en>Tedsanders: Reworded awkward sentence

2014-01-04T22:05:56Z

Reworded awkward sentence

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	I'~~m Luther~~ but ~~I never truly favored~~ that ~~name~~. ~~Delaware has always been my residing place and will by no means move. What she loves~~ ~~extended~~ [~~http~~://~~jamsuham.net~~/xe/~~joinqna~~/~~670399 auto warranty~~] ~~performing~~ [~~http~~:/~~/sticker~~-~~shocked.com~~/~~UserProfile~~/~~tabid~~/43/~~userId~~/~~404~~/~~Default~~.~~aspx extended car warranty]~~ is ~~bottle tops collecting and she is attempting to make it~~ a ~~profession~~. ~~She works~~ as a ~~[http~~://~~Www.Consumerreports.org~~/~~cro~~/~~magazine~~/~~2014~~/~~04/extended-warranties-~~for~~-cars-~~are-~~an-expensive-game~~/ ~~monetary officer~~] and ~~she will not change it whenever soon~~.<br><br>~~Feel free to surf auto warranty to my blog~~ [http://~~Auto~~.~~Howstuffworks~~.~~com~~/~~buying-selling~~/~~cg-extended-warranty-tips~~.~~htm extended~~] ~~car warranty post :: extended auto warranty (~~[http://~~der~~-~~erste~~-~~clan.de~~/~~index.php?mod=users&action=view&id=11094 click over here now~~])		The '''Hamiltonian''' of [[Optimal control\|optimal control theory]] was developed by [[Lev Semyonovich Pontryagin\|L. S. Pontryagin]] as part of his [[Pontryagin's minimum principle\|minimum principle]].<ref>I. M. Ross ''A Primer on Pontryagin's Principle in Optimal Control'', Collegiate Publishers, 2009.</ref> It was inspired by, but is distinct from, the [[Hamiltonian mechanics\|Hamiltonian]] of classical mechanics. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to minimize the Hamiltonian. For details see [[Pontryagin's minimum principle]].

			== Notation and Problem statement ==

			A control <math>u(t)</math> is to be chosen so as to minimize the objective function

			:<math>
			J(u)=\Psi(x(T))+\int^T_0 L(x,u,t) dt
			</math>

			where <math>x(t)</math> is the system state, which evolves according to the state equations

			:<math>
			\dot{x}=f(x,u,t) \qquad x(0)=x_0 \quad t \in [0,T]
			</math>

			and the control must satisfy the constraints

			:<math>
			a \le u(t) \le b \quad t \in [0,T]
			</math>

			== Definition of the Hamiltonian ==

			: <math>
			H(x,\lambda,u,t)=\lambda^T(t)f(x,u,t)+L(x,u,t) \,
			</math>

			where <math>\lambda(t)</math> is a vector of [[Costate equations\|costate]] variables of the same dimension as the state variables <math>x(t)</math>.

			For information on the properties of the Hamiltonian, see [[Pontryagin's minimum principle]].

			== The Hamiltonian in discrete time ==

			When the problem is formulated in discrete time, the Hamiltonian is defined as:

			:<math>
			H(x,\lambda,u,t)=\lambda^T(t+1)f(x,u,t)+L(x,u,t) \,
			</math>

			and the [[costate equations]] are

			:<math>
			\lambda(t)=-\frac{\partial H}{\partial x}
			</math>
			(Note that the discrete time Hamiltonian at time <math>t</math> involves the costate variable at time <math>t+1.</math><ref>Varaiya, Chapter 6</ref> This small detail is essential so that when we differentiate with respect to <math>x</math> we get a term involving <math>\lambda(t+1)</math> on the right hand side of the costate equations. Using a wrong convention here can lead to incorrect results, i.e. a costate equation which is not a backwards difference equation).

			==The Hamiltonian of control compared to the Hamiltonian of mechanics==

			[[William Rowan Hamilton]] defined the [[Hamiltonian mechanics\|Hamiltonian]] as a function of three variables:

			:<math>\mathcal{H} = \mathcal{H}(p,q,t) = \langle p,\dot{q} \rangle -L(q,\dot{q},t)</math>

			where <math>\dot{q}</math> is defined implicitly by

			:<math>p = \frac{\partial L}{\partial \dot{q}}</math>

			Hamilton then formulated his equations as

			:<math>\frac{ d}{ dt}p(t) = -\frac{\partial}{\partial q}\mathcal{H}</math>
			:<math>\frac{ d}{ dt}q(t) =~~\frac{\partial}{\partial p}\mathcal{H}</math>

			In contrast the Hamiltonian of control theory (as defined by [[Pontryagin]]) is a function of 4 variables

			:<math>H(q,u,p,t)= \langle p,u \rangle -L(q,u,t)</math>

			and the associated conditions for a maximum are

			:<math>\frac{dp}{dt} = -\frac{\partial H}{\partial q}</math>

			:<math>\frac{dq}{dt} = ~~\frac{\partial H}{\partial p}</math>

			:<math>\frac{\partial H}{\partial u} = 0</math>

			This difference is somewhat confusing, nevertheless a specific problem, such as the [[Brachystochrone]] problem, can be solved by either method. For details, see the article by Sussmann and Willems.<ref>[https://netfiles.uiuc.edu/liberzon/www/teaching/sussmann-willems.pdf Sussmann, Willems: 300 Years of Optimal Control, IEEE Control Systems, June 1997]</ref>

			==References==

			{{reflist}}

			==External links==
			* P. Varaiya: ''Lecture Notes on Optimization'', 2d. ed. (1998) [http://paleale.eecs.berkeley.edu/~varaiya/papers_ps.dir/NOO.pdf]

			* [[I. Michael Ross\|I. M. Ross]], Pontryagin's Hamiltonian Illustrated with Examples, 2009, [http://www.elissarglobal.com/home/get-chapter-2-free/ Chapter 2 download]

			{{DEFAULTSORT:Hamiltonian (Control Theory)}}
			[[Category:Optimal control]]

en>Newwhist: /* Number of possible deals */ improve math presentation

2012-07-10T20:59:57Z

Number of possible deals: improve math presentation

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en>Tedsanders: Reworded awkward sentence

en>Newwhist: /* Number of possible deals */ improve math presentation