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To address feasibility issues in model predictive control (MPC), mostimplementations relax hard state constraints using additional slack variableswith a suitable penalty. We propose an alternative strategy for open-loopasymptotically/Lyapunov stable nonlinear systems by relaxing the initial stateconstraint with a suitable penalty. The proposed MPC framework is globallyfeasible, ensures (semi-)global asymptotic stability, and (approximately)recovers the closed-loop properties of the nominal MPC on the feasible set. Theproposed framework can be naturally combined with a robust formulation toensure robustness subject to bounded disturbances while retaininginput-ot-state stability in case of arbitrarily large disturbances. We alsoshow how the overall design can be simplified in case the nonlinear system isexponentially stable. In the special case of linear systems, the proposed MPCformulation reduces to a quadratic program and the offline design and onlinecomputational complexity is only marginally increased compared to anominaldesign. Benefits compared to classical soft contrained MPC formulations aredemonstrated with numerical examples.

Globally stable and locally optimal model predictive control using a softened initial state constraint -- extended version