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[[File:Block scheme of control system.jpg|thumb|
Fig. 1. Block scheme of control system.  ''G''(''s'') – linear transfer function, ''f''(''e'') – single-valued, continuous, differentiable function]]


'''Kalman's conjecture''' or '''Kalman problem''' is a [[conjecture]] on absolute stability of
[[nonlinear control]] system  with one scalar nonlinearity, which belongs to the sector of linear stability.  Kalman's conjecture is a strengthening of [[Aizerman's conjecture]]
and is a special case of [[Markus–Yamabe conjecture]].  This conjecture was proven false but led to the (valid) [[Nonlinear_control#Absolute_stability_problem|sufficient criteria on absolute stability]].


== Mathematical statement of Kalman's conjecture (Kalman problem)==
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In 1957 [[R. E. Kalman]] in his paper
<ref name=1957-Kalman>{{cite journal
| author = Kalman R.E.
| year = 1957
| title = Physical and Mathematical mechanisms of instability in nonlinear automatic control systems
| journal = Transactions of ASME
| volume = 79
| number = 3
| pages = 553&ndash;566
}}
</ref> stated the following:
<blockquote>
If ''f''(''e'') in Fig. 1 is replaced by constants ''K'' corresponding to all possible values of ''f''<nowiki>'</nowiki>(''e''), and it is found that the closed-loop system is stable for all such ''K'', then it intuitively clear that the system must be monostable; i.e., all transient solutions will converge to a unique, stable critical point.
</blockquote>
 
Kalman's statement can be reformulated in the following conjecture:<ref name=2011-DAN-Kalman>{{cite journal
| author = Leonov G.A., Kuznetsov N.V.
| year = 2011
| title = Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems
| journal = Doklady Mathematics
| volume = 84
| number = 1
| pages = 475&ndash;481
| url = http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf
| doi = 10.1134/S1064562411040120}}
</ref>
<blockquote>
Consider a system with one scalar nonlinearity''
:<math>
  \frac{dx}{dt}=Px+qf(e),\quad e=r^*x \quad x\in R^n,
</math>
where ''P'' is a constant ''n''×''n'' matrix, ''q'', ''r'' are constant ''n''-dimensional vectors, ∗ is an operation of transposition, ''f''(''e'') is scalar function, and ''f''(0)&nbsp;=&nbsp;0. Suppose, ''f''(''e'') is a differentiable function and the following condition''
:<math>
k_1 < f'(e)< k_2. \,
</math>
is valid. Then Kalman's conjecture is that the system is stable in large (i.e. unique stationary point is global [[attractor]]) if all linear systems with ''f''(''e'')&nbsp;=&nbsp;''ke'', ''k''&nbsp;∈&nbsp;(''k''<sub>1</sub>,&nbsp;''k''<sub>2</sub>) are asymptotically stable.
</blockquote>
 
In [[Aizerman's conjecture]] in place of condition on the derivative of nonlinearity it is required that the nonlinearity itself belongs to linear sector.
 
Kalman's conjecture is true for ''n''&nbsp;≤&nbsp;3 and for ''n''&nbsp;>&nbsp;3 there are effective methods for construction of counterexamples:<ref>{{cite journal
| author = Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A.  
| year = 2011
| title = Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits
| journal = Journal of Computer and Systems Sciences International
| volume = 50
| number = 5
| pages = 511&ndash;543
| url = http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf
| doi = 10.1134/S106423071104006X}}
</ref><ref name=2011-IJBC-Hidden-attractors>{{cite journal |
author = Leonov G.A., Kuznetsov N.V. |
year = 2013 |
title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits |
journal = International Journal of Bifurcation and Chaos |
volume = 23 |
issue = 1 |
pages = art. no. 1330002|
url = http://www.worldscientific.com/doi/pdf/10.1142/S0218127413300024|
doi = 10.1142/S0218127413300024}}
</ref> nonlinearity derivative belongs to the sector of linear stability, and unique stable equilibrium coexists with a stable periodic solution ([[hidden oscillation]]).
 
== References ==
{{Reflist}}
 
== Further reading ==
*{{cite journal
| author = Leonov G.A., Kuznetsov N.V.
| year = 2011
| title = Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems
| journal = IFAC Proceedings Volumes (IFAC-PapersOnline)
| volume = 18
| number = 1
| pages = 2494&ndash;2505
| url = http://www.math.spbu.ru/user/nk/PDF/2011-IFAC-Hidden-oscillations-control-systems-Aizerman-problem-Kalman.pdf
| doi = 10.3182/20110828-6-IT-1002.03315}}
 
== External links ==
* [http://www.math.spbu.ru/user/nk/PDF/Harmonic_balance_Absolute_stability.pdf Analytical-numerical localization of hidden oscillation in counterexamples to Aizerman's and Kalman's conjectures]
 
[[Category:Conjectures]]
[[Category:Nonlinear control]]

Latest revision as of 08:42, 9 January 2015


If you're missing teeth or would be smart to have one or more teeth removed, you could be thinking about finding the simplest way to replace them. Old-fashioned bridgework and dentures usually come to mind, however they are not the only option.

Choose a dentist who you trust and feel more comfortable with. Speaking to other patients can be a way to enable you to decide whether a dentist is actually going to right a person personally.

A third option is microbrasion. The dentist grinds a thin layer there's lots of surface of one's teeth. Method works good to enamel defects like white or liver spots from fluorosis (a harmless condition that results from swallowing a good deal fluoride while your teeth are still developing). The surface spots are removed, revealing a smooth layer of enamel. Fee of this program is up to $150 per tooth.

The dentist will review your records and ask you some questions relating to your dental practices. The dentist might take some x-rays of your mouth in order to get a good examine the associated with your mouth and the way your teeth have formed. That they see any issues they'll discuss ways to handle those. The dentist in most cases do an in-depth cleaning of the teeth and take away any plaque that they find. Incredibly help give you a new beginning toward healthy teeth. Your young ones will also likely get x-rays from the dentist making sure that they possess a point to start from merely because they watch pearly white's grow. Include be included on the child's chart.

If specific niche market whiter teeth in a hurry and the buying dentist is not only factor, see a professional. Your teeth could look fifteen times whiter thanks in order to bleaching agent and an important light. Be ready to spend $ 500 for each session.

A mix of baking plenty hydrogen peroxide can be employed as a toothpaste. Gently brush you teeth this kind of mixture on the regular basis so in order to will capability to see whiter teeth without needing to use any harsh chemical substance. If you loved this article and you would certainly such as to get more information relating to Tampa dental Clinic kindly visit our internet site. Be careful, though, to keep from swallowing the item. If you swallow the mixture by accident, you have to have drink several glasses of water immediately following that.

OEating meals such as vegetables, cheese, plain yogurt, and fruits rich in Vitamin Celsius. Also, avoid consuming sugary foods as they quite simply would only increase bacteria and cavity build-up.