On linear ordinary differential equations with functionally commutative coefficient matrices

An analytical model for dispersion of pollutants from a continuous source in the atmospheric boundary layer

“…In equation (2.12), F is an N × N matrix of the real-valued continuous functions of real variable x. The system of first-order ordinary linear differential equations (2.12) will have a unique solution (Martin 1967;Zhu 1992) 15) provided the matrix F (x) satisfies the condition 16) where the column matrix A(0) is the value of the matrix A(x) at x = 0. The functional forms of K z (x, z) used in the dispersion modelling for which the corresponding matrix F satisfies the condition (2.16) are deduced in appendix A.…”

A note on extension of the eigenvalue concept

“…Notable (but by no means exhaustive) results with introductive references include: Floquet characteristic exponents for LP systems [5]; Lyapunov characteristic exponents [6]; the X-eigenvalue concept [7]; the concept of time-varying poles and zeros [8]; the co-eigenvalue concept for proper and semiproper LTV systems [9]; and the SDand PD-eigenvalue concepts [lo].…”

Model-Based Adaptive Tracking Control of Linear Time-Varying System with Uncertainties

“…Over the past several years, trajectory tracking issue as a high-level control of a nonlinear system has been received a wide attention from control community. Hence, the discussion here is principally devoted to model-based adaptive trajectory tracking control algorithm of linear time-varying (LTV) systems in the presence of uncertainty [4,5].…”

“…In general all systems are time-varying in principle and a large number of systems arising in practice are time-varying. Time variation is a result of system p a r a m e t e r s c h a n g i n g a s a f u n c t i o n o f t i m e [ 5 ] , s u c h a s a e r o d y n a m i c c o e f f i c i e n t s o f aircrafts, hydrodynamic terms in marine vessels, circuit parameters in electronic circuits, and mechanical parameters in machinery. Thus, we characterize systems as time-varying if the parameter variation is happening on time scales close to that of the system dynamics.…”