Linear Switching (Local Theory)

Abstract

In this section we take a look at systems that depend only linearly on the switching multiplier s $\boldsymbol{\lambda } = (\lambda _{1},\ldots,\lambda _{m})$ and are therefore expressible in the form

$$\displaystyle{ \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x};\boldsymbol{\lambda }) = \mathbf{a}(\mathbf{x}) + \underline{\underline{B}}(\mathbf{x})\boldsymbol{\lambda }\;, }$$

where $\underline{\underline{B}}$ is an n × m matrix. In relation to (5.3), the quantities in (8.1) are

$$\displaystyle{\mathbf{a} = \mbox{ $\frac{1} {2}$}\sum _{j=1}^{m}\sum _{ \kappa _{j}=\pm }\mathbf{f}^{\kappa _{1}\ldots \kappa _{j}\ldots \kappa _{m} }\;,\qquad \mathbf{B}_{j} = \mbox{ $\frac{1} {2}$}\sum _{j=1}^{m}\sum _{ \kappa _{j}=\pm }\kappa _{j}\mathbf{f}^{\kappa _{1}\ldots \kappa _{j}\ldots \kappa _{m} }\;,}$$

where B _j is the j ^th column of $\underline{\underline{B}}$.