Talk:Pontryagin's maximum principle

Explicit dependence on t?
The definition of H has an explicit dependence on t. But the right hand side of the definition is in terms of x, u and λ so it seems to me there is no need for explicit dependence on t. Should either (1) the explicit dependence on t be removed from H or (2) an explicit dependence on t be added to L? Sigfpe (talk) 16:34, 26 February 2015 (UTC)

First equation correct?
Should it not be: H(x*,u*,lambda*)<=H(x,u,lambda)

I mean, a different control u~=u* should result in a different trajectory, i.e. x(u)~=x(u*)

//Matthijs —Preceding unsigned comment added by Matthijs (talk • contribs) 08:20, 26 February 2010 (UTC)


 * Yes, this is correct. In every point along the optimal path, the control minimizes the Hamiltonian. Note also that by changing the control in some very small neighborhood, the path does not change much.--LutzL (talk) 10:35, 2 June 2010 (UTC)

The initial section plunges in at the deep end assuming the reader has previous knowledge of the control Hamiltonian (in which case why doesnt he or she already know about the maximum principle?) This material is then repeated below in 'formal statement' which does set out the problem. So why not just keep this and delete the initial unintelligble version? I suggest that this is done.JFB80 (talk) 10:28, 2 October 2010 (UTC)

The introduction also discusses that the maximum is convention in standard Hamiltonian problems (i.e. from physics). It then goes on to state the minimum principle. Shouldn't the inequality be reversed to be consistent with the introductory paragraphs? $ynoptik_m4yh3m (talk) 13:08, 7 April 2018 (UTC)

PMP in 2-d
Is there any information anywhere about the Pontryagin maximum principle with two independent variables, e.g. when the problem to be solved is the maximisation of a double integral? Can the PMP be easily extended to two dimensions?

--Rich P —Preceding unsigned comment added by 81.104.47.24 (talk) 19:52, 12 September 2007 (UTC)

Sure. Most books on optimal control theory will cover multivariable problems. My favorite is Stengel's 'Optimal control and estimation' —Preceding unsigned comment added by 129.49.67.32 (talk) 20:36, 27 September 2007 (UTC)

Redirect needed
The PMP is also known as Pontryagin's Maximum Principle. I think we need one article named after that and re-direct it to here. --anon
 * Done, Pontryagin's maximum principle. Note that here we don't use capitals in the middle of sentence. Oleg Alexandrov (talk) 18:51, 15 November 2005 (UTC)


 * BUT IT SHOULD BE MAXIMUM PRINCIPLE. In the early days it was always called this (check the quoted bibliography of Fuller 1963).  The article is incorrect in saying that the original work applied to the minimization of a cost function. This version appeared later. The earliest application was to optimal programming of rocket thrust to achieve maximum terminal velocity.JFB80 (talk) 18:12, 12 January 2011 (UTC)

Wrong Name

 * It should indeed be MAXIMUM PRINCIPLE with a redirect from MINIMUM principle and not vice-versa. I second JFB80's comment above. — Preceding unsigned comment added by 71.197.245.110 (talk) 20:47, 2 March 2012 (UTC)
 * Does anyone know how to change the heading from minimum principle to maximum principle? I am afraid I dont.JFB80 (talk) 21:24, 3 June 2013 (UTC)


 * I second this. Google says 4:1 to Pontryagin's Maximum Principle, and that is with Wikipedia possibly diluting the results. Maximizing or minimizing is the same problem anyway, and wiki should refer to things by what they are commonly called and not try to reinterpret it. If anybody knows how to fix this, please do it. 2001:6B0:17:F026:D95E:21B7:8ED8:C9F6 (talk) 10:52, 14 April 2015 (UTC)

@JFB80 Done. Lbertolotti (talk) 14:03, 7 September 2015 (UTC)

, – Unfortunately the move was not done properly: the page history is still associated with the redirect. See WP:CUTPASTE. I will add a histmerge note to get an admin to look at it. -- JohnBlackburne wordsdeeds 02:26, 8 September 2015 (UTC)

Merge Hamiltonian (control theory) and Pontryagin's minimum principle
Please discuss it here - Nmnogueira 16:10, 6 June 2007 (UTC)

This principle is sufficiently distinct that it should be separated from Hamiltonian (Control Theory). Moreover, when it comes to optimal control theory, we give the credit only to Hamiltonian (after his name) and this way contribution of the mathematician Pontryagin is overlooked. So, I strongly recommend Pontryagin's principles to be redirected on a separate page.

Formal statement of necessary conditions for minimization problem
Conditions (3) and (4): Notation on left-hand side of the equals sign should not include the transpose operator.

Corrected conditions (3) and (4) should read:



(3) \qquad -\dot{\lambda}(t)=H_x(x^*(t),u^*(t),\lambda(t),t)=\lambda'(t)f_x(x^*(t),u^*(t))+L_x(x^*(t),u^*(t)) $$



(4) \qquad \lambda(T)=\Psi_x(x(T)) \, $$

(Here ends that contribution, which was not signed, and I have no idea who wrote it.)

Everywhere vs almost everywhere. So, I have another question: the statement in the article says that the maximum principle should hold for all times t, but in the literature I'm only able to find proofs for it holding for almost every time t, and there is considerable difference between these two cases. I hope maybe an expert can clarify which one is right, because I do not know if there is some actual counterexample (or a proof) for the everywhere case. Thanks. Zatrp (talk) 08:06, 10 May 2017 (UTC)