# Model Predictive Control stability and feasibility

For assuring stability and feasibility of linear MPC, terminal cost and terminal constraints are used. While proving, we take feasible input sequence and shift it forward at each step and this way prove feasibility. However for proving stability we need Lyapunov function and cost function is used for this purpose at each step. The problem, for me, is, assuming the same input sequence for feasibility is ok, but for stability we also need to consider that we can end up with different input sequence (optimal solution) at each next step. Doesn't it make the cost function irrelevant as a Lyapunov function?