Pontryagin's Maximum Principle
Pontryagin's maximum (or minimum) principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It was formulated in 1956 by the Russian mathematician Lev Semenovich Pontryagin and his students. It has as a special case the Euler–Lagrange equation of the calculus of variations.
The principle states informally that the Hamiltonian must be minimized over , the set of all permissible controls. If is the optimal control for the problem, then the principle states that:

where is the optimal state trajectory and is the optimal costate trajectory.
The result was first successfully applied into minimum time problems where the input control is constrained, but it can also be useful in studying state-constrained problems.
Special conditions for the Hamiltonian can also be derived. When the final time is fixed and the Hamiltonian does not depend explicitly on time , then:

and if the final time is free, then:

More general conditions on the optimal control are given below.
When satisfied along a trajectory, Pontryagin's minimum principle is a necessary condition for an optimum. The Hamilton–Jacobi–Bellman equation provides a necessary and sufficient condition for an optimum, but this condition must be satisfied over the whole of the state space.

This is an excerpt from the article Pontryagin's Maximum Principle from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
The article Pontryagin's Maximum Principle at en.wikipedia.org was accessed 97 times in the last 30 days. (as of: 06/11/2013)
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Pontryagin's minimum principle - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/Pontryagin's_minimum_principle
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Pontryagin Maximum Principle -- from Wolfram MathWorld
Then in order for a control u(t) and a trajectory x(t) to be optimal, it is necessary that there exist nonzero absolutely continuous vector function psi(t)=(psi_0(t) ...
mathworld.wolfram.com/PontryaginMaximumPrinciple.html
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Pontryagin's Maximum Principle - San Jose State University
Pontryagin's Maximum Principle applies to a particular type of problem called a Bolzano Problem. Most optimization problems can be put into the form of a ...
www.sjsu.edu/faculty/watkins/pontryag.htm
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16 Pontryagin's maximum principle
16 Pontryagin's Maximum Principle. This is a powerful method for the computation of optimal controls, which has the crucial advantage that it does not require ...
www.statslab.cam.ac.uk/~james/Lectures/oc16.pdf
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Pontryagin's maximum principle - Washington - University of ...
Pontryagin's Maximum Principle. Emo Todorov. Applied Mathematics and Computer Science & Engineering. University of Washington. Winter 2012 ...
homes.cs.washington.edu/~todorov/courses/amath579/Maximum.pdf
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Elements of Optimal Control Theory Pontryagin's Maximum Principle
Pontryagin's Maximum Principle gives a necessary condition ... It can be formulated as a Pontryagin Maximum Principle problem as follows. Write the second ...
www.uccs.edu/~rcascava/Math4480/PontryaginSP12.pdf
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IDEA: Introduce adjoint variables λ(t) ˆ= ∂J. ∂x. (x(t),t)T. ∈ R nx and get controls from Pontryagin's Maximum Principle (historical name) u∗(x, λ) = arg min u ...
homes.esat.kuleuven.be/~mdiehl/TRONDHEIM/Pontryagin.pdf
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The Pontryagin Maximum Principle: From Necessary ... - Springer
Chapter 2. The Pontryagin Maximum Principle: From Necessary Conditions to the Construction of an Optimal Solution. We now proceed to the study of a ...
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Optimal Control Lectures 25-27: Maximum Principles Variational - LA
Pontryagin Maximum Principle: Statement. Theorem. ... Pontryagin Maximum Principle: Remarks (cont'd) .... Pontryagin Maximum Principle: Extensions (cont'd) ...
la.epfl.ch/files/content/sites/la/files/shared/import/migration/IC_32/Slides25-27.pdf
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Course on Pontryagin's Maximum Principle
May 9, 2006 ... Course on Pontryagin's Maximum Principle. Instructor ... The intent of the course is to introduce students to Pontryagin's Maximum Principle.
www.mast.queensu.ca/~andrew/teaching/MP-course/
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Pontryagin's Maximum Principle in science
Pontryagin's Maximum Principle - San Jose State University
Pontryagin's Maximum Principle applies to a particular type of problem called a Bolzano Problem. Most optimization problems can be put into the form of a ...
9 - Optimization by Pontryagin's Maximum Principle - University ...
A number of items need to be considered before getting into a formal optimization . We must decide what to optimize. Here we shall call the objective function the ...
[PDF]Optimal Control Lectures 25-27: Maximum Principles Variational - LA
Benoıt Chachuat (McMaster University). Maximum Principles. Optimal Control. 2 / 29. Pontryagin Maximum Principle. Motivations: 1. Encompass optimal control ...
$k$-symplectic Pontryagin's Maximum Principle for some families ...
Oct 25, 2012 ... This description allows us to state and prove Pontryagin's Maximum Principle on $k$-symplectic formalism. We also consider the unified ...
[PDF]Pontryagin's Maximum Principle - eolss
Pontryagin's Maximum Principle. Alexander B. Kurzhanski. Faculty of Computational Mathematics and Cybernetics, Moscow State University,. Russia ...
Pontryagin's minimum principle - Wikipedia, the free encyclopedia
Pontryagin's maximum (or minimum) principle is used in optimal control ... The principle states informally that the Hamiltonian must be minimized over \mathcal{ U} .... Applications in Science and Engineering, Cambridge University Press, 2013.
[PDF]Pontryagin's maximum principle - Washington - University of ...
Pontryagin's Maximum Principle. Emo Todorov. Applied Mathematics and Computer Science & Engineering. University of Washington. Winter 2012 ...
[PDF]A proof of Pontryagin's maximum principle
A proof of Pontryagin's Maximum Principle. Björn Bergstrand (bjorn@math.su.se). Stockholm University. We are given a time interval [0,T] and two functions: f(t, x, ...
[PDF]A Discrete Version of Pontryagin's Maximum Principle - jstor
A DISCRETE VERSION OF PONTRYAGIN'S. MAXIMUM PRINCIPLE*. C. L. Hwang and L. T. Fan. Kansas State University, Manhattan, Kansas. (Received ...
Books on the term Pontryagin's Maximum Principle
Optimization techniques, with applications to aerospace systems
Leitmann, 1962
261 7.2 The Adjoint System and the Pontryagin Maximum Principle _ _ _ _ _ __ 262 7.21 The Adjoint System and Stationary Solutions _ _ _ _ . _ _ _ _ _ _ . A _ - _ 262 7.22 A Derivation of the Maximum Principle from the Adjoint System ...
Stochastic Controls: Hamiltonian Systems and Hjb Equations
Jiongmin Young, Xun Yu Zhou, 1999
The maximum principle, formulated and derived by Pontryagin and his group in the 1950s, is truly a milestone of optimal control theory. It states that any optimal control along with the optimal state trajectory must solve the so-called (extended) ...
Pontryagin's maximum principle and optimal control
Stanford University. Division of Engineering Mechanics, I. Flugge-Lotz, H. Halkin, 1961
The mathematical formulation of the most general problem of Optimal Control can be considered as a problem of Mayer subjected to unilateral constraints, i.e., to certain restrictions expressible in terms of inequalities.
Optimal Control of Nonlinear Processes: With Applications in ...
2008
In the following sections we present the central result of Pontryagin's approach to optimal control theory, known as Pontryagin's Maximum Principle and also the Hamilton–Jacobi–Bellman (HJB) equations, proved by Bellman (1957).

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Pontryagin's Maximum Principle
Lecture 4 Maximum Principle
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Optimal Control Strategy for SEIR with Latent Period and a Saturated Incidence Rate
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Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE » DOWNEU
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Malware attacks constitute a serious security risk that threatens our ever-expanding wireless networks. Developing reliable security measures against outbreaks of malware facilitate the proliferation of wireless technologies. The first step toward this goal is to investigate potential attack strategies and the extent of damage they can incur. Given the flexibility that software-based operation provides, it is reasonable to expect that new malware will not demonstrate a fixed behavior over time. Instead, malware can dynamically change the parameters of their infective hosts in response to the dynamics of the network, in order to maximize their overall damage. ^ We first consider propagation of malware in a battery-constrained mobile wireless network by an epidemic model in which the worm can dynamically control the transmission ranges and/or the media scanning rates of the infective nodes. The malware at each infective node may seek to contact more susceptible nodes by amplifying the transmission range and the media scanning rate and thereby accelerate its spread. This may however lead to (a) easier detection of the malware and thus more effective counter-measure by the network, and (b) faster depletion of the battery which may in turn thwart further spread of the infection and/or exploitation of that node. We prove, using Pontryagin Maximum Principle from optimal control theory, that the maximum damage in this case can be attained using simple three-phase strategies: in the first phase, infective nodes use maximum transmission ranges and media access rates to amass infective nodes. In the next phase, infective nodes reduce their access attempts and enter a stealth-mode to preserve their battery and hide from detection. In the last phase, they once again use maximum transmission attempts with largest rates but this time the primary effect is killing the infective nodes by draining their batteries. ^ In an alternative attack scenario, we consider the case in which the malware can control the rate of killing the infective nodes as an independent parameter of control. At each moment of time the worm at each node faces the following decisions: (i) choosing the transmission ranges and media scanning rates so as to maximize the spread of infection subject to not exhausting its batteries by the end of the operation interval; and (ii) whether to kill the node to inflict a large cost on the network, however at the expense of losing the chance of infecting more susceptible nodes at later times. We
repository.upenn.edu/dissertations/AAI3498003/
PONTRYAGIN & CONTROL THEORY | Cambridge Forecast Group Blog
Analyzing globalization, the Middle East & the world-system
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SDE's Mathematical Economics: Pontryagin's Maximum (or Minimum Principle)
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In his paper with the title, What makes for a beautiful problem in Science? (Journal of Political Economy, Vol. 78, No.
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