Solution for IVP $x'(t)=\alpha x(t) + \beta- \sum_{i=1}^{m} x(t) \delta_{T_{i}}(t)$ with $x(t_0)=x_0.$

Question

I am studying some fuzzy applications and i end needing to solve this IVP, but i dont remember much about differential equations, it seems to be a hard problem and with what i remember i couldn't solve this, i tried studying again from book but i had no clue. Can you please help me with this one. \begin{equation} \left\{ \begin{array}{lcc} x'(t)=\alpha x(t) + \beta- \sum_{i=1}^{m} x(t) \delta_{T_{i}}(t), \\ x(t_0)=x_0. \end{array}% \right. \end{equation} $\alpha$ and $\beta$ are constants, $\delta$ is Dirac delta, and $T_i$ are certain times.

Answer 1

One approach might be as follows: Let $\mu(t)$ denote the unit step function. Then the dynamics are equivalent to \begin{equation*} \dot{x}(t) = \alpha x(t) + \beta\mu(t-t_0) - \sum_{i=1}^m x(t)\delta(t-T_i). \end{equation*} Define $t' = t-t_0$. Then \begin{equation*} \dot{x}(t'+t_0) = x(t'+t_0) + \beta\mu(t') - \sum_{i=1}^m x(t'+t_0)\delta(t' - (T_i-t_0)). \end{equation*} Define $y(t') = x(t'+t_0)$ so that $y(0) = x(t_0) = x_0$. Also, define $T_i'=T_i-t_0$. Then we obtain the ODE starting at $t'=0$ given by \begin{equation*} \dot{y}(t') = y(t') + \beta\mu(t') - \sum_{i=1}^m y(t') \delta(t'-T_i'), ~ y(0) = x_0. \end{equation*} Now, assuming $T_i'\ge 0$ for all $i$ and applying Laplace transforms and the sifting property yields \begin{align*} sY(s) - y(0) ={}& Y(s) + \frac{\beta}{s} - \sum_{i=1}^m \int_{t=0}^\infty y(t)\delta(t-T_i')e^{-st}dt \\ ={}& Y(s) + \frac{\beta}{s} - \sum_{i=1}^m\int_{t=-\infty}^\infty y(t)e^{-st}\delta(t-T_i')dt \\ ={}& Y(s) + \frac{\beta}{s} - \sum_{i=1}^my(T_i')e^{-sT_i'}, \end{align*} so \begin{equation*} Y(s) = \frac{x_0}{s-1} + \frac{\beta}{s(s-1)} - \sum_{i=1}^m y(T_i')\frac{e^{-sT_i'}}{s-1}. \end{equation*} Taking inverse Laplace transforms gives \begin{equation*} y(t') = x_0 e^{t'}\mu(t') + \beta(e^{t'}-1)\mu(t') - \sum_{i=1}^my(T_i') e^{t'-T_i'}\mu(t'-T_i') ~ \text{for $t'\ge 0$}. \end{equation*} The issue I find is that the trajectory $y$, as written, depends on the values $y(T_i)$, which I'm not sure how to fix. I hope this at least gives you a start!