Introduction: Why Impulses?

In this introductory chapter, we discuss the motivations for using impulse controls. As a mathematical justification of impulses, we present a simple variational problem that has solution only in the form of a delta function. We further consider the issue of control for a physical system that results in the same kind of variational problem, thus indicating that impulse controls do arise in real-world applications.

[1]  Alexander N. Daryin,et al.  Dynamic programming for impulse controls , 2008, Annu. Rev. Control..

[2]  Pravin Varaiya,et al.  Dynamics and Control of Trajectory Tubes , 2014 .

[3]  Boris M. Miller,et al.  Impulsive Control in Continuous and Discrete-Continuous Systems , 2003 .

[4]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[5]  F. Rampazzo,et al.  Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls , 1995, Differential and Integral Equations.

[6]  Thomas Carter,et al.  Linearized impulsive rendezvous problem , 1995 .

[7]  Thomas Carter,et al.  A new approach to impulsive rendezvous near circular orbit , 2012, 1211.3081.

[8]  F. Clarke Generalized gradients and applications , 1975 .

[9]  Richard Bellman,et al.  Introduction to the mathematical theory of control processes , 1967 .

[10]  Alexander N. Daryin,et al.  Output feedback strategies for systems with impulsive and fast controls , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[11]  Alexander B. Kurzhanski,et al.  Dynamics and control of trajectory tubes. Theory and computation , 2014, 2014 20th International Workshop on Beam Dynamics and Optimization (BDO).

[12]  L. Neustadt Optimization, a Moment Problem, and Nonlinear Programming , 1964 .

[13]  A. Kurzhanski,et al.  Impulse Control Inputs and the Theory of Fast Feedback Control , 2008 .

[14]  F. Clarke Optimization And Nonsmooth Analysis , 1983 .

[15]  Thomas Carter,et al.  Optimal impulsive space trajectories based on linear equations , 1991 .

[16]  A. B. Kurzhanskii,et al.  Control synthesis in a class of higher-order distributions , 2007 .

[18]  R. Bellman Stability theory of differential equations , 1953 .

[20]  G. Leitmann The Calculus of Variations and Optimal Control: An Introduction , 2013 .

[21]  Alexander N. Daryin,et al.  Closed-loop impulse control of oscillating systems , 2007, PSYCO.

[22]  V. Vladimirov Generalized functions in mathematical physics , 1979 .

[23]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[24]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[25]  A. Kurzhanski,et al.  Attenuation of Uncertain Disturbances Through Fast Control Inputs , 2016 .

[26]  G. Shilov,et al.  Generalized Functions, Volume 1: Properties and Operations , 1967 .

[27]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[28]  E B Lee,et al.  Foundations of optimal control theory , 1967 .