Strong Bi-Metric Regularity

• Sbi-MR is defined as an advanced regularity property for set-valued mappings with a bi-metric structure, enabling robust uniqueness and sensitivity analysis.
• It extends classical strong metric regularity by incorporating distinct strong and weak metrics, making it ideal for control problems with discontinuous controls and nonconvex objectives.
• Sbi-MR underpins convergence of discretization schemes and stability analysis under perturbations, offering precise Lipschitz and error estimates.

Strong bi-metric regularity (Sbi-MR) is an advanced regularity property for set-valued mappings—particularly optimality maps arising in infinite-dimensional optimization and optimal control—characterized by a two-metric (“bi-metric”) structure in domain and codomain. In the contemporary analysis of first-order optimality systems, especially for control-affine dynamics with possible discontinuous (e.g., bang–bang) optimal controls and nonconvex objectives, Sbi-MR provides a quantitative and robust framework for ensuring uniqueness, Lipschitz dependence, and well-posedness in both qualitative and numerical aspects. The property generalizes classical strong metric regularity (SMR) by incorporating distinct "strong" and "weak" metrics into the regularity estimates, enabling precise control of stability under perturbations and forming a rigorous basis for convergence of discretization schemes.

1. Formal Definition

Let $(Y, d_Y)$, $(Z, d_Z)$, and $(\widetilde Z, d_{\widetilde Z})$ be metric spaces with $\widetilde Z \subset Z$ and $d_Z \leq d_{\widetilde Z}$ on $\widetilde Z$. For a set-valued map $\Phi: Y \rightrightarrows Z$, Sbi-MR at $\hat y \in Y$ for $\hat z \in \widetilde Z$ (with constants $\kappa \geq 0$, $a > 0$, $b > 0$) is defined as follows (Corella et al., 18 Nov 2025, Jork et al., 2024):

• $(\hat y, \hat z) \in \operatorname{graph}\, \Phi$.
• For every $z \in B_{\widetilde Z}(\hat z; b)$, there exists exactly one $y \in B_Y(\hat y; a)$ with $z \in \Phi(y)$.
• For all $z, z' \in B_{\widetilde Z}(\hat z; b)$, the Lipschitz-type estimate holds:

$$d_Y\left( \Phi^{-1}(z) \cap B_Y(\hat y; a),\, \Phi^{-1}(z') \cap B_Y(\hat y; a) \right) \le \kappa\, d_Z(z, z').$$

A distinctive aspect is that Sbi-MR measures disturbances $z$ in the larger metric $d_{\widetilde Z}$, but the Lipschitz constant $\kappa$ is for the smaller metric $d_Z$.

This concept extends to strong bi-metric (sub)regularity with Hölder-type estimates in the weaker norms and allows, in infinite-dimensional contexts, for greater flexibility in matching the geometry of the problem to the regularity result (Jork et al., 2024).

2. Motivations and Key Principles

The motivation for Sbi-MR arises from limitations of classical single-metric SMR and SMsR in infinite-dimensional settings typical for control problems governed by ODEs or PDEs. In these cases, natural differentiability properties are associated with strong norms (e.g., $W^{1,1}$), while stability or convergence properties for solutions are more naturally expressed in weaker norms (e.g., $L^1$, $L^2$), and perturbations may enter in even stronger spaces such as $W^{1,\infty}$.

Sbi-MR exploits this structure by considering two metrics in both domain and codomain. This yields estimates where solvability and Lipschitz continuity of inverse mappings are established in weak metrics, tuned to the analytic and geometric structure of the control problem, and potentially broader classes of admissible perturbations (Corella et al., 18 Nov 2025, Jork et al., 2024).

3. Application in Affine Optimal Control Problems

In the affine-in-control Lagrange-type optimal control problem

$$\min_{u(\cdot)} \int_0^T \left[w(t, x(t)) + s(t, x(t))\right] \cdot u(t)\,dt$$

subject to

$$\dot x(t) = a(t, x(t)) + B(t, x(t)) u(t),\quad x(0) = x^0,\quad u(t) \in U \subset \mathbb{R}^m,$$

the first-order optimality system (Pontryagin Maximum Principle) is encoded via the mapping $F: Y \rightrightarrows Z$, where

$$Y = W^{1,1}([0,T];\mathbb{R}^n)\times W^{1,1}([0,T];\mathbb{R}^n)\times L^1([0,T];\mathbb{R}^m),$$

$$Z = L^\infty \times L^\infty \times L^\infty,\quad \widetilde Z= L^\infty \times L^\infty \times W^{1,\infty}.$$

Here, the strong norm in codomain ($\widetilde Z$) corresponds to higher regularity for the control perturbations, while the uniqueness and Lipschitz estimates for the solution are established in the weaker $L^1$-type norm ($Z$) (Corella et al., 18 Nov 2025).

4. Sufficient Conditions and Proof Strategies

The central result (Theorem 2.5 in (Corella et al., 18 Nov 2025)) provides sufficient second-order conditions for Sbi-MR of the optimality map $F$:

• Compact, convex control set $U$.
• Data ($f$, $g$) are $C^2$ in $(t,x)$ with globally Lipschitz derivatives.
• A strong positivity (“second variation”) condition combining the Hamiltonian's second derivatives and a multiplier function $\sigma$, formalized as follows: for all $u', u$ near the reference control and all admissible $\sigma$,

$$\int_0^T \sigma(t) \cdot (u'(t) - u(t))\,dt + \Gamma(u'-u) \ge c_0 \|u'-u\|_{L^1}^2,$$

where the second variation $\Gamma$ (involving the solution to the linearized state equation) measures the curvature of the system.

• A mild symmetry condition: $\hat H_{ux}(t)\hat B(t)$ is symmetric a.e. in $t$.

Proof proceeds by:

1. Partial linearization, showing Sbi-MR for the nonlinear optimality map $F$ is essentially determined by its linearization.
2. Reduction, eliminating state ($x$) and adjoint ($p$) variables and focusing on a variational inequality in the control variable $u$.
3. Connecting the variational inequality for controls with the second variation.
4. Establishing existence, uniqueness, and Lipschitz-type dependence by leveraging the coercivity provided by the second-order condition, even when $\Gamma$ can be negative as in nonconvex integrands.
5. Demonstrating that convexity of the cost is not necessary: the contribution from the first-order term in $\sigma$ and the second variation $\Gamma$ suffices (Corella et al., 18 Nov 2025).

5. Role of Bi-metric Structure in Analysis

A critical role is played by the choice of metrics. Typically, the “strong” metric ensures differentiability and fine linearization estimates for the optimality map, while the “weak” metric enables more permissive and natural stability/Lipschitz estimates for solution mappings. For ODE control, prototypical choices are:

• Strong: $d_X = W^{1,1} \times L^\infty$, $d_Y = L^1 \times L^\infty$.
• Weak: $\bar d_X = W^{1,1} \times L^1$, $\bar d_Y = L^1 \times L^2$. The distinction allows Sbi-MR to capture solution sensitivity even when second variation is only coercive in the weak norm, which is common in infinite dimensions (Jork et al., 2024).

6. Implications for Numerical and Perturbation Analysis

Sbi-MR implies robust qualitative and quantitative behaviors for solutions of optimal control problems:

• Local uniqueness and Lipschitz dependence of solutions under right-hand side disturbances measured in stronger metrics.
• Uniform $O(h)$ convergence rates for Euler discretization schemes applied to families of affine optimal control problems in a neighborhood of reference data, thus covering cases with bang–bang structure and nonconvex costs (Corella et al., 18 Nov 2025).
• Stability of regularity properties under small Lipschitz perturbations, as formalized by bi-metric Robinson-type perturbation lemmas (Jork et al., 2024).

A key implication is that the separation of strong and weak metrics enables the derivation of error and stability estimates tailored to the analytical structure of control problems—facilitating both theoretical results (e.g., in existence and perturbation theory) and practical outcomes (e.g., convergence rates of discretization algorithms).

7. Relationship to Other Regularity Notions and Applications

Sbi-MR is a generalization of classical SMR and SMsR, which are typically formulated in a single metric. Its bi-metric adaptation is essential when dealing with the functional-analytic complexity of infinite-dimensional spaces and when the differentiability properties of the optimality mapping and the desired stability estimates are not naturally expressed in the same topology.

Sbi-MR underpins the modern analysis of solution-mapping sensitivity, Newton-type method convergence, and discretization error bounds across ODE and PDE control problems. Verification of Sbi-MR in practice requires constraint qualifications (e.g., MFCQ), second-order sufficient conditions, and adequate smoothness in strong metrics (Jork et al., 2024).

Table: Summary of Regularity Notions in Optimal Control

Regularity Notion	Metric Structure	Canonical Estimate
SMR	Single metric	$d_X(x, x') \le \kappa\, d_Y(y, y')$
SMsR	Single metric	$d_X(x, \bar x) \le \kappa\, d_Y(\bar y, F(x))$
Sbi-MR	Bi-metric	$d_{\bar X}(x, x') \le \kappa\, d_{\bar Y}(y, y')$

Sbi-MR thus provides a versatile and potent framework for the theory and numerics of optimal control, facilitating sharp stability and convergence results even in challenging settings where classical second-order regularity is insufficient.