Normal forms for control systems at singular points (Q1418863)

The main concern of the paper is the derivation of normal forms for open loop nonlinear affine systems defined by ordinary differential equations \[ \dot y= f_0(y)+\sum^m_{i=1} u_i(t)f_i(y),\;y\in\mathbb{R}^d. \] More precisely, it is assumed that the linearization appears in a block-diagonalized form, and that the nonlinear terms are \(C^k\)-bounded. The system is associated with a skew product flow on \(\mathbb{R}^d\times{\mathcal U}\), where \({\mathcal U}\subset L_\infty(\mathbb{R},\mathbb{R}^m)\) is the set of admissible inputs \(u:\mathbb{R}\to U\) \((U\) being a convex, compact subset of \(\mathbb{R}^m\), with \(0\in U)\) endowed with the weak \(*\)-topology. For a given singular point, a notion of local \(C^k\)-conjugacy is defined. Finally, certain nonresonance conditions are pointed out, which allow us to eliminate higher-order terms of the Taylor expansion of the right-hand side under a conjugacy transformation. Two illustrative examples are given.