Extension to more general objective functions

Determine whether the finite-horizon reduction, zero-extension optimality, and receding-horizon implementation developed for the infinite-horizon sparse optimal control problem with objective function given by the sum of input 1-norms can be extended to a more general class of objective functions.

Background

The paper establishes that, for the infinite-horizon sparse optimal control problem whose cost is the sum of the input 1-norms, an optimal infinite-horizon input can be obtained by solving a sufficiently long finite-horizon subproblem and extending the resulting sparse input by zeros. The same framework also supports an optimal and sparse receding-horizon implementation.

The conclusion explicitly identifies extension beyond the sum-of-input-1-norm objective as unresolved. Such an extension would determine whether the paper’s finite-horizon solvability, finite-support sparsity, and receding-horizon optimality results depend essentially on the 1-norm structure or can hold for a broader family of objective functions.