Controllability and observability notions aligned with fractional OC and games

Develop controllability and observability notions and tests for fractional-order systems that are aligned with optimal control and differential game formulations, accounting for nonlocal memory and initialization effects.

Background

The paper reviews PBH/Gramian-type results for fractional LTI systems and notes additional complications from non-exponential decay and initialization.

For optimal control and game-theoretic settings, the existing notions need to be reconciled with memory-dependent optimality principles and dynamic programming constructs.

References

Open problems: We identify gaps in (i) existence and uniqueness of equilibria with memory and refined Isaacs-type conditions; (ii) controllability and observability notions aligned with OC and games; (iii) stability under constraints; (iv) scalable FO--MPC and multi-agent solvers; and (v) integrated data-driven identification and learning with FO control and game design.

— Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications -- A Survey (2512.12111 - Mojahed et al., 13 Dec 2025) in Section 1 (Introduction), Positioning and scope, bullet list “Open problems”