New upper bounds for free linear and nonlinear vibration systems with applications of the differential calculus of norms

Computing the maximum transient energy growth

“…The measure has also been studied for vibration systems [8], climate modeling [16] and ground vehicle dynamics [5]. It has also been used as a cost objective in determining feedback controllers for fluid flow and other systems [2,9,15,23], and controller design methods have been proposed to minimize this measure [11,21,22].…”

Two-Sided Bounds on the Displacement y(t) and the Velocity ý(t) of the Vibration Problem Mÿ+Bý + Ky = 0,y(t0)= y0,ý(t0) ý0= with Application of the Differential Calculus of Norms

“…the IVP x = A(t)x,x(t 0 ) = x 0 with A(t) = A(t + t p ) in a similar way as for x = Ax,x(t 0 ) = x 0 , more precisely, it was possible to derive an upper bound on x(t) of the same form as for the case of a constant matrix. Further, in [7], the same held for the quasilinear IVP x = Ax + h(t, x),x(t 0 ) = x 0 . Therefore, one can be optimistic to carry over the results of the present paper to the case of an IVP with periodic system matrix and to the case of a quasilinear IVP.…”

“…The upper bound y = 1,2 (t) 2 depends on x 0 and adapts faster to the curve y = y(t) 2 than the upper bound y = Y 1,2 e 0 (t t 0 ) ; but, in the initial domain, y = 1,2 (t) 2 is worse than y = Y 1,2 e 0 (t t 0 ) . This can be remedied, however, by the method described in [7].…”

“…The References [1,2,[4][5][6][7][8][9][10][11][12][13][14][15][16] contain the most important contributions to the author's current research area. In [3,[17][18][19][20][21][22] the reader finds dynamical problems of interest with respect to the present paper.…”

Solution of the matrix eigenvalue problem VA+A*V=μV with applications to the study of free linear dynamical systems

“…We compare old upper bounds with the new upper bounds depending on x 0 , and also give new lower bounds on x(t). The optimal constants in the bounds are computed by the differential calculus of norms developed by the author in earlier work, see [14][15][16][17][18][19][20][21]. In Section 6, we draw some conclusions.…”