Axially dispersed reactors with simultaneous convection, dispersion, and reaction might be used to approximate real reactors. While the steady-state solution to an axially dispersed reactor is well-described in the literature, its dynamic behavior is not often discussed. Studying this problem could provide insights into a class of industrial concentration transition problems, such as the product change in a glass melting furnace where one glass product is gradually replaced by another with a different composition. In this work, the dynamics in a tubular reactor with significant axial dispersion are explored following a state-space approach. The state-space models can be constructed either analytically or numerically. In the analytical approach, a continuous-time state-space model is derived from eigenfunction expansion followed by solving a Sturm-Liouville problem. In the numerical approach, a high-dimensional discrete-time state-space model is formulated by the discretization of the process PDE using the implicit finite-difference scheme. The proper orthogonal decomposition (POD) technique is then used to reduce the order of the state-space model. In either case, an optimal trajectory of the feed composition with bounded constraints to achieve a transition in the outlet concentration is solved based on the reduced models using quadratic programming. Computer simulations are used to demonstrate that substantial dispersions, if they exist in a reactor, can significantly affect the reactor dynamics (i.e., a long time might be required to make a concentration transition). In such a case, an optimal feed trajectory solved by dynamic optimization can be used to significantly reduce the transition time. Potential applications to the glass product transition are discussed.