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Abstract and Applied Analysis, Volume 2011, Article ID 545264, 19 Pages, 2011

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Abstract and Applied Analysis, Volume 2011, Article ID 545264, 19 Pages, 2011 Nikolai Azbelev −−− the Giant of Causal Mathematics Efim A. Galperin Universite du Quebec a Montreal, Canada One of the greatest mathematicians of all times, Nikolai Viktorovich Azbelev has died. He is one of those great scholars who rise to lasting prominence after leaving this world of fast market values in science. He left immortal ideas that are changing mathematics. I met him 20 years ago at the first world Congress of Nonlinear Analysts (1992) in Tampa, USA, and later in Ariel and Athens. We talked a lot about differential equations with deviating arguments, the branch of mathematics not duly recognized at the time and, I am afraid, not clearly understood right now. The classical theories of ODEs and PDEs well presented in all textbooks, monographs and high level articles in mathematics are physically invalid, and Nikolai Azbelev felt it strongly although he could not tell it from the podium of a congress. Even now, it is not common to talk about it. To put things straight, consider the 2nd law of Newton usually formulated in textbooks as follows: ma = mx’’ = F(t, x, v), v = x’ = lim[x(t+ dt) – x(t)]/dt as dt→ 0. Here the mass m is presumed to be constant. For m ≠ const, Georg Buquoy proposed (1812) another formula: mdv + (v – w)dm = F(t, x, v)dt where v – w is the relative velocity with which dm is ejected from a moving body. Both formulas are however non-causal, thus physically invalid, since at any current moment t, the value x(t + dt), dt > 0, does not exist and cannot be known (measured) at a future moment t + dt > t not yet realized. The Newton-Leibnitz right time and partial derivatives produce non-causal equations of rigid frozen evolutions as if the future values x(t + dt) were actually known in the real physical processes. But it is the consideration of the left time derivatives v* = lim[x(t) – x(t− dt)]/dt as dt→ 0, or delayed arguments in the right-hand side as v = x’(t − δ1), v’ = x’’(t − δ2), etc., with δi > 0, dt < δi all i (Azbelev) that presents the causal ODEs and PDEs as valid descriptions of the realistic physical processes that allow us to use controls dependent on the higher order left or delayed derivatives in the right-hand side F(t, x, v, v’, v’’, v’’’, ...) in order to control the motion and effectively alleviate actual disturbances and uncertainties always present in nature. For example, the autopilot control systems for airplanes are now constructed without the use of acceleration assisted control, according to the current textbook formula for the 2nd law of Newton. Such autopilots are dangerous if applied at take-off, at landing, or in bad weather, in which cases the pilot has to take controls since he feels the acceleration and sudden changes of velocity of the plane even if the Pitot tubes fail in flight as already happened on May 31, 2009 in the Airbus A330 flight between Rio de Janeiro and Paris. It is these ideas of causality and their implementation in mathematics and engineering that present the everlasting legacy of Nikolai Viktorovich Azbelev for further progress in science and technology. Functional Differential Equations and Applications 2012 On LQR and control of structures using canonical representations G. Agranovich Department of Electrical and Electronics Engineering I. Halperin Department of Electrical and Electronics Engineering Department of Civil Engineering Y. Ribakov Department of Civil Engineering Faculty of Engineering, Ariel University Center of Samaria, Ariel, 40700, Israel AMS Subject Classi cation: 93C05, 93C15, 93C95, 49K15, 70E55 Keywords and Phrases: modal control, LQR, optimal structural control, algebraic Ric- catis equation. According to the seismic modal approach, optimal modal control design should derive a control signal a¤ecting only the dominant modal coordinates. However, existence of an optimal state feedback, a¤ecting only a selected set of modes, is questionable. This study proves the existence of optimal modal feedback for the case of in nite horizon LQR and control methods. It also formulates a modal design method, based on a corresponding se- lection of LQR weighting and output matrices. New state-space similarity transformations are introduced. E¢ciency of the theoretical results is demonstrated in a numerical example, presenting a modal design of a typical 4 story residential building Functional Differential Equations and Applications 2012 A binomial identity via di¤erential equations D. Aharonov and U. Elias Technion, Haifa, Israel In the following we discuss a well known binomial identity. Many proofs by di¤erent methods are known for this identity. Here we present another proof which uses linear ordinary di¤erential equations of the rst order. Functional Differential Equations and Applications 2012 A Numerical method to solve the axisymmetric static Maxwell equations in singular domains F. Assous Ariel University Center, 40700 Ariel, Israel. I. Raichik Bar Ilan University, 52900, Ramat Gan, Israel. We propose a new numerical method to solve the axisymmetric static Maxwell equations in singular domains, as for example a non convex polygonal domain ! belonging to the meridian half-space (r; z). In these conditions, computing for instance the static magnetic eld B = (Br;Bz) consists in nding the divergence-free solution to @Br @Bz curlB := = f : @z @r together with an ad hoc boundary condition. This problem is singular in the sense that for a non-convex axisymmetric domain !, the space of solutions (says W ) is not a subspace of the Sobolev space H1. Nevertheless, W can be decomposed into two subspaces, i.e. 1 W = WR WS, where WR is a regular subspace, that is a subspace of H in which one can easily compute a numerical solution. The di¢culty comes from the singular subspace WS, that is a nite-dimensional subspace, the dimension depending on the number of reentrant corners of the domain !. Moreover, a basis function wS of WS can be characterized as the solution to curl wS = PS 2 !; div wS = 0 2 !; wS = 0 2 @! : 2 1 Here, the right-hand side PS is singular, that it belongs to L but not to H , and this solves 2 2 @ PS @ PS 1 @PS PS PS := + + = 0 in !; 0 @r2 @z2 r @r r2 PS = 0 on @! ; Hence, the key point is to compute PS, which can not be solved by a standard nite element method, which would give PS = 0. In this talk, we propose a new method to e¢ciently compute PS and consequently wS. It consists in decomposing the domain ! into 2 subdo- mains, and to derive an ad hoc variational formulation, in which the interface conditions are imposed through a method deduced from a Nitsche approach. Examples to illustrate our method will be shown in the talk. Functional Differential Equations and Applications 2012 On estimates of alternating-sign solutions to nonlinear di¤erential equations I. Astashova Moscow State University of Economics, Statistics and Informatics, Moscow Lomonosov State University ast@di¢ety.ac.ru AMS Subject Classi cation: 34C10, 34C11 Keywords and Phrases: nonlinear ordinary di¤erential equation of higher order, uniform estimates of solutions. Uniform estimates and qualitative behavior of solutions to quasi-linear ordinary di¤er- ential equations of the higher order are described. In particular, to the equation n 1 (n) (j) k y + X aj(x) y + p(x) jyj sgn y = 0 (1) j=0 with n 1; real (not necessary natural) k > 1; and continuous functions p(x) and aj(x); uniform estimates for positive solutions with the same domain ([1]) are obtained. For alternating-sign solutions the uniform estimates are obtained to the equation n 1 (n) (j) k y + X aj(x) y + p(x) jyj = 0 (2) j=0 and for some special cases of equation (1). References [1] Astashova, I.V. Uniform estimates for positive solutions of quasi-linear ordinary di¤er- ential equations, (English. Russian original) Izv. Math. 72, No. 6, (2008), 11411160; translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 6, (2008), 85104. Functional Differential Equations and Applications 2012 Oscillation criterion for one class of discrete equations J. Batinec, J. Diblík Brno University of Technology,Brno Czech Republic, [email protected], [email protected] AMS Subject Classi cation: 39A10, 39A11. Keywords and Phrases: Discrete delayed equation, oscillating solution, positive solution, asymptotic behavior. We consider the delayed (k + 1)-order linear discrete equation x(n) = p(n)x(n k) (1) where n 2 Za1 := fa; a + 1;::: g, a 2 N := f1; 2;::: g is xed, x(n) = x(n + 1) x(n), p: Za1 ! R, k 2 N. A solution x = x(n): Za1 ! R of (1) is positive (negative) on Za1 if x(n) > 0 (x(n) < 0) for every n 2 Za1. A solution x = x(n): Za1 ! R of (1) is oscillating on Za1 if it is not positive or negative Z a1 Z on a11 for an arbitrary 2 a1. Let us de ne the expression lnq t, q 1, by lnq t = ln(lnq 1 t), ln0 t t. The equation (1) is known to have a positive solution if the sequence p(n) satis es an inequality. Our aim is to show that, in the case of the opposite inequality for p(n), k k 1 k k k p(n) + + + + ; k + 1 k + 1 8n2 8(n ln n)2 8(n ln n : : : ln n)2 q assuming > 1, all solutions of the equation (1) are oscillating for n ! 1. Acknowledgement: This research was supported by the grants P201/10/1032 and P201/11/0768 of the Czech Grant Agency (Prague) and by the project FEKT-S-11-2(921) Brno University of Technology.
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