Affine Optimal Control Problems

• Affine optimal control problems are defined by systems with control-affine dynamics and cost functionals that are linear or convex in the control variables.
• The Pontryagin Maximum Principle yields bang–bang and singular arcs, with metric regularity and stability properties crucial for convergence analysis.
• Recent advances extend computational methods and theoretical insights to ODE, PDE, and geometric settings, enhancing robustness and sensitivity analysis.

Affine optimal control problems are a broad and structurally important class of dynamic optimization tasks characterized by control-affine system dynamics and cost functionals that are either affine or convex in the control variables. These problems include classical and modern applications in engineering, applied mathematics, and geometric control theory, encompassing both ordinary differential equations (ODE), partial differential equations (PDE), and manifold-valued systems. The analytic and numerical properties of affine problems are deeply influenced by the noncoercivity and potential discontinuity of optimal controls—most notably, the emergence of bang–bang and singular arcs. Recent advances in metric regularity theory, stability, and algorithmic techniques have sharply expanded the understanding and computational tractability of this class.

1. Structure and Variants of Affine Optimal Control Problems

Affine optimal control problems are defined by state equations where the control appears linearly, while the cost functional may be linear or nonlinear (often convex or quadratic) in the control:

• ODE paradigm: $\dot{x}(t) = a(t, x(t)) + B(t, x(t))\,u(t)$ with $u:[0,T] \to U \subset \mathbb{R}^m$, $U$ convex and compact, cost $$J(u) = \int_0^T w(t, x(t)) + s(t, x(t))\cdot u(t)\ dt$$ (Corella et al., 18 Nov 2025).
• PDE and ensemble systems: State $y$ and control $u$ satisfy an elliptic PDE with affine control, e.g., $A(x)\nabla y + d(x, y) = \beta(x) u$; or parametrized ODEs $x(t;\theta)$, $\dot{x}(t;\theta) = f(x(t;\theta), u(t), \theta)$ affine in $u$ (Corella et al., 2022, Scagliotti, 2022).
• Control-affine, geometric, or mechanical systems: Dynamical systems with second-order equations on manifolds, $\ddot{q} = F(q,\dot{q}) + G(q)u$ (Leyendecker et al., 2023), or affine connection control systems on Lie groups (Abrunheiro et al., 2014).

Variants include deterministic or stochastic settings, the presence of state and control constraints (including mixed constraints or box-bound sets), and the possibility of partially affine structures (e.g., some controls appearing nonlinearly) (Aronna, 2012).

2. Pontryagin Maximum Principle, Singularity, and Problem Order

The Pontryagin Maximum Principle (PMP) provides necessary conditions for optimality, providing a Hamiltonian $H(x,p,u)$ that is affine in $u$. The resulting first-order conditions typically yield switching functions $\sigma(t)$ whose sign structure determines bang–bang arcs, and, in cases where $\sigma(t) = 0$ on nontrivial intervals, singular arcs arise (Corella et al., 18 Nov 2025, Oda et al., 2013):

• Switching function and singularity: For affine systems, the optimal control is typically discontinuous (bang–bang), determined by $\sigma(t) \not= 0$, with singular arcs characterized by vanishing higher derivatives until control dependence appears at some $k$th order. The intrinsic (problem) order $q=k/2$ encodes the minimal order at which controls enter explicitly, which is always an integer for single-input systems but can be fractional for multiple inputs (Oda et al., 2013).
• Second-order conditions: For singular arcs, generalized Legendre–Clebsch conditions and second-order sufficient conditions (Goh transformation) are pivotal for both stability analysis and the convergence of indirect (shooting) algorithms (Aronna, 2012, Aronna, 2013).
• Manifold and geometric settings: In geometric control, e.g., affine connection or control-affine systems on manifolds, the PMP is formulated via generalized (Lie algebroid) Hamiltonian flows and Poisson geometry (Abrunheiro et al., 2014).

3. Regularity, Stability, and Sensitivity of the Solution Map

The regularity of the optimality system—mapping problem data to optimal state, adjoint, and control—is central for both qualitative control theory and numerical approximation.

• Strong bi-metric regularity (Sbi-MR): Sufficient conditions (requiring only second-order growth along the extremal, not global convexity) guarantee local Lipschitz continuity of the solution map in the topology of state-adjoint-control triples, even in the presence of discontinuous (bang–bang) archetypes (Corella et al., 18 Nov 2025). Sbi-MR underpins robust sensitivity results, well-posedness of Newton-type and direct–indirect hybrid numerical solvers, and sharp error bounds for discretization.
• Hölder metric subregularity (HSMs-R): In bang–bang regimes, classical Lipschitz metric regularity fails. Instead, a refined Hölder-type subregularity is characterized using a bang–bang-sensitive metric $d^*$ in control space, quantifying the effect of jump locations: $d_Y(y, \hat{y}) \le \kappa \|\Phi(y)\|_Z^{1/\nu^2}$ for residual $\Phi(y)$, where $\nu$ is the order of the switching function zero. This allows quantification of convergence rates (e.g., for Euler discretization: $O(h^{1/\nu^2})$) in the vicinity of non-Lipschitz points (Corella et al., 18 Nov 2025).
• Stability and genericity of bang–bang minimizers: In broad function-analytic settings (including ODE/PDE-constrained and abstract measure-space problems), every desirable stability property—local uniqueness, continuity, strong metric subregularity—holds if and only if the minimizer is bang–bang. Generic linear perturbations of the objective yield unique bang–bang minimizers (Corella et al., 2023).

4. Numerical Analysis and Algorithms

The structure of affine optimal control problems admits both direct methods (finite-dimensional or variational discretizations) and indirect methods (shooting, Newton, operator-splitting), with convergence and accuracy governed by the regularity properties of the optimality system.

• Discretization (Euler, finite elements, variational discretization): Under Sbi-MR or suitable local joint growth conditions, finite differences or finite element schemes achieve provable first-order convergence for controls and state-adjoint variables ($O(h)$) in $W^{1,1} \times W^{1,1} \times L^1$, even without convexity (Corella et al., 18 Nov 2025, Jork, 2023). New results extend this to less regular (Hölder-exponent) regimes when the switching function is degenerate.
• Interior-point and operator splitting methods: For state and mixed-constrained affine problems, log-barrier interior-point methods and operator-splitting (Douglas–Rachford) algorithms are competitive, with explicit convergence results and efficient handling of non-smooth (box or polyhedral) constraints (Malisani, 2023, Burachik et al., 15 Jan 2024).
• Indirect methods and shooting: For singular arcs, convergence of Gauss–Newton shooting algorithms is guaranteed under second-order sufficiency (Legendre–Clebsch, Goh transformation), with analysis extended to partial or fully affine problems with endpoint constraints (Aronna, 2012, Aronna, 2013).
• Data-driven synthesis and learning: Infinite-horizon affine dynamic programming admits linear programming characterizations that can be built directly from finite datasets, provided appropriate richness (persistency of excitation) conditions: this enables model-free policy extraction and Q-function learning for stochastic affine systems (Martinelli et al., 2022).

5. Value Function Regularity and Examples

The regularity properties of the value function—continuity, differentiability—reflect the underlying control system, attainable set geometry, and presence or absence of singular or abnormal extremals.

• Regularity strata: For affine systems with quadratic cost, the value function is continuous on an open-dense set of tame endpoints and $C^\infty$ on a (possibly smaller) open-dense subset of "smooth" points—those admitting unique, strictly normal minimizers with no conjugate points (Barilari et al., 2016). Discontinuities and non-smoothness are thus confined to subsets of measure zero, even without excluding singular minimizers.

Property	Valid on (typical) set	Description
Continuity $V_T$	Open-dense "tame" points	All minimizers regular, endpoint map maximal rank
$C^\infty$-smooth $V_T$	Open-dense "smooth" points	Unique strictly normal minimizer, no conjugacy

• Singular and L1-cost problems: For $L^1$-type costs (e.g., minimum-fuel), sufficient optimality conditions and strong-local minimality have been characterized for bang–inactive–bang concatenations under natural controllability, strict Hamiltonian conditions, and transversality at switches (Chittaro et al., 2017).
• Affine monotonic dynamic programming: Infinite-horizon affine models admit a Bellman equation that can accommodate undiscounted, exponential, multiplicative, and risk-sensitive costs. Solution uniqueness, optimality, and the validity of PI/VI schemes are characterized by the contractiveness of policy-induced affine maps; non-contractive policies can lead to weak or nonclassical solutions (Bertsekas, 2016).

6. Extensions: Geometric, PDE-constrained, and Mechanical Systems

The control-affine paradigm has been extended and unified across a broad range of geometric and functional-analytic settings:

• Geometric (Lie group, manifold, mechanical) models: The Lie algebroid formalism generalizes the PMP and exposes the underlying symplectic/Hamiltonian structure of affine connection control systems—including the derivation of splines and interpolation problems on matrix Lie groups (Abrunheiro et al., 2014). Novel Lagrangian approaches recast affine mechanical systems in a symplectic variational framework, yielding structure-preserving integrator constructions (Leyendecker et al., 2023).
• Semilinear PDE-constrained optimization: The full paradigm (state: elliptic semilinear PDE, control: $u$ affine, cost: tracking-type or affine in $u$) now admits local growth, regularity, and error theory under minimal assumptions—weakening the previous necessity for strong structural or second-order conditions. Error exponents and convergence rates for finite element and variational discretizations have been sharply improved in line with the metric regularity of the optimality map (Jork, 2023, Corella et al., 2022).
• Ensemble control and $\Gamma$-convergence: Strong results establish that optimal control for infinite ensembles of affine systems can be systematically approximated by optimization over finite sub-ensembles, via $\Gamma$-convergence, for both ODE/PDE models (Scagliotti, 2022).

7. Impact, Open Problems, and Future Directions

The recent resolution of strong metric and bi-metric regularity for affine control problems, and their generic stability properties, provide a foundation for robust sensitivity analysis, error estimation, and algorithmic convergence—across classical calculus-of-variations, hybrid direct/indirect numerical methods, and model-based or data-driven learning paradigms. The principle that bang–bang optimal controls underlie both theoretical regularity and practical algorithmic guarantees underscores their central role (Corella et al., 2023).

Future challenges include the extension of sharp regularity bounds to higher-order singularity types, the systematization of model-predictive control (MPC) error analysis in discontinuous settings, and further integration of geometric and variational discretization techniques for advanced mechanical and infinite-dimensional control systems.