Young measure relaxation gaps for controllable systems with smooth state constraints

In this article, we tackle the problem of the existence of a gap corresponding to Young measure relaxations for state-constrained optimal control problems. We provide a counterexample proving that a gap may occur in a very regular setting, namely for a smooth controllable system state-constrained to the closed unit ball, provided that the Lagrangian density (i.e., the running cost) is non-convex in the control variables. The example is constructed in the setting of sub-Riemannian geometry with the core ingredient being an unusual admissible curve that exhibits a certain form of resistance to state-constrained approximation. Specifically, this curve cannot be approximated by neighboring admissible curves while obeying the state constraint due to the intricate nature of the dynamics near the boundary of the constraint set. This example therefore demonstrates the impossibility of Filippov-Wazewski type approximation in the presence of state constraints. Our example also presents an occupation measure relaxation gap.

💡 Research Summary

The paper investigates whether the optimal cost of a state‑constrained control problem coincides with its Young‑measure relaxation. Denoting by M_c the infimum of the original problem over measurable controls and by M_y the infimum after relaxing the control to a family of probability measures (Young measures), the central question is whether M_c = M_y holds. While previous work identified two mechanisms that can create a gap—lack of reachability and failure of the Filippov‑Ważewski approximation—the latter had only been observed in artificial examples.

The authors construct a natural counterexample in a smooth, controllable sub‑Riemannian setting. The dynamics are linear in the control: f(t,x,u)=u₁f₁(x)+u₂f₂(x), where (f₁,f₂) generate a free, step‑(d‑2) distribution that spans ℝ^d at every point (d≥4). The state constraint set Ω is the closed unit ball, with the initial point x₀ in its interior.

The key idea is to design a special admissible trajectory η that follows only f₁, i.e. η′=f₁(η), which is a straight line. The domain Ω is then smoothly deformed (via a diffeomorphism) into a spiral that wraps around η so that the sub‑Riemannian velocity distribution at points of η is tangent to the boundary of Ω. Consequently, any ordinary admissible curve that tries to approximate η while staying inside Ω must deviate significantly, leading to higher control effort and cost.

The running cost L(t,x,u) is chosen to be smooth but non‑convex in u, possessing four wells located at (±1,±1). For the straight‑line control u=(1,0) the cost is relatively high, whereas a Young measure ν_t =½(δ_{(1,1)}+δ_{(1,−1)}) yields an average velocity equal to f₁ while incurring the lower cost associated with the wells at (1,±1). Hence the relaxed problem attains a strictly lower value: M_y < M_c.

Four families of optimal‑control problems are treated:

1. Lagrange problem (running cost only). Theorem 1 shows M_c(Ω,X) > M_y(Ω,X) for any non‑empty open target X⊂Ω, while equality holds when the state constraint is removed (Ω=ℝ^d).

2. Bolza problem without terminal constraints. By adding a suitable terminal penalty g, Theorem 2 demonstrates that the gap persists even when the endpoint constraint is replaced by a cost term.

3. Mayer problems (no running cost). Theorems 3 and 4 construct examples in dimensions d≥5 where a non‑convex control set U and a non‑linear velocity map f produce a gap both with and without terminal constraints.

4. Filippov‑Ważewski breakdown. Theorem 5 provides a curve γ that belongs to the convexified differential inclusion S^r_Ω(x₀) but cannot be approximated by any genuine admissible curve in S_Ω(x₀). This violates the Soner‑type inward‑pointing condition required in the classical Filippov‑Ważewski theorem, showing that the theorem cannot be extended to state‑constrained settings without additional hypotheses.

The proofs rely on a geometric construction: the admissible curve η is kept on the boundary of a spirally deformed Ω, forcing any nearby admissible trajectory to oscillate increasingly as it approaches η. The non‑convex Lagrangian then makes the oscillatory trajectories more expensive than the Young‑measure mixture. The authors also verify that when the state constraint is removed, the gap disappears, confirming that the phenomenon is purely due to the interaction between the dynamics and the boundary geometry.

An important corollary concerns occupation‑measure relaxations, which are linear programs equivalent to Young‑measure relaxations. Since M_o = M_y, the same gap appears for the occupation‑measure formulation. Consequently, numerical schemes based on moment‑SOS hierarchies that compute M_o may return a value strictly lower than the true optimal cost M_c in the presence of such state constraints.

In summary, the paper provides the first natural example where a Young‑measure relaxation gap arises solely from the interplay of smooth sub‑Riemannian dynamics and a smooth state‑constraint boundary, without relying on reachability issues. This result deepens the theoretical understanding of state‑constrained optimal control, highlights limitations of existing approximation theorems, and cautions against blind application of occupation‑measure based numerical methods in constrained settings.