We are considering an optimal control problem for a type of
hybrid system involving ordinary differential
equations and a discrete time feature. One state variable has
dynamics in only one season of the year and has a jump condition
to obtain the initial condition for that corresponding season in
the next year. The other state variable has continuous dynamics.
Given a general objective functional, existence, necessary
conditions and uniqueness for an optimal control are established.
We apply our approach to a tick-transmitted disease model with age
structure in which the tick dynamics changes seasonally while
hosts have continuous dynamics. The goal is to maximize
disease-free ticks and minimize infected ticks through an optimal
control strategy of treatment with acaricide. Numerical examples
are given to illustrate the results.