Time switched differential equations and the Euler polynomials

Abstract

We discuss differential equations depending non-smoothly on the integration time of the form $$\begin{aligned} y^{(n)}=\mathop {\text{ sgn}}\nolimits (\sigma (t)) + F(t) \end{aligned}$$where \(n\in \mathbb N , n>0\), and \(F, \sigma \) are piecewise-\({\fancyscript{C}}^{\infty }\) periodic functions. The main results deal with the existence of periodic solutions of such equations as well as their computation by explicit formulas. No infinite series appear, and it is indeed established that these periodic solutions are explicitly computable by means of finitely many Euler polynomials. We also introduce a wider class of piecewise-\({\fancyscript{C}}^{\infty }\) equations where the problem of finding periodic solutions is finitely solvable as well.