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.
Functional Differential Equations and Applications 2012 On global stability for some nonlinear functional di¤erential systems
Leonid Berezansky
Ben Gurion University of the Negev Beer-Sheva, Israel
2000 MSC: 34K20
For the vector functional di¤erential equatuion dx A t x t F x t : dt = ( ) ( ) + ( )( ) (1) where F is a nonlinear causal (Volterra) operator we discuss some new global stability results. There results are applied for nonautonomous Mackey-Glass models.
Functional Differential Equations and Applications 2012 Integro-di¤erential equations in high energy scattering
S.Bondarenko
Department of Physics, Ariel University Center, Ariel, Israel
We will talk about the special class of integro-di¤erential equations arising in high-energy physics. These equations we can de ne as a generalization of Lotka-Volterra equations at the case of non-local interactions between the functions when two-value boundary conditions are given. We discuss numerical solutions of our equations for some special case and properties of the solutions on an example of oversimpli ed model of the same equations in zero transverse dimensions.
Functional Differential Equations and Applications 2012 On the best constants in the solvability conditions of the periodic problem
Eugene Bravyi
Perm State Technical University, Perm, Russia
AMS Subject Classi cation: 34K06, 34K10. Keywords and Phrases: Functional di¤erential equations, boundary value problems.
In 1995 A. Lomtatidze and S. Mukhigulashvili obtained the following result: the periodic boundary value problem
x (t) = (T x)(t) + f(t); t 2 [0;!]; x(0) = x(!); x_(0) =x _(!); has a unique solution for all f 2 L[0;!] and for all linear bounded positive operators T : C[0;!] ! L[0;!] with a given norm kT k = T > 0 i¤ T 16=!. Here we consider the periodic problem for a perturbed (a > 0) equation with negative feedback:
x (t) + a2x(t) = (T x)(t) + f(t); t 2 [0;!]; x(0) = x(!); x_(0) =x _(!):
This problem has a unique solution for all f 2 L[0;!] and for all linear bounded positive operators T : C[0;!] ! L[0;!] with a given norm kT k = T i¤
a! a! 16 4 cot( 4 ) for 0 < a! < 2; T a! a! N ! ( 8 jsin( 2 )j for 2 < a! 6= 2k; k 2 :
Supported by Grant 10-01-96054-r-ural-a of The Russian Foundation for Basic Research
Functional Differential Equations and Applications 2012 On mathematical model for the treatment of chronic myelogenous (myeloid) leukemia (CML)
Svetlana Bunimovich
Ariel University Center of Samaria, Ariel, Israel
We propose and analyze a mathematical model for the treatment of chronic myelogenous (myeloid) leukemia (CML), a cancer of the blood. We introduce combined treatment of CML based on Imatinib therapy and Immunotherapy. Imatinib therapy is a molecular targeted therapy that inhibits the cell, involved in the chronic CML pathogenesis. Immunotherapy based on interferon alfa-2a e¤ects the cancer cells mortality and leads to improvement out- come of the combined therapy. The system of di¤erential equations was used to model the interaction between CML cancer cells and e¤ector cells of the immune system in the human body. We introduced biologically motivated time-varying delays in the treatment terms. The proposed model belongs to a special class of nonlinear nonautonomous systems of ordinary di¤erential equations (ODEs). The analysis of the described system shows the existence of a unique global positive solution, existence of a unique nontrivial equilibrium, explicit local and global stability conditions for the nontrivial equilibrium.
Functional Differential Equations and Applications 2012 Multiplicity results for the periodic Hills equation
Alberto Cabada
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain. [email protected]
AMS Subject Classi cation: 34B15, 34B16, 34B27. Keywords and Phrases: Hills equation; periodic boundary value problem; second order singular equation.
In this talk we present some existence and multiplicity results for the periodic boundary value problem
x00(t) + a(t) x(t) = g(t) f(x) + c(t); x(0) = x(T ); x0(0) = x0(T ); where is a positive parameter. The function f : (0; 1) ! (0; 1) is allowed to be singular at x = 0 and the related Greens function does not change its sign. Moreover we will make a survey on the known results that imply that the Greens function is nonnegative on its square of de nition.
This work was partially supported by Ministerio de Educación y Ciencia, Spain, project MTM2010-15314.
References
[1] A. Cabada and J. A. Cid, Existence and Multiplicity of Solutions for a Periodic Hills Equation with Parametric Dependence and Singularities. Abstract and Applied Analysis, Volume 2011, Article ID 545264, 19 pages, 2011.
Functional Differential Equations and Applications 2012 pLaplacian discrete boundary value problems on bounded and unbounded intervals
Alberto Cabada Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain. [email protected]
AMS Subject Classi cation: 39A10, 39A12, 47J30, 58E05 Keywords and Phrases: Di¤erence equations, Discrete pLaplacian, Variational methods, Heteroclinic solutions. In this talk we present some applications of the critical point theory to deduce the existence of solutions for certain discrete problems. We will speak about the existence of some > 0 for which there are nontrivial solutions of the non autonomous equation px(k 1) = f(k; x(k)); (1) coupled with one of the following boundary value conditions x(0) = x(T + 1) = 0; (2) or x(0) = 0; x(1) = 1: (3) Here p > 1 is a given real number, and p 2 px(k 1) = jx(k 1)j x(k 1) 'p(x(k 1)); is the classical p Laplacian operator. Concerning the problem (1) (2), under suitable assumptions on the function f, in [1] it is proved the existence of a positive for which this problem admits at least three solutions. The proof follows from critical point theory. Problem (1) (3) has been considered in [2]. In this case it is showed that, by using some sign assumptions in function f, there is a positive solution for all > 0. The solution is attained as the limit of a sequence of solutions of related homogeneous Dirichlet problems in bounded intervals.
References
[1] A. Cabada, A. Iannizzotto and S. Tersian, Multiple solutions for discrete boundary value prob- lems, J. Math. Anal. Appl. 356 (2009), 418428. [2] A. Cabada and S. Tersian, Existence of heteroclinic solutions for discrete p laplacian problems with a parameter, Nonlinear Anal. Real World Appl. (2011). doi:10.1016/j.nonrwa.2011.02.022 This work was partially supported by Ministerio de Educación y Ciencia, Spain, project MTM2010- 15314.
Functional Differential Equations and Applications 2012 Smooth solutions of some linear functional di¤erential equations
Valery Cherepennikov
Melentiev Energy System Institute of Sib. Dep. RAS. Irkutsk. Russia
AMS Subject Classi cation: 34K06, 34K10.
The paper considers initial and boundary value problems for the following scalar func- tional di¤erential equations: x_(t) = a(t)x(t ) + f(t); 0 const; x_(t) + p(t)_x(t 1) = a(t)x(t 1) + b(t)x(t) + f(t); x_(t) = a(t)x(t (1 "t)) + b(t)x(t) + f(t); x_(t) = a(t)x(t 1) + b(t)x(t + 1) + d(t)x(t) + f(t); x_(t) = a(t)x(t 1) + b(t)x(t=s) + d(t)x(t) + f(t); s > 1: Here t 2 R, the coe¢cients of equations are represented in the form of polynomials. One can investigate other equations as well. Smooth solutions to these problems are studied by the method of polynomial quasiso- lutions (PQ-solutions) [1, 2]. The method is based on the representation of the unknown N n function in the form of polynomial x(t) = Pn=1 xnt , whose substitution into each of the N equations results in the residuals (t)s = O(t ). The paper investigates the problems of existence of PQ-solutions of di¤erent degrees, an algorithm for nding unknown coe¢cients xn as well as exact formulas for the residuals, which allow one to characterize the measure of disturbance for the considered problems. The results obtained are illustrated by examples.
References
[1] V.B. Cherepennikov. Analytic solutions of some functional di¤erential equations linear systems, Nonlinea Analysis, Theory, Methods & Applications. 30/5, (1997), 2641-2651. [2] V.B. Cherepennikov, P.G. Ermolaeva, Polynomial quasisolutions of linear di¤erential dif- ference equations, Opuscula Mathematica, 26/3, AGH Univ. of Science and Technology, Kracow, (2006), 47-57.
Functional Differential Equations and Applications 2012 Classical solutions on a cylindrical domain to quasilinear hyperbolic functional di¤erential equations
W. Czernous
Institute of Mathematics, University of Gdansk, Gdansk, Poland
AMS Subject Classi cation: 35R10, 35L45. Keywords and Phrases: partial functional di¤erential equations, classical solutions, local existence, bicharacteristics, cylindrical domain, well-posedness.
We break the tradition of setting the domain as a Cartesian product of real intervals, and we give a new set of conditions on the possibly unbounded domain with Lipschitz di¤erentiable boundary. Well-posedness is then relying on a variant of normal vector condition. Consequently, negative invariance of a neighbourhood of is shown, which enables us to use the method of bicharacteristics. With local assumptions on equation coe¢cients, we prove local existence and continuous dependence on data of classical solutions on initial boundary value problem. Regularity of solutions matches this of the domain, and the proof uses the Banach xed- point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-di¤erential equations. Nevertheless, presented existence result is new also in the case without functional dependence.
Functional Differential Equations and Applications 2012 Oscillation criterion for one class of discrete equations
J. Diblík, M. R°uµziµcková, Z. utá
Zilinaµ University, Zilina,µ Slovak Republic,
AMS Subject Classi cation: 39A10, 39A11. Keywords and Phrases: Discrete equation, delay, asymptotical convergence, increasing solution.
A system of s discrete equations
y(n) = (n)[y(n j) y(n k)] is considered where k and j are integers, k > j 0, (n) is a real s s square matrix de ned for n n0 k, n0 2 Z with non-negative elements ij(n), i; j = 1; : : : ; s such s T s that Pj=1 ij(n) > 0, y = (y1; y2; : : : ; ys) : fn0 k; n0 k + 1;::: g ! R and y(n) = y(n + 1) y(n) for n n0. A method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically convergent under some conditions, i.e., for every solution y(n) de ned for all su¢ciently large n, there exists a nite limit limn y(n). Moreover, the asymptotic convergence of all solutions is equivalent to the existence!1 of one asymptotically convergent solution with increasing coordinates. A relation to the so-called critical case known for scalar equations will be discussed as well. Acknowledgement: This research was supported by the Grant No 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).
Functional Differential Equations and Applications 2012 Maximum principles and stability of delay di¤erential equations
Alexander Domoshnitsky
Ariel University Center, Ariel, Israel, [email protected]
In this talk we discuss maximum principles for systems of functional di¤erential equa- tions. A connection of maximum principles with nonoscillation and positivity of the Cauchy functions is demonstrated. The method to compare only one component of the solution vector of linear functional di¤erential systems, which does not require heavy sign restric- tions on their coe¢cients, is proposed. Necessary and su¢cient conditions of the positivity of elements in a corresponding row of the Cauchy and Greens matrices are obtained in the form of theorems about di¤erential inequalities. Tests of the exponential stability of functional di¤erential systems are obtained on this basis. The main idea of our approach is to construct a rst order functional di¤erential equation for one of the components of the solution vector and then to use assertions about positivity of its Greens functions.
Functional Differential Equations and Applications 2012 Stability of equilibrium of di¤erential systems with quadratic right-hand side
Irada Dzhalladova
Kyiv National Economical University, Ukraine [email protected]
AMS Subject Classi cation: 93D05, 34D05 Keywords and Phrases: Di¤erence Lotka-Volterra equations, stability analysis, di¤erence systems with quadratic right-hand side, stability estimates.
Nonlinear systems of di¤erence Lotka-Volterra equations with quadratic right-hand non- linearity and asymptotically stable linear part are considered. One of the basic methods for stability analysis of trivial solution of nonlinear systems are methods of linearization and stability studying based on stability results for linear approximation system. If the trivial solution of linear approximation system is asymptotically stable, then trivial solution of the initial nonlinear system will be also stable in a su¢ciently small neighborhood of the equi- librium. Di¤erence systems with the quadratic right-hand side are considered in the report. The systems are presented in the uniform vector-matrix form. An algorithm of estimation of stability region in the phase space of trivial equilibrium of the system is proposed.
Functional Differential Equations and Applications 2012 Time evolution of spin exchange with a time delay
D. Gamliel, Department of Medical Physics, Ariel University Center of Samaria, Israel,
A. Domoshnitsky, Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel
R. Shklyar Department of Computer Science and Mathematics, Ariel University Center of Samaria, Israel
AMS Subject Classi cation: 34K06,34C26 Keywords and Phrases: Magnetic resonance, delay di¤erential equations, Lambert func- tion, Cauchy matrix
The NMR (nuclear magnetic resonance) spectrum of a spin system is a¤ected by exchange processes in the system, and thus enables investigation of the exchange process. In a previous work we considered a generalization of spin exchange in which the jump process takes a non- negligible time, so the system is described by di¤erential equations with a time delay. In this paper some characteristics of the solutions are studied, rst by using the Lambert function and then by using the Cauchy matrix approach.
Functional Differential Equations and Applications 2012 Normal forms and asymptotic solution of nonhomogeneous, nonautonomous quasilinear di¤erential equations
Ya. Goltser
Ariel University Center of Samaria, Ariel, Israel
We present a method for asymptotic expansion of the solution of the quasi-linear paramet- rically perturbed systems. Subject under consideration: i) Formal equivalent systems, ii) Normal forms of quasilinear asymptotic expansion of the solution, iv) Cases that the linear homogeneous part of di¤erential system has exponential di- chotomy or exponential trichotomy.
Functional Differential Equations and Applications 2012 Fredholm type theorem for systems of functional-di¤erential equations with positively homogeneous operators
Robert Hakl
Institute of Mathematics, Academy of Sciences of the Czech Republic, branch in Brno, Ziµ µzkova 22, 616 62 Brno, Czech Republic
AMS Subject Classi cation: [2000]34K10 Keywords and Phrases: Functional-di¤erential equations, boundary value problems, Fredholm type theorems Consider the system of functional-di¤erential equations
ui0 (t) = pi(u1; : : : ; un)(t) + fi(u1; : : : ; un)(t) for a. e. t 2 [a; b](i = 1; : : : ; n) (1) together with boundary conditions
`i(u1; : : : ; un) = hi(u1; : : : ; un)(i = 1; : : : ; n): (2)
n Here, pi; fi : C [a; b]; R 2 [a; b]; R are continuous operators satisfying Carathéodory condition, i.e., they are bounded on every ball by an integrable function, and `i; hi : n C [a; b]; R ! R are continuous functionals which are bounded on every ball by a con- stant. Furthermore, we assume that pi and `i satisfy the folowing condition: there exist positive real numbers ij and i such that ijjm = im whenever i; j; m 2 f1; : : : ; ng, and for every c > 0 and uk 2 [a; b]; R (k = 1; : : : ; n) we have
i1 in cpi(u1; : : : ; un)(t) = pi(c u1; : : : ; c un)(t) for a. e. t 2 [a; b];
i1 in c i `i(u1; : : : ; un) = `i(c u1; : : : ; c un):
By a solution to (1), (2) we understand an absolutely continuous vector-valued function n n (ui)i=1 :[a; b] ! R satisfying (1) almost everywhere in [a; b] and (2).
Functional Differential Equations and Applications 2012 Partial stability of linear stochastic functional di¤erential equations and N.V. Azbelevs w-transforms
Ramazan I. Kadiev1 and Arcady Ponosov2
1Dagestan Scienti c Center, Russian Academy of Sciences, Makhachkala 367005, Russia 2Department of Mathematical Sciences and Technology Norwegian University of Life Sciences, NO-1432 Ås, Norway
AMS Subject Classi cation: 34K50, 34D20. Keywords and Phrases: Partial stability, stochastic di¤erential equations, aftere¤ect, semimartingales, integral transforms.
The concept of partial Lyapunov stability was introduced by A. M. Lyapunov himself. Informally speaking, partial stability means that only a part of the variables in a system of di¤erential equations is Lyapunov stable, while the remaining variables may behave arbi- trarily. A mainstream framework of studying partial stability is based on modi cations of the direct (second) Lyapunov method. In our analysis we demonstrate how partial Lyapunov stability for linear stochastic func- tional di¤erential equations can be formulated in terms of a di¤erent kind of stability: partial input-to-state stability. We also show how N.V.Azbelevs W-transformations can be used to study partial input-to-state stability. As a result of applying this technique, e¢cient conditions of asymptotic and exponential partial Lyapunov stability for general and speci c stochastic functional di¤erential equations are o¤ered.
Functional Differential Equations and Applications 2012 Representation of solutions for linear stationary systems with one delay
D. Khusainov Taras Shevchenko National University of Kyiv, Ukraine Y. Bastinec, G. Piddubna Brno University of Technology, Czech Repablic [email protected]
Systems with commuting matrices. The Cauchy problem is considered for a system x_ (t) = Ax (t) +Bx (t ) +f (t) ; x (t) = ' (t) ; t 0; with matrices A and B satisfying the commutativity condition AB = BA. A special matrix function, called a delayed exponential, is used.
De nition. The delayed exponential function exp fB; tg is a matrix function de ned as ; 1 < t <
exp fB; tg = I; t < 0 2 k 8 t 2 (t ) k [t (k 1)] < I + B 1! + B 2! + + B k ! ; (k 1) t < k where k = 0; 1; 2; :::,: and is a zero matrix. By using this function, a solution of the Cauchy problem can be written as
x (t) = exp fA (t t0)g exp fB1; t g ' ( ) 0
+ exp fA (t s)g exp fB1; t sg ['0(s) A'(s)] ds Z t
+ exp fA (t s)g exp fB1; t sg f(s)ds: Z0
General systems. For general system a fundamental matrix of solutions X0 (t) for homoge- neous systems of linear equations can be represented as follows
n 1 X0(t) = I + 'k(t); (n 1) < t n; k X=0 k+1 i (t k) 1 i (t k) k where 'k(t) = k! A (k+ i+1)! B (A + B): A solution of the Cauchy problem for i=0 inhomogeneous system is P 0 t
x(t) = X0(t)'( ) + X0(t s)'0(s)ds + X0 (t s) f (s) ds: Z Z0
Functional Differential Equations and Applications 2012 Two-point boundary balue problems for second order linear di¤erential equation
Roman Koplatadze
Department of Mathemetics of Tbilisi State University Tbilisi, Georgia
AMS Subject Classi cation: 34K10.
Keywords and Phrases: Boundary value problem; Singular di¤erential equation.
Consider the problem
u00 + p(t)u = f(t); (1) u(0) = ; u(1) = ; (2) where p 2 C((0; 1); R), f 2 C([0; 1]; R), ; 2 R. The existence of a solution of the singular di¤erential equation (1), satisfying the condi- tion (2) is established.
Functional Differential Equations and Applications 2012 Sharp real-part theorems for derivatives of analytic functions
Gershon Kresin Department of Computer Science and Mathematics Ariel University Center of Samaria, Israel [email protected]
It is assumed that the boundary values of the real part of analytic functions in the unit disk
p D and the upper half-plane C+ are in L . Representations for the sharp coe¢cient in an estimate of the modulus of the n-th deriv- ative of analytic functions in D and C+ are obtained. The maximum of a bounded factor in the representation of the sharp coe¢cient for analytic functions in D is found. Thereby, a pointwise estimate of the modulus of the n-th derivative of an analytic function in D with a best constant is given. The representation for the sharp coe¢cient in the estimate of the modulus of the n-th derivative of analytic functions in C+ is concretized for some n and p. In particular, for p = 1 and for derivatives of odd order of analytic functions in C+, an explicit formula for the sharp coe¢cient is found. Also, explicit formulas for the sharp coe¢cient in the estimate of the modulus of the rst derivative of analytic functions in D and C+ are derived. In the case of analytic functions in D under the assumption that p 2 (1; 1), the coe¢cient is represented as the product of monotonic functions of jzj. A limit relation for the sharp coe¢cient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in C+. As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary.
Functional Differential Equations and Applications 2012 Control in di¤erence-di¤erential equations with distributed parameters
Oleksandra Kukharenko
Taras Shevchenko National University of Kyiv, Ukraine e-mail: [email protected]
AMS Subject Classi cation: 35Q93, 35K20, 35R10 Keywords and Phrases: Control problem, delay wave equation, steady state control.
The control problem for di¤erence-di¤erential wave equation with distributed parameters is considered in the presented paper. Preliminary, an equation without delay is considered. The rst boundary value problem is solved using the method of separation of variables. Next, delay wave equation is considered. The rst boundary value problem is solved using the Fourier method and special functions, called the delay sine and cosine functions. Finally, the steady state control problem is solved.
Functional Differential Equations and Applications 2012 On positivity of the Greens operator of two-point boundary value problem for functional-di¤erential equation
S. Labovskiy Moscow State University of Economics, Statistics and Informatics Moscow, Russia
Positive de niteness of the quadratic functional l 2 2 E(u) = (u0 + qu )dx 0 Z under boundary conditions u(0) = u(l) = 0 is equivalent to positiveness of the Green function for the Euler equation