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Gronwall used an integral bound to extend direct parameter-differentiability arguments from a single differential equation to a system, and treated derivatives with respect to initial values and initial time. A differential equation describes how a changing quantity evolves. To understand how a small change in an input affects the trajectory, one must justify differentiating the solution itself. Gronwall’s1919 note assumes that the solutions already exist uniquely on a shared interval and that the right-hand side and the relevant state and parameter derivatives are continuous. It bounds the largest difference quotient by a simple integral estimate, then applies the estimate again to show convergence to the solution of a linear variation system. The same strategy treats initial values and, with an extended existence region, the initial time. This direct system argument extends a scalar argument explicitly credited to Ritt. Its integral lemma became a reusable way to control accumulating changes. The original bound uses constant nonnegative coefficients on a finite interval; the broader time-varying, nonlinear and measure versions often called Gronwall–Bellman inequalities belong to later work.
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Scientific scope
The integral inequality uses the stated sign, continuity/integrability and domain assumptions. It controls solutions relative to a specified comparison function and coefficient; it does not automatically give uniform-in-time stability or convergence when the resulting exponential grows. Generalized variants and nonlinear applications require their own hypotheses.
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People and systems
T. H. Gronwall
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English catalogue narrative
Gronwall: control how differential-equation solutions change with their inputs
Gronwall used an integral bound to extend direct parameter-differentiability arguments from a single differential equation to a system, and treated derivatives with respect to initial values and initial time.
A differential equation describes how a changing quantity evolves. To understand how a small change in an input affects the trajectory, one must justify differentiating the solution itself. Gronwall’s1919 note assumes that the solutions already exist uniquely on a shared interval and that the right-hand side and the relevant state and parameter derivatives are continuous. It bounds the largest difference quotient by a simple integral estimate, then applies the estimate again to show convergence to the solution of a linear variation system. The same strategy treats initial values and, with an extended existence region, the initial time. This direct system argument extends a scalar argument explicitly credited to Ritt. Its integral lemma became a reusable way to control accumulating changes. The original bound uses constant nonnegative coefficients on a finite interval; the broader time-varying, nonlinear and measure versions often called Gronwall–Bellman inequalities belong to later work.
Scientific scope
The integral inequality uses the stated sign, continuity/integrability and domain assumptions. It controls solutions relative to a specified comparison function and coefficient; it does not automatically give uniform-in-time stability or convergence when the resulting exponential grows. Generalized variants and nonlinear applications require their own hypotheses.
People and systems
T. H. Gronwall