Saturated differential equation
The bound on ODI solutions is based on the Ordinary Differential Equation (ODE)
obtained by saturating the inequality constraint, that is to say, by replacing
the inequality sign by an equality:
Theorem (Petrovitsch, 1901): if u satisfies the differential
inequality u′(t)≥f(u(t),t), and y is the solution to
the ODE y′(t)=f(y(t),t) under the boundary condition
u(t0)=y(t0), then:
∀t<t0, u(t)≤y(t)∀t>t0, u(t)≥y(t)Gronwall's inequality
is a specific case of Petrovitsch's theorem in the case of first-order linear
differential inequalities:
Theorem (Gronwall, 1919): if u satisfies the differential
inequality u′(t)≤β(t)u(t), then it is bounded by the
solution of the saturated differential equation y′(t)=β(t)y(t):
u(t) ≤ u(a)exp(∫atβ(s)ds)Both results follow the same approach. Thanks to the standard trick to reduce
n-th order differential equations to first-order ones,
Petrovitsch's theorem can be applied to our second-order example.
Solution of the example
In the example above, the saturated ODE is given by:
ω2y¨ = y+bIt is well known that the set of solutions to such second-order differential
equations, whose characteristic polynomial has real roots, are linear
combinations of hyperbolic functions, plus the non-homogeneous
part of the solution:
y = αcosh(ω(t−t0))+βsinh(ω(t−t0))−bWe now choose the boundary conditions such that y(t0)=x(t0)=−b
and y˙(t0)=x˙(t0), resulting in:
y = ωx˙(t0)sinh(ωt)−bApply Petrovitsch's theorem to the two-dimensional u(t)=(x,x˙), we
consider the difference δ between x (ODI solution) and
y (ODE solution). Then:
δ¨δ(t0)δ˙(t0)≥ω2δ=0=0Due to the initial condition, either δ is uniformly 0 for
all t≥t0 (in which case the bound is tight), or δ is
increasing for t≥t0 and thus positive. In both cases, this
difference ends up being positive, so that ∀t≥t0,x(t)≥y(t) and y is indeed an analytical lower-bound to all solutions of the
ODI.