# 2-Dimensional LTV System

We consider a 2-dimensional Linear Time-Varying (LTV) system with Time-Varying (TV) and Time-Invariant (TI) uncertainties. The dynamics is defined by the following ODE:

$\left\{\begin{array}{lcl} \dot{x} & = & -x - t y + t + u_1 + v_1 \\ \dot{y} & = & t^2 x + y - t + u_2 + v_2 \end{array}\right.$

such that $v_1\in [-0.1,0.1], v_2\in [-0.1,0.1]$ are TV uncertainties and $u_1\in [-0.5,0.5], u_2\in [-0.5,0.5]$ are TI uncertainties.

We compute the symbolic flowpipes over the time horizon $[0,5]$. The time cost is around 0.2 seconds. The following figure shows the octagon overapproximations for the concretized flowpipes w.r.t. the initial set $x(0) \in [1,1.5], y(0) \in [5,5.5]$.

By changing the initial set in the output file (.flow file), we are able to reuse the symbolic flowpipes and obtain the concretized flowpipes for another initial set $x(0) \in [-1.5,-1], y(0) \in [-5.5,-5]$.