Differential (infinitesimal)

About: Differential (infinitesimal) is a research topic. Over the lifetime, 747 publications have been published within this topic receiving 8182 citations.

Finite-Time Stability of Linear Time-Varying Systems: Analysis and Controller Design

Abstract: The note deals with the finite-time analysis and design problems for continuous-time, time-varying linear systems. Necessary and sufficient conditions and a sufficient condition for finite-time stability are devised. Moreover, sufficient conditions for the solvability of both the state and the output feedback problems are stated. Such results require the feasibility of optimization problems involving Differential Linear Matrix Inequalities. Some numerical examples illustrate the effectiveness of the proposed approach.

Discrete differential forms for computational modeling

Abstract: The emergence of computers as an essential tool in scientific research has shaken the very foundations of differential modeling. Indeed, the deeply-rooted abstraction of smoothness, or differentiability, seems to inherently clash with a computer's ability of storing only finite sets of numbers. While there has been a series of computational techniques that proposed discretizations of differential equations, the geometric structures they are supposed to simulate are often lost in the process.

A Differential Geometric Approach to Motion Planning

Abstract: We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added to the right-hand side. The main point then is to use formal calculations based on the product expansion relative to a P. Hall basis, to produce another control that achieves the desired result on the formal level. It then turns out that this control provides an exact solution of the original problem if the given system is nilpotent. When the system is not nilpotent, one can still produce an iterative algorithm that converges very fast to a solution. Using the theory of feedback nilpotentization, one can find classes of non-nilpotent systems for which the algorithm, in cascade with a precompensator, produces an exact solution in a finite number of steps. We also include results of simulations which illustrate the effectiveness of the procedure.

Differential Games of Pursuit

Abstract: The classical optimal control theory deals with the determination of an optimal control that optimizes the criterion subject to the dynamic constraint expressing the evolution of the system state under the influence of control variables. If this is extended to the case criteria (payoff function) it is possible to begin to explore differential games. Zero-sum differential games, also called differential games of pursuit, constitute the most developed part of differential games and are rigorously investigated. In this book, the full theory of differential games of pursuit with complete and partial information is developed. Numerous concrete pursuit-evasion games are solved ("life-line" game, simple pursuit games, etc) and new time-consistent optimality principles in the n-person differential game theory are introduced and investigated.

A primer of infinitesimal analysis

Abstract: Introduction 1. Basic features of smooth worlds 2. Basic differential calculus 3. First applications of the differential calculus 4. Applications to physics 5. Multivariable calculus and applications 6. The definite integral: Higher order infinitesimals 7. Synthetic geometry 8. Smooth infinitesimal analysis as an axiomatic system Appendix Models for smooth infinitesimal analysis.