Undecidability for Scalar-Input, Simple-Constraint MPC

Determine whether global asymptotic-stability undecidability persists for a few-state linear MPC plant with scalar input or under a fixed simple constraint structure.

Background

The paper proves that deciding global asymptotic stability for constrained finite-horizon linear model predictive control is undecidable, even for horizon-one controllers with identity state, input, and terminal weights, unique optimizers, and global feasibility. The reductions use predicted-state constraints, hard input boxes, quadratically softened input constraints, and, in a separate construction, fixed state and input dimensions of three and six, respectively.

The remaining unresolved issue is whether these undecidability results continue to hold under more restrictive structural conditions than those used in the fixed-dimensional construction—specifically, for a plant with only a few state variables and a scalar input, or for a fixed and simple form of the constraints. Establishing persistence under either restriction would further narrow the structural conditions under which global asymptotic-stability verification might remain undecidable.