Series expansions for analytic systems linear in control
For the solution \(x(t,\varepsilon)\) of the initial value problem \(x(0,\varepsilon)=0\) of the system \[ \dot x=f(x)+\varepsilon g(t)\quad (x\in\mathbb R^n,\;t\in\mathbb R_+,\;\varepsilon\in\mathbb R_+), \] where \(f(x)\) is analytic in a neighborhood of the origin, \(f(0)=0\), \(g(t)\) is piecewise continuous and uniformly bounded, the paper presents a series expansion \[ x(t,\varepsilon)=\sum_{k=1}^{+\infty}x_k(t)\varepsilon ^k \tag{1} \] that can be computed via explicit recursive expressions. Conditions sufficient for the uniform convergence of (1) over finite- and infinite-time intervals are obtained. For the second-order polynomial \(f(x)\), a simplified series and convergence analysis are presented. Some open problems are discussed.
scientific article; zbMATH DE number 1797126
scientific article