# An implicit function theorem: Comment

@article{Kumagai1980AnIF,
title={An implicit function theorem: Comment},
journal={Journal of Optimization Theory and Applications},
year={1980},
volume={31},
pages={285-288}
}
• S. Kumagai
• Published 1 June 1980
• Mathematics
• Journal of Optimization Theory and Applications
In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:Rn ×Rm →Rn, withF(x0,y0)=0, that requires neither differentiability ofF nor nonsingularity of ∂xF(x0,y0). In the proof, the local one-to-one condition forF(·,y):A ⊂Rn →Rn for ally ∈B is consciously or unconsciously treated as implying thatF(·,y) mapsA one-to-one ontoF(A, y) for ally ∈B, and the proof is not perfect. A proof can be given directly, and the theorem is shown to be the strongest, in the sense that… Expand
62 Citations
Brockett ’ s Theorem and Coron ’ s Condition
• 2020
ABSTRACT. Feedback asymptotic stabilization of nonlinear control systems is an important topic of control theory and applications. Broadly speaking, if the system • x = f(x, u) is locallyExpand
The Implicit Function Theorem when the matrix $\frac{\partial F}{\partial y}(x,y)$ is only continuous at the base point
This article presents an elementary proof of the Implicit Function Theorem for differentiable maps F(x,y), defined on a finite-dimensional Euclidean space, with $\frac{\partial F}{\partial y}(x,y)$Expand
THE IMPLICIT FUNCTION THEOREM AND IMPLICIT PARAMETRIZATIONS
We discuss a dierential equations treatment of the implicit functions problem. Our approach allows a precise and complete description of the solution, of continuity and dierentiability properties.Expand
On Lipschitz Continuous Optimal Stopping Boundaries
• Mathematics, Computer Science
• SIAM J. Control. Optim.
• 2019
A probabilistic proof of the local Lipschitz continuity for the optimal stopping boundary of a class of problems with state space, which is the only existing proof that relies exclusively upon stochastic calculus. Expand
Control from an interior hypersurface
• Mathematics, Physics
• 2017
We consider a compact Riemannian manifold $M$ (possibly with boundary) and $\Sigma \subset M\setminus \partial M$ an interior hypersurface (possibly with boundary). We study observation and controlExpand
Measure-valued differentiation for finite products of measures : theory and applications
In this dissertation we perform a comprehensive analysis of measure-valued differentiation, in which weak differentiation of parameter-dependent probability measures plays a central role. We developExpand
Numerical Analysis of Nonlinear Partial Differential-Algebraic Equations: A Coupled and an Abstract Systems Approach
Various mathematical models in many application areas give rise to systems of partial differential equations and differential-algebraic equations (DAEs). These systems are called partial or abstractExpand
On the best Lipschitz extension problem for a discrete distance and the discrete ∞-Laplacian
• Mathematics
• 2012
Abstract This paper concerns the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discreteExpand
On finite-time stabilization of evolution equations: A homogeneous approach
• Mathematics, Computer Science
• 2016 IEEE 55th Conference on Decision and Control (CDC)
• 2016
A universal homogeneous feedback control for a finite-time stabilization of linear evolution equation in a Hilbert space is designed using homogeneity concept and the design scheme is demonstrated for distributed finite- time control of heat and wave equations. Expand
On the Cauchy problem for planar autonomous Hamiltonian Systems with a discontinuous right-hand side
Using the implicit function theorem for non-differentiable functions and structural properties of the dynamical system, we prove a local existence and uniqueness result for a class of planarExpand