2.1b.1 Scalar Model

State Trajectory in Scalar Continuous-Time LTI Differential Equation Model

The solution to the scalar (vector-less) continuous-time LTI model
$\frac{d}{dt}x(t)=ax(t)+bu(t)$
$x(0)=x_0$
for the state scalar as a function of time $x:{\mathbb{R}}_{+}\to \mathbb{R}$, the control input as a function of timestep $u:{\mathbb{R}}_{+}\to \mathbb{R}$ and scalar coefficients $a,b\in \mathbb{R}$ is
$x(t)=e^{at}{x}_0+{\int }_0^t{e}^{a(t-\tau )}bu(\tau )d\tau .$

Proof

Since this is a first-order differential equation, we can solve it using the solution for non-homogeneous first-order differential equations $x(t)=e^{\lambda t}{x}_0+{\int }_0^t{e}^{\lambda (t-\tau )}u(\tau )d\tau$ (see Note 2).