In dynamical systems, adjoint state is an elegant theory to compute sensitivities and gradients with respect to some parameters. However, this requires the knowledge and the storage of the internal state within the whole time interval. That can become irrelevant for large-scale high-dimensional problems. A strategy is to use dimensional reduction, also known as model-order reduction in this context. For solutions of dynamical systems with a small Kolmogorov $n$-width,  an empirical proper orthogonal decomposition (POD) reduced basis method allows us to approximate the internal state under a very condensed form (summarization) which becomes very convenient for approximating adjoint states, and then gradients of cost functions within an optimization loop.