Applied Optimal Control: Optimization, Estimation, and Control

Bryson, Arthur E.; Ho, Yu-Chi

doi:10.1109/tsmc.1979.4310229

Cited by 630 publications

(439 citation statements)

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“…The minimisation of errors through gradient descent (Hadamard, 1908) in the parameter space of complex, nonlinear, differentiable (Leibniz, 1684), multi-stage, NN-related systems has been discussed at least since the early 1960s (e.g., Kelley, 1960;Bryson, 1961;Bryson and Denham, 1961;Pontryagin et al, 1961;Dreyfus, 1962;Wilkinson, 1965;Amari, 1967;Bryson and Ho, 1969;Director and Rohrer, 1969), initially within the framework of Euler-LaGrange equations in the Calculus of Variations (e.g., Euler, 1744).…”

Section: -1981 and Beyond: Development Of Backpropagation (Bp) Fomentioning

confidence: 99%

“…Steepest descent in the weight space of such systems can be performed (Bryson, 1961;Kelley, 1960;Bryson and Ho, 1969) by iterating the chain rule (Leibniz, 1676;L'Hôpital, 1696)à la Dynamic Programming (DP) (Bellman, 1957). A simplified derivation of this backpropagation method uses the chain rule only (Dreyfus, 1962).…”

Section: -1981 and Beyond: Development Of Backpropagation (Bp) Fomentioning

confidence: 99%

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Deep learning in neural networks: An overview

2015

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In recent years, deep artificial neural networks (including recurrent ones) have won numerous contests in pattern recognition and machine learning. This historical survey compactly summarises relevant work, much of it from the previous millennium. Shallow and deep learners are distinguished by the depth of their credit assignment paths, which are chains of possibly learnable, causal links between actions and effects. I review deep supervised learning (also recapitulating the history of backpropagation), unsupervised learning, reinforcement learning & evolutionary computation, and indirect search for short programs encoding deep and large networks.LATEX source: http://www.idsia.ch/˜juergen/DeepLearning8Oct2014.tex Complete BIBTEX file (888 kB): http://www.idsia.ch/˜juergen/deep.bib Preface This is the preprint of an invited Deep Learning (DL) overview. One of its goals is to assign credit to those who contributed to the present state of the art. I acknowledge the limitations of attempting to achieve this goal. The DL research community itself may be viewed as a continually evolving, deep network of scientists who have influenced each other in complex ways. Starting from recent DL results, I tried to trace back the origins of relevant ideas through the past half century and beyond, sometimes using "local search" to follow citations of citations backwards in time. Since not all DL publications properly acknowledge earlier relevant work, additional global search strategies were employed, aided by consulting numerous neural network experts. As a result, the present preprint mostly consists of references. Nevertheless, through an expert selection bias I may have missed important work. A related bias was surely introduced by my special familiarity with the work of my own DL research group in the past quarter-century. For these reasons, this work should be viewed as merely a snapshot of an ongoing credit assignment process. To help improve it, please do not hesitate to send corrections and suggestions to juergen@idsia.ch.

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Section: -1981 and Beyond: Development Of Backpropagation (Bp) Fomentioning

confidence: 99%

Section: -1981 and Beyond: Development Of Backpropagation (Bp) Fomentioning

confidence: 99%

Deep learning in neural networks: An overview

2015

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“…First-order necessary conditions for optimality are given by the Pontryagin Maximum Principle [23,5]. It can easily be shown that extremals are normal (in the sense of optimal control) and therefore these conditions reduce to the following statement: If u * is an optimal control with corresponding trajectory (N * , c * ), then there exists an absolutely continuous function (λ, µ), which we write as row-vectors λ :…”

Section: Necessary Conditions For Optimalitymentioning

confidence: 99%

“…In principle, this order can vary with time over the interval I. If it is constant on the interval I, then it is a necessary condition for optimality of a singular arc of order k, the so-called generalized Legendre-Clebsch condition [12,5], that…”

Section: Singular Extremalsmentioning

confidence: 99%

Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy

Ledzewicz

Schättler

2007

Mathematical Biosciences

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A mathematical model for the depletion of bone marrow under cancer chemotherapy is analyzed as an optimal control problem. The control represents the drug dosage of a single chemotherapeutic agent and pharmacokinetic equations which model its plasma concentration are included. The drug dosages enter the objective linearly. It is shown that optimal controls are bang-bang, i.e. alternate the drug dosages at full dose with rest-periods in between, and that singular controls which correspond to treatment schedules with varying dosages at less than maximum rate are not optimal. Numerical simulations are given to illustrate the effect of the pharmacokinetic equations on the dosages.

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“…On the other hand, modern guidance laws are developed based on optimal control [19], [11], sliding mode control [20], differential geometry [21], and so on. Interestingly, in a linearized engagement geometry against a moving but non-maneuvering target, TPN was shown to be optimal in [22]. And, for aerodynamically driven vehicles like fixed-wing UAVs PPN is more suited option than TPN.…”

Section: Introductionmentioning

confidence: 99%

Development and testing of the intercept primitives for planar UAV engagement

Ghosh

Davis

Chung

et al. 2017

2017 International Conference on Unmanned Aircraft Systems (ICUAS)

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Abstract-With the advance in technologies and applications involving unmanned aerial systems cooperation among the autonomous agents within such an unmanned system as well as preparedness for responding to adversarial threat to such a system has become of pivotal importance. A multiphase operational scenario of such cooperative and adversarial engagement, motivated by manned aerial engagement scenarios, is presented in this paper. A Proportional navigation-based integrated guidance methodology is proposed and investigated as a candidate strategy for intercept primitives in such realistic multi-phase UAV engagements. Numerical examples are presented to demonstrate the performance of the proposed guidance strategy for an UAV in such an engagement. Finally, implementation of these guidance laws in a waypoint-based manner, well suited for use in modern-day autopilots of UAVs, is demonstrated in software-in-the-loop simulations, further accelerating future live-fly capabilities for autonomous aerial engagements.

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Applied Optimal Control: Optimization, Estimation, and Control

Cited by 630 publications

References 0 publications

Deep learning in neural networks: An overview

Deep learning in neural networks: An overview

Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy

Development and testing of the intercept primitives for planar UAV engagement

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