For distributions, we build a theory of higher order pointwise differentiability comprising, for zero, {\L}ojasiewicz's notion point value. Results include Borel regularity differentials, rectifiability the associated jets, Rademacher-Stepanov type theorem, and Lusin approximation. A substantial part this development is new also zeroth order. Moreover, establish Poincar\'e inequality involving ...

This letter develops, theoretically justifies, and experimentally implements an optimization-based nonlinear control methodology for stabilizing quadrupedal locomotion. framework utilizes virtual constraints Lyapunov functions (CLFs) in the context of quadratic programs (QPs) to robustly stabilize periodic orbits hybrid models robots. Properties proposed QP are studied wherein sufficient condit...

Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of G~teaux, Fr6chet, and Hadamard are singled out from the general framework of cr-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equ...

Abstract In this paper, the concept of generalized saddle point(GSP) is employed to discuss the optimization problems of a set of convex functions on a normed linear space X , which presents an equivalence under a special condition between GSP and its optimum solution. A study on integrated convex optimization problem by using Gâteaux and Fréchet differentiability respectivly, and the equivalen...

Abstract Artificial neural networks are referred to as universal approximators due their inherent ability reconstruct complex linear and nonlinear output maps conceived input-output relationships from data sets. This can be done by reducing large via regularization in order establish compact models containing fewer parameters aimed at describing vital dependencies In situations where the sets c...