Pattern-preserving optimal control problems with increasing time horizon

We establish a general framework that guarantees the preservation of optimal control patterns as the time horizon $[0,T]$ increases and becomes unbounded. A concept of pattern-preserving family of optimal control problems is introduced and the goal is achieved by analyzing the $\mathrm{\Gamma }$-convergence of the corresponding variational formulations as $T\to \mathrm{\infty }$. Special attention is given to scenarios involving state constraints. To illustrate the results, examples and applications are provided, with particular focus on switched systems and epidemic control.