A note on asymptotic stability

H. A. Antosiewicz
1951 Quarterly of Applied Mathematics  
1. In this note we shall develop a stability criterion for a vector differential equation of the form I = *0*. (« where the elements of the matrix A(t) -(a< ,■(<)), i, j = 1, 2, • ■ • n, are real continuous and uniformly bounded functions for all positive t ^ t0 . A. Wintner** recently established the following criterion: Let \,(t) be the greatest, and X2(0 the least characteristic value of the matrix + A'(<)], and let || x(t) || denote the Euclidean length of the vector x(t). If f°° Ai(t) dt <
more » ... , f '° X2(<) dt < °°, then || x(t) || -> k ^ 0 as t ->°o for every non-trivial solution x(t) of (1). It is to be noted that the condition of integrability of Ai (t), \2(t) over (ta, 00) implies J"°° [trace A (£)] dt <<*>. Furthermore, this condition automatically excludes the important case A (t) = const, unless A (t) = const, is skew-symmetric. In the following we shall establish a stability criterion which is free of the above objection, i.e. which will also apply to the general case A{t) -const. We shall consider a condition to be satisfied by the matrix A(t) which will suffice to insure that || x(t) || of every non-trivial solution x{t) of (1) tends to zero as /->«>. According to Liapounofff, the trivial solution x(t) = 0 is then said to be asymptotically stable. 2. Consider a function V(x, t) which is defined and continuous for all x and t in It: ] | ^ c, t 2: T (i = 1, 2, • • • n). If for equation (1) there exists in R a function V(x, t) which is of fixed sign and admits of an infinitely small upper bound, and for which dV/dt by virtue of (1) is opposite in sign to V(x, t) in R, then the trivial solution x{t) = 0 of (1) is asymptotically stable. Liapounoff proved that the existence of such a function *Received April 16, 1951. **A. Wintner, On free vibrations with amplitudinal limits, Quart. Applied Math. 8, 102-10-i (1950). fA. Liapounoff, Problbme general de la stability du mouvement, Ann. Math. Studies, No. 17, 1949. 318 NOTES [Vol. IX, No. 3 V(x, t) is sufficient for asymptotic stability; it is, however, not necessary as was shown by J. Malkin.* We shall make use of Malkin's results to establish the following theorem : Let Ai (f) be the greatest, and X2(0 the least characteristic value of the matrix J /or every non-trivial solution x(t) of (1), i.e. the trivial solution x(t) = 0 is asymptotically stable. Note that now J" [trace A(t)]dt diverges. 3. First, we transform (1) into diagonal form. Let xx, x2, • • • xn be a base of solutions of (1), and use this base to construct an orthogonal matrix C(t). If y = C'1{t)x, then (1) reduces to ft -B(t)y, B(t) = C~lAC + ^C (2) where the matrix B(t) = (bu(t)), i, j = 1, 2, • • • n, is diagonal, i.e. 6,-,(0 = 0 for all i > j. If Vi(t), y2{t), ■ • • yn(f) is that base of solutions of (2) for which yi{tQ) = F, the i-th column vector of the identity matrix I, then || x,(i) |[ = || yi(t) || as is easily verified. Evidently, C(t) and C-1(0 have bounded elements and | C(t) \ = | C_1(i) | = 1; hence stability properties are preserved in both directions. Observing that C~1(t) = C'(t) by construction, we find by differentiating the identity C(£)C_1(<) = I that (dC'1/dt)C is skew-symmetric. Therefore B(t) + B'(t) = C-1[A(£) + A'(t)]C, and thus the characteristic values of §[£(£) + B'(t)] are identical with those of |[A(t) + A'{t)]. Hence it is sufficient to prove our theorem for the reduced equation (2). We shall show that there exists a function V (y, t) which satisfies Liapounoff's criterion for asymptotic stability. Consider the diagonal elements bu(t), i = 1, 2, ■ ■ • n, of the matrix B{t). Since (dC~1/dt)C is skew-symmetric, trace (dC~1/dt)C = 0, and thus MO = (OiAC = (CyAC' = i(C')'[A(0 + A'(t)]C\ (3) All diagonal elements of B(t) are quadratic forms in the components of the column vectors & of the matrix C(t) for which we evidently have || C' || = 1. These quadratic forms attain their maximum and minimum on the unit sphere || C" || = 1 (compact set); if Xi(0 is the greatest, X2(<) the least characteristic value of \[B(t) + B'(t)], then Xx(0 is the maximum, X2(0 the minimum. From (3) we then obtain \i(t) S 6"(t) ^ X2(0 whence for all t ^ ta exp (/!" Xi(t) dr) ^ exp (Jj" bu(r) dr) ^ exp (Jj" X2(r) dr). By hypothesis /' X4(r) dr -> -oo as t ->oo, fc = 1, 2, and thus Vi(t) = exp (/!" 6,<(r) dr) »0 as t ->«>. As Malkin has shown, (6) involves for all t 2; t0 (7) Jt. <pM *J. Malkin, Certain questions on the theory of the stability of motion in the sense of Liapounoff, American Math. Soc., Translation No. 20, 1950. 1951] J. L. SYNGE 319 and (6) and (7) together, in turn, imply f°° dt <°Hence the functions UO = [<pM'2 [ WMf dr (8) exist for all t ^ t0 and are uniformly bounded; in fact, a2 :S ^,(<) ^ 62 where a and 6 are certain constants. Now consider the function V(y, t) = 'MO2/1 + ^2(02/2 + • • * + . It evidently satisfies Liapounoff's criterion for asymptotic stability; it is a positive definite quadratic form, admitting of an infinitely small upper bound, and its derivative, by virtue of (2), becomeŝ = ~{y\ + y\ + • • • + yl) + W(y, t) where W(y, t) is a quadratic form whose coefficients depend upon those elements bu(t) of Bit) for which i < j, i, j = 1, 2, • • • n. Since these elements can always be made sufficiently small by a transformation with constant coefficients (which will not affect stability properties) the derivative dV/dt will be a negative definite quadratic form. Hence the trivial solution y(t) = 0 of (2) is asymptotically stable, and therefore the trivial solution x(t) = 0 of (1) is asymptotically stable. This establishes our theorem. O. Perron* was the first to prove directly that the conditions Ik c1 [' ^f-<C2 Jto <PAV are necessary and sufficient for the trivial solution x(t) = 0 of (1) to be asymptotically stable. *0. Perron, Die Stabilitaetsfrage bei Differentialgleichungen, Math. Zeitschrift 32, 703-728 (1930).
doi:10.1090/qam/42580 fatcat:dkjcl47sxfb3dja5lm2odtinfq