Asymptotic stability for 2 x 2 linear dynamic systems on time scales.AbstractWe prove asymptotical stability and instability instability /in·sta·bil·i·ty/ (-stah-bil´i-te) lack of steadiness or stability. detrusor instability theorems This is a list of theorems, by Wikipedia page. See also
AMS AMS - Andrew Message System subject classification: 34A45, 39A11. Keywords: Asymptotic stability, linear dynamic equations on time scales, asymptotic solutions, integral representations, first-order systems of differential and difference equations. 1. Main Result In this paper we study asymptotic stability of a system of linear dynamic equations on a time scale [T.sub.[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ]] = T [intersection intersection /in·ter·sec·tion/ (-sek´shun) a site at which one structure crosses another. intersection a site at which one structure crosses another. ] ([t.sub.0],[infinity]): [u.sup.[nabla Nabla may refer to one of the following:
where [u.sup.[nabla]] is the nabla derivative derivative: see calculus. derivative In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function. (see [7]), u(t) is a 2-vector function, and [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (1.2) is a 2 x 2 matrix-function ld-differentiable on T.sub.[infinity]. Exponential decay Noun 1. exponential decay - a decrease that follows an exponential function exponential return decay, decline - a gradual decrease; as of stored charge or current and stability of solutions of dynamic equations on time scales was investigated in the recent papers [1, 8-11, 16, 17] by using Lyapunov's method. We use a different approach based on integral representations of solutions via asymptotic solutions and error estimates developed in [3, 12, 13, 15]. Denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. Tr A(t) = [a.sub.11](t) + [a.sub.22](t), [absolute value of A(t)] = det(A(t)). (1.3) A time scale T is an arbitrary nonempty closed subset A group of commands or functions that do not include all the capabilities of the original specification. Software or hardware components designed for the subset will also work with the original. of the real numbers. We assume Sup T = [infinity]. For t [member of] T we define the backward jump operator [rho] : T [right arrow] T by [rho](t) = sup{s [member of] T : s < t} for all t [member of] T. (1.4) The backward graininess graininess a fault in x-ray films in which there is clumping together of the silver particles in the emulsion, causing the image to lose its homogeneous appearance and to give an impression of lumpiness. function i : T [right arrow] [0,[infinity]] is defined by i(t) = t - [rho](t). If [rho](t) < t (or v(t) > 0) we say that t is left-scattered. If t > inf T and [rho](t) = t, then t is called left-dense. If T has a right-scattered minimum m, define [T.sub.k] = T \ {m}. For f : T [right arrow] R and t [member of] [T.sub.k] define the nabla derivative of f at t denoted [f.sup.[nabla]] (t) to be the number (provided it exists) with the property that given any [epsilon] > 0, there is a neighborhood U of t such that [absolute value of f ([rho](t)) - f (s) - [f.sup.[nabla]] (t)([rho] - s)] [less than or equal to] [epsilon][absolute value of [rho](t) - s] (1.5) for all s [member of] U. The rest state u(t) = 0 of the system (1.1) is called stable if for any [epsilon] > 0 there exists [delta](T, [epsilon]) > 0 such that if [absolute value of u(T)] [less than or equal to] [delta](T, [epsilon]), then [absolute value of u(t)] [less than or equal to] [epsilon] for all t [greater than or equal to] T. The rest state u(t) = 0 of the system (1.1) is called asymptotically stable if it is stable, and attractive: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6) To prove asymptotic stability we establish stability estimates for the dynamic system (1.1) by using integral representations of the fundamental matrix of (1.1) via asymptotic solutions, and calculus on time scales [6, 7]. A function f : T [right arrow] R is called ld-continuous (f [member of] [C.sub.ld](T)) provided it is continuous at left-dense points in T and its right-sided limits exist (finite finite - compact ) at right-dense points in T. [C.sup.k.sub.ld](T) is the class of functions for which nabla derivatives derivatives In finance, contracts whose value is derived from another asset, which can include stocks, bonds, currencies, interest rates, commodities, and related indexes. Purchasers of derivatives are essentially wagering on the future performance of that asset. of order k exist and are ld-continuous on T. Denote by [L.sub.ld](T) the class of functions f : T [right arrow] R that are ld-continuous on T and Lebesgue nabla integrable on T. Let [R.sup.+.sub.ld] := {K : T [right arrow] R, K(t) [greater than or equal to] 0, 1 - vK(t) > 0, and K [member of] [C.sub.ld](T)}. (1.7) We assume that A [member of] [C.sub.ld]([T.sub.[infinity]]) and [a.sub.12](t) [not equal to] 0 for all t [member of] [T.sub.[infinity]]. The main idea of this paper is a special construction of the phase functions [[theta Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ].sub.1,2] of asymptotic solutions of the nonautonomous Adj. 1. nonautonomous - (of peoples and political bodies) controlled by outside forces nonsovereign unfree - hampered and not free; not able to act at will system (1.1). From a given nontrivial nontrivial - Requiring real thought or significant computing power. Often used as an understated way of saying that a problem is quite difficult or impractical, or even entirely unsolvable ("Proving P=NP is nontrivial"). The preferred emphatic form is "decidedly nontrivial". function [theta] [member of] [C.sup.2.sub.ld] ([T.sub.[infinity]]) we construct the function k(t) = [a.sub.12](t)/2[[theta].sup.2](t) [([theta](t)/[a.sub.12](t)).sup.[nabla]]. (1.8) Here and further in the text we often suppressed sup·press tr.v. sup·pressed, sup·press·ing, sup·press·es 1. To put an end to forcibly; subdue. 2. To curtail or prohibit the activities of. 3. dependance on t for simplicity. Assuming 1-2k(t)[theta](t)i(t) [not equal to] 0 for all t [member of] [T.sub.[infinity]] we choose a phase function [[theta].sub.1](t) as a solution of the equation v[[theta].sup.2.sub.1] - 2[[theta].sub.1](1 + v[theta]) + 2[theta] + Tr A - v[absolute value of A] - 2k[theta]/1 - 2k[theta]v = 0, (1.9) which is the version of Liouville's formula. If v > 0, then [[theta].sub.1] is the solution of the quadratic equation quadratic equation Algebraic equation of particular importance in optimization. A more descriptive name is second-degree polynomial equation. Its standard form is ax2 + bx + c [[theta].sub.1] = 1/v + [theta] + [square root of (D)], D = [[theta].sup.2] + 1 - v Tr A + [v.sup.2][absolute value of A]/ (1 - 2k[theta]v)[v.sup.2], v > 0. (1.10) If v = 0, then (1.9) reduces to the linear equation 2[[theta].sub.1] - Tr A + 2[theta](k - 1) = [[theta].sub.1] + [[theta].sub.2] - Tr A + [a.sub.12]/[theta] ([theta]/[a.sub.12])' = 0, and in this case the function [[theta].sub.1](t) is defined by the formula [[theta].sub.1](t) = [theta](t) - [a.sub.12](t)/2[theta](t) ([theta](t)/[a.sub.12](t))' + Tr A(t)/2, v(t) = 0. (1.11) Define auxiliary auxiliary In grammar, a verb that is subordinate to the main lexical verb in a clause. Auxiliaries can convey distinctions of tense, aspect, mood, person, and number. functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13) [Q.sub.o](t) = [Hov HOV High Occupancy Vehicle HOV Hover HOV Heckscher-Ohlin-Vanek (trade model) HOV Haunch of Venison HOV Heat Of Vaporization HOV Hand of Virtue (Everquest gaming) HoV Hill of Vision .sub.1](t) - [Hov.sub.2](t)/2[theta](t) (1.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16) where [parallel]x[parallel] id the Euclidean Eu·clid·e·an also Eu·clid·i·an adj. Of or relating to Euclid's geometric principles. Euclidean or Euclidian Adjective matrix norm In mathematics, a matrix norm is a natural extension of the notion of a vector norm to matrices. Properties of matrix norm In what follows,  will denote the field of real or complex numbers. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and
[[??].sub.[theta]] (t, [t.sub.0]) is the nabla exponential function exponential functionIn mathematics, a function in which a constant base is raised to a variable power. Exponential functions are used to model changes in population size, in the spread of diseases, and in the growth of investments. on a time scale (see [7, 10]). Note that [[theta].sub.1] and [[theta].sub.2] can be used to form the approximate fundamental matrix [PSI] of system (1.1) in form (1.12). Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. 1.1. Assume [a.sub.12](t) [not equal to] 0, and there exists a nontrivial function [theta] [member of] [C.sup.2.sub.ld] ([T.sub.[infinity]]) such that [M.sub.j] [member of] [R.sup.+.sub.ld], 1 - v Tr A + [v.sup.2][absolute value of A](t) [not equal to] 0, 1 - 2kv[theta](t) [not equal to] 0 for all t [member of] [T.sub.[infinity]], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.17) Then equation (1.1) is asymptotically stable if and only if the condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.18) is satisfied. Remark 1.2. If one can find two different phase functions [[theta].sub.j], j = 1, 2 such that the generalized gen·er·al·ized adj. 1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain. 2. Not specifically adapted to a particular environment or function; not specialized. 3. characteristic equations [Hov.sub.j] (t) = 0 are satisfied, then from (1.15) we get [M.sub.j] = 0, condition (1.17) disappears, and formula (1.12) with the above phase functions defines the exact fundamental solution of (1.1). Note also that for a constant matrix A, equations [Hov.sub.j] (t) = 0 turn to the usual characteristic equations of system (1.1). Condition (1.17) of Theorem 1.1 is complicated and it is very restrictive when one of the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has exponential growth Extremely fast growth. On a chart, the line curves up rather than being straight. Contrast with linear. as t [member of] [infinity]. In Theorem 1.3 below we replace the condition (1.17) by the less restrictive and simple condition (1.19) under some additional conditions. Theorem 1.3. Assume [a.sub.12](t) [not equal to] 0, 1 - v Tr A + [v.sup.2][absolute value of A](t) [not equal to] 0, and there exists a nontrivial function [theta] [member of] [C.sup.2.sub.ld]([T.sub.[infinity]]) such that K [member of] [R.sup.+.sub.ld], 1 - 2kv[theta](t) [not equal to] 0, for all t [member of] [T.sub.[infinity]], and there exist some constants [beta] > 0 and [sigma] > 1 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23) Then equation (1.1) is asymptotically stable. Note that if v = 0, then condition (1.20) becomes the classical stability condition Re[[[theta].sub.j] (t)] [less than or equal to] 0. Condition (1.19) means that the error of the chosen asymptotic solution is small enough (compare with the well-known Levinson integrability condition from [15]). The next three lemmas This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. (see [1,10,17]) are useful tools for checking condition (1.23). Lemma lemma (lĕm`ə): see theorem. (logic) lemma - A result already proved, which is needed in the proof of some further result. 1.4. [1, 10] Let [theta] be a complex valued function from [C.sub.ld](T) such that 1 - [theta](t)v(t) [not equal to] 0 for all t [member of] [T.sub.[infinity]]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24) if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.25) The following lemma gives simpler sufficient conditions of decay The reduction of strength of a signal or charge. decay - [Nuclear physics] An automatic conversion which is applied to most array-valued expressions in C; they "decay into" pointer-valued expressions pointing to the array's first element. of the nabla exponential function. Lemma 1.5. [1, 10] Assume [theta] [theta] [C.sub.ld](T), and for some [epsilon] > 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.27) Then (1.24) is satisfied. Remark 1.6. [1] The first condition (1.27) for v > 0 means that values of [theta](t) are located in the exterior of the ball with the center 1/[v.sup.*] and radius [e.sup.[epsilon]/[v.sup.*]: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.28) and it may be written in the form 2Re[[theta](t)] < v(t)[[absolute value of [theta](t)].sup.2]. (1.29) Remark 1.7. In view of Lemma 1.5, conditions (1.20), (1.23) of Theorem 1.3 can be replaced by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.30) 2Re[[[theta].sub.j] (t)] < v(t)[[absolute value of [[theta].sub.j] (t)].sup.2], t [member of] [T.sub.[infinity]], j = 1, 2. (1.31) The scalar scalar, quantity or number possessing only sign and magnitude, e.g., the real numbers (see number), in contrast to vectors and tensors; scalars obey the rules of elementary algebra. Many physical quantities have scalar values, e.g. equation [x.sup.[nabla]](t) = [theta](t)x(t) (1.32) is called exponentially ex·po·nen·tial adj. 1. Of or relating to an exponent. 2. Mathematics a. Containing, involving, or expressed as an exponent. b. stable if there exists a constant [alpha] > 0 such that for every [t.sub.0] [member of] T there exist a N = N([t.sub.0]) [greater than or equal to] 1 with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.33) If the constant N([t.sub.0]) from (1.33) can be chosen independent of [t.sub.0], then equation (1.32) is called uniformly exponentially stable. Lemma 1.8. [17] Equation (1.32) is exponentially stable if and only if one of following conditions is satisfied for arbitrary [t.sub.1] [member of] T: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.34) for every [tau] [member of] T : there exist t [member of] T with t > [tau] such that 1 - v(t)[theta](t) = 0, (1.35) where we use the convention log 0 = -[infinity] in (1.34). Remark 1.9. In order to apply Theorem 1.3 to the study of exponential 1. (mathematics) exponential - A function which raises some given constant (the "base") to the power of its argument. I.e. f x = b^x If no base is specified, e, the base of natural logarthims, is assumed. 2. stability of the dynamic system (1.1), one can replace condition (1.23) by the necessary and sufficient conditions
2. Fundamental Matrix and Error Estimates If we seek a solution of (1.1) in the form u = [PSI]v, (2.1) then from (1.1) we get [[PSI].sup.[nabla]]v + [PSi][v.sup.[nabla]] - v[[PSI].sup.[nabla]] [v.sup.[nabla]] = A[PSI]v, [PSI](1 - v[[PSI].sup.-1] [[PSI].sup.[nabla]]) [v.sup.[nabla]] = (A[PSI] - [[PSI].sup.[nabla]])v, or [v.sup.[nabla]] (t) = H(t)v(t), (2.2) where H = [(1 - [[PSI].sup.-1] [[PSI].sup.[nabla]]).sup.-1] [[PSI].sup.-1] (A[PSI] - [[PSI].sup.[nabla]]). (2.3) Assume we can find an exact solution of an auxiliary system [[PSI].sup.[nabla]] (t) = [A.sub.1] (t)[psi](t), t [member of] [T.sub.[infinity]], (2.4) with a matrix-function [A.sub.1] close to the matrix-function A, which means that condition (2.6) below is satisfied. Note that if A = [A.sub.1], then H = 0 and (2.6) is satisfied. Let [PSI](t) be the fundamental matrix of the auxiliary system (2.4). If the matrix-function [A.sub.1] is regressive re·gres·sive adj. 1. Having a tendency to return or to revert. 2. Characterized by regression. re·gres and ld-continuous, then [PSI](t) exists ([6]). The solutions of (1.1) can be represented in the form u(t) = [PSI](t)(C + [epsilon](t)), (2.5) where u(t), [epsilon](t),C are the 2-vector columns: u(t) = column(u1(t), [u.sub.2](t)), [epsilon](t) = column ([[epsilon].sub.1](t), [[epsilon].sub.2] (t)), C = column ([C.sub.1], [C.sub.2]), [C.sub.j] are arbitrary constants (Math.) a quantity of function that is introduced into the solution of a problem, and to which any value or form may at will be given, so that the solution may be made to meet special requirements. . We can consider (2.5) as a definition of the error vector-function [epsilon](t). In [12, 14] the following theorem was proved. Theorem 2.1. Assume there exists a matrix function [PSI] [member of] [C.sup.1.sub.ld] ([T.sub.[infinity]]) such that [parallel]H[parallel] [member of] [R.sup.+.sub.ld], the matrix function [PSI] - v[[PSI].sup.[nabla]] is invertible in·vert v. in·vert·ed, in·vert·ing, in·verts v.tr. 1. To turn inside out or upside down: invert an hourglass. 2. , and the following exponential function is bounded: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6) Then every solution of (1.1) can be represented in the form (2.5) and the error vector-function [epsilon](t) can be estimated by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7) where [parallel]x[parallel] is the Euclidean vector (or matrix) norm. To find the fundamental matrix function let us seek solutions of equation (1.1) [u.sup.[nabla].sub.1] = [a.sub.11][u.sub.1] + [a.sub.12] [u.sub.2], u.2 = [a.sub.21][u.sub.1] + [a.sub.22][u.sub.2], (2.8) of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10) By differentiation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11) and from (2.8) assuming [a.sub.12] [not equal to] 0 we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13) and u(t) = [PSI](t)C, (2.14) where the fundamental matrix [PSI](t) of system (1.1) is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15) Denote [B.sub.j] = ([a.sub.11] - [[theta].sub.j])(1 - v[[theta].sub.j])[U.sup.[nabla].sub.j]/[U.sub.j] = [a.sub.12] (1 - v[[theta].sub.j])[([a.sub.11] - [[theta].sub.j]/[a.sub.12]).sup.[nabla]], j = 1, 2. (2.16) Then from (1.13) we get [Hov.sub.j] = [E.sub.j] - [B.sub.j], [E.sub.j] = [[theta].sup.2.sub.j] - [[theta].sub.j] Tr(A) + [absolute value of A], j = 1, 2, (2.17) where [E.sub.j] is the usual characteristic polynomial This article is about the characteristic polynomial of a matrix. For the characteristic polynomial of a matroid or graded poset, see Graded poset. In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. of (1.1). Lemma 2.2. Assume [a.sub.12] [not equal to] 0, 1 - v Tr A + [v.sup.2][absolute value of A] [not equal to] 0, 1 - 2kv[theta] [not equal to] 0 for all t [member of] [T.sub.[infinity]], [PSI] [member of] [C.sub.ld](T) is invertible and nabla differentiable dif·fer·en·tia·ble adj. 1. That can be differentiated: differentiable species. 2. Mathematics Possessing a derivative. . Then the following formulas are true: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.23) Tr A - v[absolute value of A] = 2k[theta] + ([[theta].sub.1] + [[theta].sub.2] - v[[theta].sub.1][[theta].sub.2]) (1 - 2k[theta]v). (2.24) Note that (2.24) is the version of Liouville's formula. Proof. From (2.13) we have [U.sub.2] - [U.sub.1] = [[theta].sub.2] - [[theta].sub.1]/[a.sub.12] = -2[theta]/[a.sub.12], (2.25) and formula (2.18) follows from (2.15) and (2.25). From (2.15) we get the inverse (mathematics) inverse - Given a function, f : D -> C, a function g : C -> D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold. matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Formula (2.19) follows from (2.17). From the time scales calculus we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The nabla derivative of the [PSI] matrix function is given by the formula [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.26) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.27) Formulas (2.20), (2.21) are proved by direct calculations: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] From (2.13), (2.17) we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where P = [absolute value of A] + [a.sub.11]([[theta].sub.1] + [[theta].sub.2] - Tr A) - [[theta].sub.1][[theta].sub.2]. (2.28) Further we prove formulas (2.22): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in view of (2.19). For j = 1,2 we have [a.sub.12]/2[theta] ([Q.sub.1] + [Q.sub.0][U.sub.j]) = [a.sub.12]/2[theta] ([U.sub.1] [Hov.sub.2] - [U.sub.2] [Hov.sub.1]/2[theta] + ([Hov.sub.1] - [Hov.sub.2])[U.sub.j]/2[theta] = [Hov.sub.j]/2[theta], from which formula (2.23) is deduced: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Formula (2.24) is proved by using the well-known Liouville formula for (1.1) (see [7, Theorem 3.9.4]): [[absolute value of [PSI](t)].sup.[nabla]]/[absolute value of [PSI](t)] = Tr A(t) - v [absolute value of A(t)], (2.29) or in view of (2.18): [a.sub.12]/[theta][[theta].sub.1][[theta].sub.2] [([theta][[theta].sub.1][[theta].sub.2]/[a.sub.12]).sup.[nabla]] = Tr A - v[absolute value of A]. From this formula using notation notation: see arithmetic and musical notation. How a system of numbers, phrases, words or quantities is written or expressed. Positional notation is the location and value of digits in a numbering system, such as the decimal or binary system. (1.8) we get formula (2.24): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This completes the proof. Proof of Theorem 1.1. First note that from the assumption [M.sub.j] [member of] [R.sup.+.sub.ld] it follows that the exponential functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exist ([7]). From assumptions 1-v Tr A + [v.sup.2][absolute value of A](t) [not equal to] 0, 1 - 2kv[theta](t) [not equal to] 0 it follows that 1 - v[[theta].sub.j] [not equal to] 0 and the exponential functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] exist. Consider the system (1.1) or the equivalent system (2.2). From (2.3), (1.15) and (2.23) it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and from condition (1.17) of Theorem 1.1 it follows that condition (2.6) of Theorem 2.1 is satisfied. Applying Theorem 2.1 we obtain representation (2.5) for the solutions of (1.1) and the estimate (2.7) for the error function a. From (2.5), (2.7) we get the stability inequality inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved. [parallel]u(t)[parallel] [less than or equal to] const[parallel][PSI](t)[parallel]. (2.30) From (1.18) it follows that [parallel][PSI](t)[parallel] [right arrow] 0 as t [right arrow] [infinity], and using (2.30) we obtain asymptotic stability of (1.1). Lemma 2.3. If the conditions of Lemma 2.2 are satisfied and for some number [sigma] > 1 1 + [absolute value of [[theta].sub.j] - [a.sub.11]/[a.sub.12]] [less than or equal to] [sigma], j = 1, 2, t [member of] [T.sub.[infinity]], (2.31) [absolute value of 1 - v[[theta].sub.j]] = [square root of ([(1 - vRe[[[theta].sub.j]]).sup.2] + [(v[??][[[theta].sub.j]]).sup.2] [greater than or equal to] 1, j = 1, 2, t [member of] [T.sub.[infinity]], (2.32) then [parallel][PSI](s)[parallel] [less than or equal to] const, (2.33) [parallel][PSI](t)[[PSI].sup.-1](s)[parallel] [less than or equal to] C [absolute value of [a.sub.12](s)/[theta](s)], s [less than or equal to] t, (2.34) [parallel]A - [[PSI].sup.[nabla]] [[PSI].sup.-1][parallel] [less than or equal to] [absolute value of [Q.sub.0]] + [absolute value of [Q.sub.1]] [less than or equal to] [absolute value of [Hov.sub.1]/[a.sub.12]] + [sigma] [absolute value of [Q.sub.0]]. (2.35) Proof. From (2.32) it follows that the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (t, [t.sub.0])|, j = 1, 2, are nonincreasing. Indeed, if v > 0, then from (2.32) it follows that Log [absolute value of 1 - v(t)[[theta].sub.j] (t)]/-v(t) [less than or equal to] 0, (2.36) so the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are nonincreasing. If i [equivalent to] 0, then the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are nonincreasing in view of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.37) Because the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are nonincreasing we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.38) From condition (2.31) it follows that [absolute value of [U.sub.j]] [less than or equal to] C and inequality (2.33) is true. Inequality (2.34) follows from the formula [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.39) Further using (2.20), (2.22) we get estimate (2.35): [parallel]A - [[PSI].sup.[nabla]][[PSI].sup.-1][parallel] = [absolute value of [Q.sub.1]] + [absolute value of [Q.sub.0]] = [absolute vale of [Hov.sub.1]/[a.sub.12] - [U.sub.1][Q.sub.0]] + [absolute value of [Q.sub.0]] [less than or equal to] [absolute value of [Hov.sub.1]/[a.sub.12]] + [sigma] [absolute value of [Q.sub.0]]. (2.40) This completes the proof. Lemma 2.4. If the conditions of Lemma 2.3 and (1.21) are satisfied, then [parallel][PSI](t)H(s)[[PSI].sup.-1](s)[parallel] [less than or equal to] K(s), s [member of] [T.sub.[infinity]] [intersection] [[t.sub.0], t], (2.41) where K(s) is defined in (1.16). Proof. Denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.42) Then [parallel][OMEGA 1. (programming) Omega - A prototype-based object-oriented language from Austria. ["Type-Safe Object-Oriented Programming with Prototypes - The Concept of Omega", G. Blaschek, Structured Programming 12:217-225, 1991]. 2. ][parallel] [less than or equal to] 1 + v([parallel]A[parallel] + [absolute value of [Q.sub.1]] + [absolute value of [Q.sub.0]]), [parallel][[OMEGA].sup.co] [less than or equal to] 1 + v([parallel]A[prallel] + [absolute value of [Q.sub.1]] + [absolute value of [Q.sub.0]]), where [[OMEGA].sup.co] is the adjoint Ad´joint n. 1. An adjunct; a helper. of the matrix [OMEGA]. Using (2.22) we have [a.sub.11][Q.sub.0] - [a.sub.12][Q.sub.1] = [a.sub.11][Q.sub.0] - [a.sub.12] ([Hov.sub.1]/[a.sub.12] - [U.sub.1][Q.sub.0]) = [[theta].sub.1][Q.sub.0] - [Hov.sub.1]. (2.43) From (2.20) we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] In view of (2.43) [absolute value of det([OMEGA])] = [absolute value of 1 - v([Q.sub.0] + Tr A) + [v.sup.2]([absolute value of A] + [[theta].sub.1][Q.sub.0] - [Hov.sub.1])], and from assumption (1.21) we have [absolute value of det([OMEGA])| [greater than or equal to] [beta] > 0. (2.44) Further [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and using (2.34), (2.35) we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The proof is complete. Lemma 2.5. [6, 12] Assume y, f [member of] [C.sub.ld](T), f, y [greater than or equal to] 0, K [member of] [R.sup.+.sub.v]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.45) implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.46) Proof of Theorem 1.3. Integrating both sides of (2.2) we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.47) Multiplying mul·ti·ply 1 v. mul·ti·plied, mul·ti·ply·ing, mul·ti·plies v.tr. 1. To increase the amount, number, or degree of. 2. Mathematics To perform multiplication on. by [PSI](t) we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.48) and using u(t) = [PSI](t)v(t) we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.49) In view of (2.41) we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.50) Applying Gronwall's inequality (2.46) to (2.50) we get the stability estimate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.51) From (1.22), (1.23) it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.52) so for any [epsilon] > 0 there exists [t.sub.0] such that for t [member of] [T.sub.[infinity]] we have [parallel][PSI](t)C[parallel] [less than or equal to] [epsilon]. (2.53) Hence it follows from (2.51) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.54) Further t [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.55) and so [parallel]u(t)[parallel] [less than or equal to] [epsilon][[??].sub.K](t, [t.sub.0]) [less than or equal to] C[epsilon], (2.56) from which we get asymptotic stability of the dynamic system (1.1). Example 2.6. Consider the system (1.1) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.57) Tr A = - [??] = -a/[rho], [absolute value of A] = [??] = b/t[rho]. From (1.15) it follows that if there exist two different phase functions such that the generalized characteristic equation (see (1.13)) Hov(t) = [[theta].sub.2] - [theta] Tr(A) + [absolute value of A] - [a.sub.12](1 - v[theta])[([a.sub.11] - [theta]/[a.sub.12]).sup.[nabla]] = 0, is satisfied, then [M.sub.j] [equivalent to] 0, j = 1, 2 and condition (1.17) of Theorem 1.1 disappears. For the Euler system In mathematics, an Euler system is a technical device in the theory of Galois modules, first noticed as such in the work around 1990 by Victor Kolyvagin on Heegner points on modular elliptic curves. (1.1) with the matrix A(t) given by (2.57) this equation Becomes Hov(t) = [[theta].sub.2](t) + a[theta](t)/[rho](t) + b/t[rho](t) + (1 - i(t)[theta](t))[[theta].sup.[nabla]](t) = 0. (2.58) We seek a solution of this nonlinear A system in which the output is not a uniform relationship to the input. nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input. Riccati equation in the form [theta](t) = [lambda]/t. In view of [[theta].sup.[nabla]](t)/[theta](t) = -1/[rho](t), v(t) = t - [rho](t), [[theta].sub.2.sub.j] - v[[theta].sup.[nabla].sub.j] [[theta].sub.j] = t[[theta].sub.2.sub.j]/[rho] the characteristic equation (2.58) becomes Hov(t) = [[lambda].sub.2] - (1 - a)[lambda] + b/t[rho](t) = 0 Or [[lambda].sub.2] - (1 - a)[lambda] + b = 0, which is the usual characteristic quadratic equation with solutions [[lambda].sub.1,2] = 1 - a/2 [+ or -][lambda], [theta] = [lambda] = [square root of ([(1 - a).sup.2]/4 - b]. (2.59) Choosing the phase functions [[theta].sub.j] (t) = [[theta].sub.j]/t, j = 1, 2, (2.60) we have [Hov.sub.j] (t) = [M.sub.j] (t) [equivalent to] 0. Well-known exact solutions of the Euler system can be constructed by using the phase functions (2.60). So condition (1.17) is satisfied and from Theorem 1.1 it follows that the system (1.1) with matrix A(t) defined by (2.57) is asymptotically stable if and only if the condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.61) is satisfied. Example 2.7. Consider system (1.1) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.62) Tr A = -a/[rho], [absolute value of A] = tb/[rho](t)(1 + [t.sup.2]). For this system we cannot solve the generalized characteristic equation Hov(t) = [[theta].sub.2](t) + a[theta](t)/[rho](t) + tb/[rho](t)([t.sup.2] + 1) + (1 - v(t)[theta](t))[[theta].sup.[nabla]](t) = 0. (2.63) Choosing the phase functions by formula (2.60), the same way as in Example 2.6, from (1.13) we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.64) Condition (1.21) becomes the condition [absolute value of 1 + atv(t) + b[v.sup.2](t)/t[rho](t)] [greater than or equal to] [beta] > 0, for all t [member of] [T.sub.[infinity]]. (2.65) Condition (1.22) is satisfied and conditions (1.30), (1.31) turn to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.66) Using the estimate (2.64) one can simplify condition (1.19): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.67) So from conditions (2.65), (2.66), (2.67) it follows from Theorem 1.3 that the system (1.1) with matrix A(t) given by (2.62) is asymptotically stable. Note that if v(t) [equivalent to] 0, then conditions (2.65), (2.67) are satisfied and (2.66) becomes Re[[[lambda].sub.j]] < 0. Received November 15, 2006; Accepted May 30, 2007 References [1] B. Aulbach and S. Hilger, Linear dynamic processes with inhomogeneous Adj. 1. inhomogeneous - not homogeneous nonuniform heterogeneous, heterogenous - consisting of elements that are not of the same kind or nature; "the population of the United States is vast and heterogeneous" time scale, In Nonlinear Dynamics nonlinear dynamics, study of systems governed by equations in which a small change in one variable can induce a large systematic change; the discipline is more popularly known as chaos (see chaos theory). and Quantum Dynamical Systems Dynamical Systems A system of equations where the output of one equation is part of the input for another. A simple version of a dynamical system is linear simultaneous equations. Non-linear simultaneous equations are nonlinear dynamical systems. (Gaussig, 1990), volume 59 of Math. Res., pages 9-20. Akademie Verlag, Berlin, 1990. [2] Z. Benzaid and D.A. Lutz, Asymptotic representation of solutions of perturbed per·turb tr.v. per·turbed, per·turb·ing, per·turbs 1. To disturb greatly; make uneasy or anxious. 2. To throw into great confusion. 3. systems of linear difference equations, Stud stud 1. purebred. 2. a place, usually a farm, at which purebred animals are maintained and reproduced. stud animal an animal registered in a stud book. . Appl. Math., 77:195-221, 1987. [3] G.D. Birkhoff, Quantum mechanics quantum mechanics: see quantum theory. quantum mechanics Branch of mathematical physics that deals with atomic and subatomic systems. It is concerned with phenomena that are so small-scale that they cannot be described in classical terms, and it is and asymptotic series, Bull. Amer. Math. Soc., 32:618-700, 1933. [4] S. Bodine and D.A. Lutz, Exponential functions on time scales: Their asymptotic behavior and calculation, Dynam. Systems Appl., 12:23-43, 2003. [5] M. Bohner and D.A. Lutz, Asymptotic behavior of dynamic equations on time scales, J. Differ. Equations Appl., 7(1):21-50, 2001. Special issue in memory of W. A. Harris, Jr. [6] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001. [7] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. [8] J.J. DaCunha, Stability for time varying linear dynamic systems on time scales, J. Comput. Appl. Math., 176(2):381-410, 2005. [9] T. Gard and J. Hoffacker, Asymptotic behavior of natural growth on time scales, Dynam. Systems Appl., 12(1-2):131-147, 2003. Special issue: dynamic equations on time scales. [10] S. Hilger, Analysis on measure chains--a unified approach to continuous and discrete calculus, Results Math., 18:18-56, 1990. [11] J. Hoffacker and C.C. Tisdell, Stability and instability for dynamic equations on time scales, Comput. Math. Appl., 49(9-10):1327-1334, 2005. [12] G. Hovhannisyan, Asymptotic stability for dynamic equations on time scales, Adv. Difference Equ., pages Art. ID 18157, 17, 2006. [13] G. Hovhannisyan, Asymptotic stability and asymptotic solutions of second-order differential equations differential equation Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions. , J. Math. Anal anal (a´n'l) relating to the anus. a·nal adj. 1. Of, relating to, or near the anus. 2. . Appl., 327(1):47-62, 2007. [14] G. Hovhannisyan, Error estimates for asymptotic solutions of dynamic equations on time scales, Electron. J. Differential Equations, pages 159-162, 2007. Sixth Mississippi Mississippi, state, United States Mississippi (mĭs'əsĭp`ē), one of the Deep South states of the United States. It is bordered by Alabama (E), the Gulf of Mexico (S), Arkansas and Louisiana, with most of the border formed by State Conference on Differential Equations and Computational Having to do with calculations. Something that is "highly computational" requires a large number of calculations. Simulations, Conference 15. [15] N. Levinson, The asymptotic nature of solutions of linear differential equations (Math.) an equation which is of the first degree, when the expression which is equated to zero is regarded as a function of the dependent variable and its differential coefficients. See also: Linear , Duke Math. J., 15:111-126, 1948. [16] A.C a.c., adv the abbreviation for ante cibum, a Latin phrase meaning “before eating.” . Peterson and Y.N. Raffoul, Exponential stability of dynamic equations on time scales, Adv. Difference Equ., (2):133-144, 2005. [17] C. Potzsche, S. Siegmund, and F. Wirth, A spectral spectral /spec·tral/ (spek´tral) pertaining to a spectrum; performed by means of a spectrum. spec·tral adj. Of, relating to, or produced by a spectrum. characterization A rather long and fancy word for analyzing a system or process and measuring its "characteristics." For example, a Web characterization would yield the number of current sites on the Web, types of sites, annual growth, etc. of exponential stability for linear time-invariant systems A time-invariant system is one whose output does not depend explicitly on time.
[18] R.A. Smith, Sufficient conditions for stability of a solution of difference equations, Duke Math. J., 33:725-734, 1966. Gro Hovhannisyan Kent State University, Stark Campus, 6000 Frank Ave AVE Avenue AVE Average AVE Alta Velocidad Espanola (train between Madrid and Seville) AVE Alta Velocidad Española (Spanish: High Speed Train) AVE Audio Video Entertainment AVE Advertising Value Equivalent . NW, Canton Canton, cities, United States Canton. 1 City (1990 pop. 13,922), Fulton co., W central Ill., in the corn belt; inc. 1849. It is a trade and industrial center for a coal and farm area. 2 Town (1990 pop. 18,530), Norfolk co. , OH 44720-7599, USA E-mail: ghovhann@kent.edu |
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