Soft Numerical Computing in Uncertain Dynamic Systems

4.3.9.2: Nonhomogeneous fuzzy linear differential systems

In this section, the following nonhomogeneous fuzzy system of first order linear fuzzy differential equation with real constant coefficients will be solved:

${\mathit{x}}_{\mathit{gH}}^{\prime }\left(t\right)=\mathit{A}\left(\mathit{t}\right)\odot \mathit{x}\left(\mathit{t}\right)\oplus \mathit{f}\left(\mathit{t}\right),\mathit{x}\left({\mathit{t}}_{\mathbf{0}}\right)={\mathit{x}}_{\mathbf{0}}\in {\mathbb{F}}_R^n$

Considering the definition of gH-difference, this equation is equivalent to the following equation:

${\mathit{x}}_{\mathit{gH}}^{\prime }\left(t\right){\ominus }_{\mathit{gH}}\mathit{A}\left(\mathit{t}\right)\odot \mathit{x}\left(\mathit{t}\right)=\mathit{f}\left(\mathit{t}\right)$

Multiplying both sides of the equation by e^− At ≥ 0, we have:

$e^{-\mathit{A}t}\odot {\mathit{x}}_{\mathit{gH}}^{\prime }\left(t\right){\ominus }_{\mathit{gH}}{e}^{-\mathit{A}t}\mathit{A}\left(\mathit{t}\right)\odot \mathit{x}\left(\mathit{t}\right)=e^{-\mathit{A}t}\odot \mathit{f}\left(\mathit{t}\right)$

Based on the gH-derivative of composite of two fuzzy functions, the left-hand side can be considered ...