of antitone functions
A function \(W\) that is continuous and strictly antitone on \((0,\infty )\), blows up at \(0^+\) and vanishes at \(+\infty \) surjects onto \((0,\infty )\), and its right inverse has the same shape. This is the mechanism behind the class \(\mathcal{KL}\) construction of the next section: \(W\) is a mean-value bound and its inverse supplies the decay rate.
Let \(W : \mathbb {R} \to \mathbb {R}\) be continuous on \((0,\infty )\) with \(W(\eta ) \to +\infty \) as \(\eta \to 0^+\) and \(W(\eta ) \to 0\) as \(\eta \to +\infty \). Then for every \(s {\gt} 0\) there exists \(c {\gt} 0\) with \(W(c) = s\).
Under the surjectivity conditions of lemma 241, \(W\bigl(\mathrm{invFunOn}\, W\, (0,\infty )\, s\bigr) = s\) for all \(s {\gt} 0\).
Under the surjectivity conditions of lemma 241, \(\mathrm{invFunOn}\, W\, (0,\infty )\, s {\gt} 0\) for all \(s {\gt} 0\).
If \(W\) is strictly antitone on \((0,\infty )\), then \(\mathrm{invFunOn}\, W\, (0,\infty )\) is strictly antitone on \((0,\infty )\).
Under the conditions of lemma 241 and with \(W\) strictly antitone, \(\mathrm{invFunOn}\, W\, (0,\infty )\, s \to 0\) as \(s \to +\infty \).
Under the conditions of lemma 241 and with \(W\) strictly antitone, \(\mathrm{invFunOn}\, W\, (0,\infty )\) is continuous on \((0,\infty )\).
Under the conditions of lemma 241 and with \(W\) strictly antitone, \(\mathrm{invFunOn}\, W\, (0,\infty )\, s \to +\infty \) as \(s \to 0^{+}\).
5.3 Class \(\mathcal{KL}\) functions and the Osgood construction
A class \(\mathcal{L}\) function on \([0,\infty )\) is continuous, strictly positive, and strictly decreasing, and satisfies \(\sigma (s) \to 0\) as \(s \to \infty \). It carries the time decay of an asymptotic stability estimate: the product of a class \(\mathcal{K}\) function of the initial deviation with a class \(\mathcal{L}\) function of elapsed time is a class \(\mathcal{KL}\) bound.
A singular class \(\mathcal{L}\) function on \((0,\infty )\) is continuous, strictly positive and antitone, with \(U(s) \to 0\) as \(s \to \infty \) and \(U(s) \to \infty \) as \(s \to 0^{+}\). It differs from class \(\mathcal{L}\) in two ways: the domain excludes \(0\), where the function is instead required to blow up, and the decrease is only required to be weak. It arises as the inverse of the sliding-window function in the class \(\mathcal{KL}\) characterization of asymptotic stability.
A class \(\mathcal{KL}\) function on \([0,a) \times [0,\infty )\) is continuous, class \(\mathcal{K}\) in the first argument, and for each fixed \(r {\gt} 0\) is strictly decreasing and tends to \(0\) as \(s \to \infty \). It arises as the bound \(\| x(t)\| \le \beta (\| x_0\| , t)\) in asymptotic stability estimates.
Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.3. Difference: Khalil says \(\beta (\cdot , s)\) is class \(\mathcal{K}\); here that is spelled out as separate fields, so no class \(\mathcal{K}\) instance is required for each \(s\).
A global class \(\mathcal{KL}\) function is a map \(\beta : [0,\infty ) \times [0,\infty ) \to \mathbb {R}\) that is class \(\mathcal{K}_{\infty }\) in the first argument (continuous, strictly increasing, zero at zero, radially unbounded) and, for each fixed \(r \ge 0\), is antitone and tends to \(0\) as \(s \to \infty \).
Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.3 with \(a = \infty \).
5.3.1 Building class \(\mathcal{KL}\) functions
Two constructions and the two composition rules, in the bounded and the global setting. The product \(\alpha (r)\gamma (s)\) of a class \(\mathcal{K}\) and a class \(\mathcal{L}\) function is the easy one; the Sontag-style cap is the one that does real work.
If \(\alpha \) is class \(\mathcal{K}\) on \([0,a)\) and \(\gamma \) is class \(\mathcal{L}\), then \(\beta (r, s) := \alpha (r)\gamma (s)\) is class \(\mathcal{KL}\) on \([0,a)\).
The cap is built in the form \(\min \bigl(\alpha (r), \sqrt{\alpha (r)\, U(s)}\bigr)\), so the decay of that particular expression is recorded first.
Let \(c \ge 0\) and let \(U : \mathbb {R} \to \mathbb {R}\) satisfy \(U(s) \to 0\) as \(s \to +\infty \). Then \(\min \bigl(c, \sqrt{c\, U(s)}\bigr) \to 0\) as \(s \to +\infty \).
Let \(\alpha \) be class \(\mathcal{K}\) on \([0,a)\) and let \(U\) be a singular class \(\mathcal{L}\) function (positive, antitone, \(U(s) \to 0\) as \(s \to \infty \) and \(U(s) \to \infty \) as \(s \to 0^{+}\)). Then
is class \(\mathcal{KL}\) on \([0,a)\).
Every field but continuity is immediate from the corresponding property of \(\alpha \) and \(U\), the decay in \(s\) being lemma 253. Continuity is checked in two regimes. For \(s {\gt} 0\) the \(\min \) branch is a composition of continuous maps. At \((r, 0)\), pick a buffer radius \(r_1 \in (r, a)\); since \(U(s) \to \infty \) as \(s \to 0^{+}\), on a neighbourhood we have \(\alpha (r_1) {\lt} U(s)\), hence \(\alpha (r') \le U(s)\) and so \(\alpha (r') \le \sqrt{\alpha (r')U(s)}\) — the minimum is locally just \(\alpha \), which is continuous.
Let \(\beta \) be class \(\mathcal{KL}\) on \([0,a)\) and \(\alpha \) be class \(\mathcal{K}\) on \([0,b)\), and suppose \(\beta (r, s) {\lt} b\) for every \(r \in [0,a)\) and \(s \ge 0\). Then \((r, s) \mapsto \alpha (\beta (r, s))\) is class \(\mathcal{KL}\) on \([0,a)\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet (the outer \(\alpha _1\)). The range hypothesis is additional: \(\alpha _1\) is defined only on \([0,b)\), so \(\beta \)’s values must land there, which Khalil leaves implicit.
If \(\beta \) is class \(\mathcal{KL}\) on \([0,a)\) and \(\alpha \) is class \(\mathcal{K}_{\infty }\), then \((r, s) \mapsto \alpha (\beta (r, s))\) is class \(\mathcal{KL}\) on \([0,a)\). This is definition 255 with the range hypothesis discharged by \(\alpha \) being defined on all of \([0,\infty )\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet, with \(\alpha _1\) of class \(\mathcal{K}_{\infty }\); no range hypothesis is then needed.
If \(\beta \) is class \(\mathcal{KL}\) on \([0,b)\) and \(\alpha \) is class \(\mathcal{K}\) on \([0,a) \to [0,b)\), then \((r, s) \mapsto \beta (\alpha (r), s)\) is class \(\mathcal{KL}\) on \([0,a)\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet (the inner \(\alpha _2\)).
If \(\beta \) is class \(\mathcal{KL}\) on \([0, a) \times [0, \infty )\) and \(0 {\lt} a' \le a\), then \(\beta \) is class \(\mathcal{KL}\) on \([0, a') \times [0, \infty )\).
If \(\alpha \) is class \(\mathcal{K}_{\infty }\) and \(\gamma \) is class \(\mathcal{L}\), then \(\beta (r, s) := \alpha (r)\gamma (s)\) is global class \(\mathcal{KL}\).
If \(\beta \) is global class \(\mathcal{KL}\) and \(\alpha \) is class \(\mathcal{K}_{\infty }\), then \((r, s) \mapsto \alpha (\beta (r, s))\) is global class \(\mathcal{KL}\). No range hypothesis is needed, since \(\alpha \) is defined on all of \([0,\infty )\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet (the outer \(\alpha _1\)), in the global case \(a = \infty \).
If \(\beta \) is global class \(\mathcal{KL}\) and \(\alpha \) is class \(\mathcal{K}_{\infty }\), then \((r, s) \mapsto \beta (\alpha (r), s)\) is global class \(\mathcal{KL}\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet (the inner \(\alpha _2\)), in the global case \(a = \infty \).
If \(\beta \) is class \(\mathcal{KL}\) on \([0,a)\) and \(\alpha \) is class \(\mathcal{K}\) on \([0,a)\), then \((r, s) \mapsto \min \bigl(\beta (r,s), \alpha (r)\bigr)\) is class \(\mathcal{KL}\) on \([0,a)\).
Capping a class \(\mathcal{KL}\) bound by a function of the initial condition alone preserves the class. Its use is to recover a range condition: a bound obtained only as a majorant — from a smoothing construction, say — carries no upper control, and capping restores it without disturbing the bound, since the minimum still dominates whatever \(\beta \) dominated.
Each field is inherited. The minimum of two strictly increasing functions of \(r\) is strictly increasing; \(\alpha \) does not depend on \(s\), so antitonicity in \(s\) survives the minimum; and \(\min (\beta (r,s), \alpha (r)) \to \min (0, \alpha (r)) = 0\) as \(s \to \infty \) because \(\alpha \ge 0\).
If \(\beta \) is global class \(\mathcal{KL}\) and \(\alpha \) is class \(\mathcal{K}_{\infty }\), then \((r, s) \mapsto \min \bigl(\beta (r, s), \alpha (r)\bigr)\) is global class \(\mathcal{KL}\) — capping a \(\mathcal{KL}\) bound by a function of the initial condition alone preserves the class.
5.3.2 Smoothing a bound into a class \(\mathcal{KL}\) function
A candidate bound that has the right qualitative behaviour — monotone in the initial deviation, decaying in time — need not itself be class \(\mathcal{KL}\): it may fail continuity or strict monotonicity. These two results replace such a candidate by a genuine class \(\mathcal{KL}\) function dominating it, which is how the stability characterizations produce their bounds. The first lemma below is the continuity criterion the construction rests on.
Let \(f : \mathbb {R} \times \mathbb {R} \to \mathbb {R}\) be nondecreasing in its first argument, nonincreasing in its second, and continuous in each argument separately. Then \(f\) is jointly continuous.
Near \((r_0, s_0)\), monotonicity gives \(f(r_-, s_+) \le f(r, s) \le f(r_+, s_-)\) whenever \(r_- {\lt} r {\lt} r_+\) and \(s_- {\lt} s {\lt} s_+\). Given \(u {\gt} f(r_0, s_0)\), continuity of \(f(r_0, \cdot )\) gives \(s_- {\lt} s_0\) with \(f(r_0, s_-) {\lt} u\), and then continuity of \(f(\cdot , s_-)\) gives \(r_+ {\gt} r_0\) with \(f(r_+, s_-) {\lt} u\); so \(f {\lt} u\) on the neighbourhood \((-\infty , r_+) \times (s_-, \infty )\). The lower bound is symmetric.
Let \(\psi : \mathbb {R} \to \mathbb {R} \to \mathbb {R}\) satisfy, for all \(r, s \ge 0\):
\(\psi (r, s) \ge 0\);
\(\psi (0, s) = 0\);
\(r \mapsto \psi (r, s)\) is monotone on \([0,\infty )\);
\(s \mapsto \psi (r, s)\) is antitone on \([0,\infty )\);
\(\psi (r, s) \to 0\) as \(s \to \infty \);
\(r \mapsto \psi (r, 0)\) is continuous at \(0\) within \([0,\infty )\).
Then there is a global class \(\mathcal{KL}\) function \(\beta \) with \(\psi (r,s) \le \beta (r,s)\) for all \(r, s \ge 0\).
Clamp both arguments of \(\psi \) at \(0\) and average it over a window in each variable: first \(\psi _1(r, s) = \int _0^1 \psi (r, s - v)\, dv\), over \([s-1, s]\), then \(\psi _2(r, s) = \int _1^2 \psi _1(ru, s)\, du\), over \([r, 2r]\). Each window sits on the side that makes the average dominate, so \(\psi \le \psi _1 \le \psi _2\), and \(\psi _2\) is still nondecreasing in \(r\) and nonincreasing in \(s\).
\(\psi _1(r, \cdot )\) is a difference of primitives of \(\psi (r, \cdot )\), so continuous; \(\psi _2(r, \cdot )\) is then continuous by dominated convergence, the integrands being bounded by \(\psi (2r, 0)\).
For \(r {\gt} 0\), \(\psi _2(r, s) = r^{-1} \int _r^{2r} \psi _1(\cdot , s)\) is a quotient of primitives, so continuous; at \(r \le 0\), \(0 \le \psi _2(r, s) \le \psi (2\max (r,0), 0) \to \psi (0, 0) = 0\) by continuity of \(\psi (\cdot , 0)\) at \(0\).
By lemma 264, \(\psi _2\) is jointly continuous. Also \(\psi _2(r, s) \le \psi (2r, s - 1) \to 0\). Finally \(\beta (r, s) = \psi _2(r, s) + r e^{-s}\) is class \(\mathcal{KL}\), the second term making it strictly increasing in \(r\).
Let \(\alpha \) be class \(\mathcal{K}_{\infty }\) and let \(U\) be a family of functions such that
\(U(r, s) {\gt} 0\) for all \(r {\gt} 0\) and \(s {\gt} 0\);
\(U(r, \cdot )\) is antitone on \((0,\infty )\) for each \(r {\gt} 0\);
\(U(r, s) \to 0\) as \(s \to \infty \), for each \(r {\gt} 0\);
\(r \mapsto U(r+1, s)\) is monotone on \([0,\infty )\) for each \(s {\gt} 0\).
Then there is a global class \(\mathcal{KL}\) function \(\beta \) such that
\(\beta (r, 0) \to \infty \) as \(r \to \infty \);
\(\alpha (r) \le \beta (r, 0)\) for all \(r \ge 0\);
\(\min \bigl(\alpha (r), \sqrt{\alpha (r)U(r+1, s)}\bigr) \le \beta (r, s)\) for all \(r \ge 0\) and \(s {\gt} 0\).
Take \(\psi (r, s) := \alpha (r)\) for \(s = 0\) and \(\min \bigl(\alpha (r), \sqrt{\alpha (r)U(r+1,s)}\bigr)\) for \(s {\gt} 0\), the radius being shifted by \(1\) so that \(U\) is only ever evaluated at a strictly positive radius. Each hypothesis of lemma 265 holds for \(\psi \): monotonicity in \(r\) from that of \(\alpha \) and of \(r \mapsto U(r+1,s)\), antitonicity in \(s\) from that of \(U(r+1,\cdot )\), decay from lemma 253, and continuity at \(r = 0\) because \(\psi (\cdot , 0) = \alpha \). The resulting \(\beta \) dominates \(\psi \), which gives the three conclusions; \(\beta (r,0) \ge \alpha (r) \to \infty \) gives the first. No property of \(\psi \) beyond these bounds appears in the statement.
5.4 Class \(\mathcal{K}\) bounds for positive definite functions
A Lyapunov function is positive definite: a pointwise condition, awkward to combine with estimates. The theorem below converts it into monotone bounds in the single variable \(\| x\| \), after which every Lyapunov argument can be run in the comparison-function calculus. It rests on two smoothing results, which replace a monotone bound by a continuous strictly increasing one.
Let \(r {\gt} 0\) and let \(\psi : \mathbb {R} \to \mathbb {R}\) satisfy \(\psi (0) = 0\), \(\psi (s) {\gt} 0\) for \(s \in (0, r]\), and \(\psi \) nondecreasing on \([0, r]\). Then there are \(b {\gt} 0\) and a function \(f\) with \(f(0) = 0\), \(f(r) = b\), \(f\) continuous and strictly increasing on \([0, r]\), and \(f(s) \le \psi (s)\) for all \(s \in [0, r]\).
That is: a positive nondecreasing function admits a class \(\mathcal{K}\) minorant on any bounded interval. This is what turns a positive-definite \(V\) into the lower comparison bound of theorem 270.
Take \(f(s) = r^{-1} \int _0^s \psi \) (with the argument of \(\psi \) clamped to \([0, r]\), so that it is monotone, hence integrable, on all of \(\mathbb {R}\)). It is continuous as a primitive, and \(f(0) = 0\). For \(s {\lt} s'\) in \([0, r]\), \(f(s') - f(s) = r^{-1} \int _s^{s'} \psi {\gt} 0\) since \(\psi {\gt} 0\) on \((s, s')\), so \(f\) is strictly increasing and \(b = f(r) {\gt} 0\). Finally, \(\psi \) is nondecreasing, so \(\int _0^s \psi \le s\, \psi (s) \le r\, \psi (s)\), i.e. \(f(s) \le \psi (s)\).
Let \(\varphi : \mathbb {R} \to \mathbb {R}\) satisfy \(\varphi (0) = 0\), be monotone on \([0, \infty )\), and be continuous at \(0\) from the right. Then there is an \(f\) with \(f(0) = 0\), continuous and strictly increasing on \([0, \infty )\), \(f(s) \to \infty \) as \(s \to \infty \), and \(\varphi (s) \le f(s)\) for all \(s \ge 0\).
That is: a nondecreasing function admits a class \(\mathcal{K}_{\infty }\) majorant. The unbounded domain is what distinguishes this from lemma 269, which it implies.
Continuity at \(0\) is necessary: the unit step has no continuous majorant vanishing at \(0\).
Take \(f(s) = s + g(s)\) with \(g(s) = \int _1^2 \varphi (su)\, du\), the average of \(\varphi \) over \([s, 2s]\) (with \(\varphi \) clamped to \(\varphi (\max (t,0))\) so that it is monotone, hence integrable, on all of \(\mathbb {R}\)). Monotonicity of \(\varphi \) gives \(\varphi (s) \le g(s) \le \varphi (2s)\) and that \(g\) is nondecreasing, so \(f\) is strictly increasing, \(f \ge \varphi \), and \(f(s) \ge s \to \infty \). For \(s {\gt} 0\), \(g(s) = s^{-1}\bigl(\Phi (2s) - \Phi (s)\bigr)\) with \(\Phi \) the (continuous) primitive of \(\varphi \), so \(g\) is continuous there; at \(0\), \(0 \le g(s) \le \varphi (2s) \to 0\) by continuity of \(\varphi \) at \(0\).
Let \(r {\gt} 0\) and let \(\varphi \) satisfy \(\varphi (0) = 0\), be monotone on \([0, r]\), and be continuous at \(0\) from the right. Then there are \(b {\gt} 0\) and an \(f\) with \(f(0) = 0\), \(f(r) = b\), \(f\) continuous and strictly increasing on \([0, r]\), and \(\varphi (s) \le f(s)\) for all \(s \in [0, r]\).
Clamp \(\varphi \) to \(\varphi _{\mathrm{ext}}(s) = \varphi (\min (s, r))\), which is monotone on all of \([0,\infty )\), still vanishes at \(0\) and agrees with \(\varphi \) near \(0\) (so is still continuous there), and apply lemma 268. Restricting the resulting majorant to \([0, r]\) gives \(f\); \(b = f(r) {\gt} 0\) because \(f\) is strictly increasing from \(f(0) = 0\).
Let \(W : \mathbb {R}^{n} \to \mathbb {R}\) be positive definite (definition 27) about \(x_{\mathrm{eq}}\) on a set containing \(\overline{B}(x_{\mathrm{eq}}, r)\). Then there are class \(\mathcal{K}\) functions \(\alpha _1, \alpha _2\) on \([0, r)\) with
The conclusion holds on the open ball: \(\alpha _1\) and \(\alpha _2\) are class \(\mathcal{K}\) on \([0, r)\), so neither is defined at \(\| x - x_{\mathrm{eq}}\| = r\).
This is what lets a Lyapunov argument be run entirely in terms of comparison functions: positive definiteness is a pointwise condition, and this converts it into monotone bounds in the single variable \(\| x - x_{\mathrm{eq}}\| \).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.3 (proved in his Appendix C.4). Differences:
The bound holds on the open ball, with \(\alpha _1, \alpha _2\) class \(\mathcal{K}\) on \([0,r)\); Khalil states it on \([0,r]\).
The radially unbounded \(\mathcal{K}_{\infty }\) case is not derived here.
Translate to the origin, \(V(y) = W(y + x_{\mathrm{eq}})\). For the lower bound take \(\psi (s) = \inf \{ V(y) : s \le \| y\| \le r\} \), the infimum of \(V\) over the annulus of inner radius \(s\): it vanishes at \(0\), is positive for \(s {\gt} 0\) by compactness and positive definiteness, and is nondecreasing. For the upper bound take \(\phi (s) = \sup \{ V(y) : \| y\| \le s\} \), similarly nondecreasing and vanishing at \(0\). Neither is continuous or strictly monotone, so smooth them: lemma 267 gives a class \(\mathcal{K}\) minorant of \(\psi \) and lemma 269 a class \(\mathcal{K}\) majorant of \(\phi \). Since \(\psi (\| y\| ) \le V(y) \le \phi (\| y\| )\) pointwise, these sandwich \(V\).
5.5 The decay system and the comparison bound
A class \(\mathcal{K}\) decay rate produces a class \(\mathcal{KL}\) bound. This is the step that turns a Lyapunov estimate — which controls only the rate of decrease — into an asymptotic stability statement, and it is where the comparison system \(\dot y = -\alpha (y)\) enters: a Lyapunov function along a trajectory is a subsolution of it.
For \(\alpha \) class \(\mathcal{K}\) on \([0, a)\) (definition 225), the decay system is the scalar time-invariant system
as a continuous-time system (definition 65) with no input. Its meaningful states are \(y \in [0, a)\).
Let \(\alpha \) be class \(\mathcal{K}\) on \([0,a)\) and at most linear near the origin, \(\alpha (x) \le Lx\) on \((0, \mathrm{base}]\). Then there is a class \(\mathcal{KL}\) function \(\sigma \) (definition 250) with \(\sigma (y_0, 0) = y_0\) such that for every \(y_0 \in [0, a)\) and every \(t_0\), \(t \mapsto \sigma (y_0, t - t_0)\) is a trajectory on \([t_0, \infty )\) (definition 83) of the decay system \(\dot y = -\alpha (y)\) (definition 271).
Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.4. Differences:
Khalil assumes \(\alpha \) locally Lipschitz; here the hypothesis is the linear bound \(\alpha (x) \le Lx\) near the origin, which is what the construction uses.
Uniqueness of the solution is not asserted.
Osgood’s construction. Let \(\eta (y) = -\int _{\mathrm{base}}^{y} \mathrm{d}x/\alpha (x)\) be the time for the solution started at \(\mathrm{base}\) to reach \(y\). Since \(\alpha {\gt} 0\) on \((0,a)\), \(\eta \) is continuous and strictly decreasing there; and since \(\alpha (x) \le Lx\) near \(0\), \(\eta (y) \ge (\log \mathrm{base} - \log y)/L \to +\infty \) as \(y \to 0^{+}\), so the origin is not reached in finite time. Setting \(\sigma (r,s) := \eta ^{-1}(\eta (r) + s)\), extended by \(\sigma (0,s) = 0\), the class \(\mathcal{KL}\) conditions follow from monotonicity and continuity of \(\eta ^{-1}\), and differentiating \(\eta (\sigma ) = \eta (r) + s\) gives \(\dot\sigma = -\alpha (\sigma )\) for \(s {\gt} 0\). At \(s = 0\) the one-sided derivative is the limit of these, \(-\alpha (r)\), by continuity of \(\sigma \) and \(\alpha \).
Let \(\alpha \) be a class \(\mathcal{K}\) function on \([0, a)\). Then there are \(b' {\gt} 0\), \(L {\gt} 0\) and a class \(\mathcal{K}\) function \(\beta : [0, a) \to [0, b')\) with \(\beta (x) \le \alpha (x)\) for \(x \in [0, a)\), \(\beta (x) \le L x\) for \(x \ge 0\), and \(\beta \) \(L\)-Lipschitz on all of \(\mathbb {R}\).
The range of \(\beta \) cannot in general be the range \([0, b)\) of \(\alpha \): \(\alpha (x) = 1 - \sqrt{1 - x}\) on \([0, 1)\) approaches \(1\) more slowly than any Lipschitz function can.
Extend \(\alpha \) to a nondecreasing \(g : \mathbb {R} \to [0, b]\) by \(0\) on the left and \(b\) from \(a\) on, and set \(\beta (x) = a^{-1} \int _0^x g\), so \(b' = \beta (a)\). Since \(0 \le g \le b\), \(\beta \) is \((b/a)\)-Lipschitz, and with \(\beta (0) = 0\) this gives \(\beta (x) \le (b/a)\, x\). Since \(g {\gt} 0\) on \((0, \infty )\), \(\beta \) is strictly increasing, hence class \(\mathcal{K}\) by definition 227. Finally \(g\) is nondecreasing, so for \(x \in [0, a)\), \(\int _0^x g \le x\, \alpha (x) \le a\, \alpha (x)\), i.e. \(\beta (x) \le \alpha (x)\).
Let \(\alpha \) be class \(\mathcal{K}\) on \([0,a)\). Then there is a class \(\mathcal{KL}\) function \(\sigma \) with \(\sigma (r,0) \le r\) such that: whenever \(v\) is continuous on \([t_0,t]\), takes values in \([0,a)\) there, has bounded forward difference quotients, and is a Dini subsolution of the decay system (definition 271),
it obeys \(v(t) \le \sigma \bigl(v(t_0),\, t - t_0\bigr)\).
A differential inequality with a class \(\mathcal{K}\) decay rate therefore yields a class \(\mathcal{KL}\) bound — which is what converts a Lyapunov decay estimate into an asymptotic stability statement. No hypothesis is placed on \(\alpha \) beyond being class \(\mathcal{K}\): the linear-growth condition the construction needs is obtained internally by passing to a Lipschitz minorant.
Replace \(\alpha \) by a class \(\mathcal{K}\) minorant \(\beta \le \alpha \) that is Lipschitz near the origin; this only weakens the Dini hypothesis, and supplies the linear bound \(\beta (x) \le Lx\) that theorem 272 requires. That theorem gives a class \(\mathcal{KL}\) \(\sigma \) whose sections solve \(\dot y = -\beta (y)\) with \(\sigma (r,0) = r\). Since \(v\) is a Dini subsolution of the same equation and starts at \(\sigma (v(t_0), 0)\), theorem 51 bounds \(v\) by it — the perturbed solutions that lemma requires coming from theorem 47, whose hypotheses hold because \(\beta \) is globally Lipschitz (lemma 273).