Pontryagin Extremals in Control Theory

• Pontryagin Extremals are candidate trajectories that satisfy necessary optimality conditions derived from the Maximum Principle using needle variations.
• They are characterized by analytical conditions involving Hamiltonian maximization and geometric interpretations in state-control spaces with nonholonomic constraints.
• Applications span conservation laws, sub-Riemannian geometry, and measure-theoretic control, with both numerical and covariant methods enhancing their analysis.

Pontryagin extremals are solutions to the necessary optimality conditions derived via the Pontryagin Maximum Principle (PMP) in optimal control theory, forming a central concept in modern variational calculus, geometric control, and sub-Finsler geometry. These extremals are characterized either analytically—through algebraic and differential conditions involving Hamiltonians and adjoint variables—or geometrically, as distinguished trajectories (or geodesics) in a state-control space, often subject to nonholonomic constraints. The notion spans classical finite-dimensional systems, infinite-dimensional problems on measure spaces, control on Lie groups with nontrivial geometry, and extends to cases with minimal regularity. Pontryagin extremals encode the essential features of candidate minimizers (or maximizers) for control problems, including regular arcs, abnormal extremals, and nontrivial phenomena such as switching or breaking through singular loci.

1. Foundational Principles: The Role of Needle Variations and the Maximum Principle

The classical construction of Pontryagin extremals proceeds by probing an optimal control process with needle variations—localized, infinitesimal changes of the control over short intervals. This methodology, as re-examined in "On the proof of Pontryagin's maximum principle by means of needle variations" (Dmitruk et al., 2014), systematically replaces infinite-dimensional variations in control functions with a smooth, finite-dimensional parameterization. Specifically, the effect of needle perturbations is described by a differentiable operator $P : (a, \varepsilon) \mapsto x_\varepsilon(t_1)$, where $\varepsilon$ encodes needle widths and $a$ the base state. By applying the Lagrange multipliers rule to the finite-dimensional proxy problem—where cost and endpoint constraints are expressed as compositions of smooth mappings in the needle-parameter space—one obtains necessary conditions of the form

$$H(\psi(\theta_i), x(\theta_i), v_i) \leq H(\psi(\theta_i), x(\theta_i), u(\theta_i)),$$

where $H$ is the Pontryagin function and $\psi$ the adjoint variable, itself governed by a backward linearized equation. These localized conditions are aggregated, via the centered family property of compact multiplier sets, into the universal maximum condition:

$$\max_{u \in U} H(\psi(t), x(t), u) = H(\psi(t), x(t), u(t)), \quad \text{a.e. } t \in [t_0, t_1].$$

Through this framework, Pontryagin extremals are precisely those trajectories (with associated controls and multipliers) that satisfy this maximization condition almost everywhere.

2. Geometric, Covariant, and Tensorial Frameworks

In geometric constrained variational calculus, Pontryagin extremals acquire a coordinate-free, tensorial character. The treatment in (Massa et al., 2015) recasts the variational calculus on fiber bundles and jet spaces, handling admissible evolutions that are piecewise smooth and subject to nonholonomic constraints. The necessary conditions for extremals arise from analyzing the first variation of the action:

$I = \int_{t_0}^{t_1} L_{\text{dt}},$

and yield both Euler–Lagrange dynamics and Erdmann–Weierstrass corner conditions at non-smoothness points. By introducing the concept of infinitesimal control—vector bundle homomorphisms splitting the tangent space of the constraint submanifold—one defines covariant derivatives $D/Dt$ acting on virtual tensors. The Hamiltonian reformulation uses the contact bundle, Liouville 1-form, and the Pontryagin Hamiltonian

$H = p_i \psi^i - L,$

generating Hamilton equations along with constraints and matching conditions. This approach enables the intrinsic classification of extremals as normal (abnormality index zero) or abnormal (index $p>0$ via co-rank of a transport map), provides an algorithmic algebraic characterization of abnormal extremals, and establishes an existence theorem for finite deformations based on non-degeneracy of a symmetric matrix built from the derivatives of constraints.

3. Analytic and Algorithmic Construction of Extremals: Conservation Laws and Control Algorithms

In concrete applications such as scalar convex conservation laws, Pontryagin extremals instantiate pointwise solution algorithms. As shown in (Kang et al., 2016), conservation laws are reinterpreted as optimal control problems via Hamilton–Jacobi theory, with solutions extracted through PMP. The corresponding extremals solve characteristic ODEs, and the solution $u(x, t)$ is found by minimizing a cost functional determined by the control and initial data:

$J^*(p) = (p F'(p) - F(p)) t + G(x - F'(p) t),$

with $p$ subject to algebraic equations arising from matching endpoint or jump conditions. Numerical implementation uses function approximation (Chebfun in MATLAB) to solve for candidate $p$, evaluate the cost, and select the minimizer. This algorithm produces entropy (viscosity) solutions pointwise and is inherently parallelizable and robust to discontinuities.

Application Domain	Key extremal form	Algorithmic tool
Conservation laws	Algebraic minimization problem for $p$	Chebfun root-finding
Sub-Riemannian geometry	Explicit control law via Hamiltonian system	Integration of PMP ODEs
Wasserstein optimal control	Measure-valued adjoints, continuity equations	Subdifferential calculus

4. Classification, Structure, and Local Optimality: Broken and Abnormal Extremals

Pontryagin extremals encompass both regular and singular trajectories. In affine control systems, as analyzed in (Agrachev et al., 2017), extremals may break through singular loci—sets where the maximized Hamiltonian becomes degenerate. For $n=3$, $k=2$ systems, contact geometry enables a geometric proof of the local optimality of broken normal extremals when the controllable fields generate a contact distribution. The transition at the switching is governed by explicit jump formulas for the control and normalization conditions:

$u(t \pm 0) = [ \pm d I + H ]^{-1} H_{or},$

with $d$ determined via normalization. When $k=2$ but without the contact condition, time-rescaling and Lie bracket conditions ensure optimality even for abnormal extremals. The argument typically involves comparison via Lagrangian submanifolds, codimension–1 conditions, and direct estimation of variations, with the central point being that any deviation results in longer travel time.

5. Extremals in Non-Euclidean and Non-Smooth Geometries: Lie Groups and Polyhedral Finsler Structures

In sub-Finsler and polyhedral contexts on Lie groups (Cartan, Engel, or general left-invariant structures), Pontryagin extremals are constructed as solutions of Hamiltonian systems coupled with maximization conditions on convex sets defining the allowed control directions (Berestovskii et al., 2020, Berestovskii et al., 2020, Prudencio et al., 2022). The vertical part of the extremal (covector in the Lie algebra dual) evolves via Euler–Arnold type equations:

$$\mathfrak{a}'(t) = -\mathrm{ad}^*(u(t))(\mathfrak{a}(t)),$$

with $u(t)$ chosen as the maximizer of the pairing over the unit sphere $S_F$. Uniqueness of $u(t)$ may fail when the norm $F$ is not strictly convex; asymptotic curvature invariants (obtained via Riemannian approximations) are used to quantify when the maximizer is unique or lies on a face of $S_F$. In the sub-Riemannian limit (Euclidean norm), the maximizer is always unique, and extremals can be integrated explicitly.

6. Generalizations: Infinite-Dimensional, Nonsmooth, and Low-Regularity Settings

Pontryagin extremals extend to problems in Wasserstein spaces, where the state is a probability measure and the adjoint variable is replaced by a measure-valued costate evolving by continuity equations (Bonnet et al., 2017). Subdifferential calculus replaces standard derivatives, and the maximization condition is imposed on an infinite-dimensional Hamiltonian. The framework accommodates non-local and nonlinear dynamics arising in distributed control and mean-field PDEs.

In problems with minimal regularity, the maximum principle and hence the theory of Pontryagin extremals remains valid under weakened differentiability assumptions. Through the use of Gâteaux and Hadamard differentials (Blot et al., 2022), value functions become differentiable in a directional sense, and envelope theorems provide explicit sensitivity formulas even when the data is only piecewise-smooth or lacks full Fréchet differentiability.

7. Extremal Structure and Loss of Optimality: Conjugate Points and Cut-Loci

In sub-Riemannian Carnot groups, Pontryagin extremals provide a basis for understanding not only geodesic structure but also the emergence of conjugate points and the cut locus (Montanari et al., 23 Feb 2025). Explicit formulas, such as

$$u(s) = a_1 \cos(\omega_1 s) + b_1 \sin(\omega_1 s) + a_2 \cos(\omega_2 s) + b_2 \sin(\omega_2 s),$$

allow computation of endpoints via exponential maps. Singularities in the exponential map's differential (e.g., when $\det M_1 = 0$ or $t^2 x \in \mathrm{span}\{x\}$) signal conjugate times, beyond which the extremal fails to be locally minimizing. The cut locus—where global minimality is lost—is conjecturally identified via such analytic conditions related to the structure of the PMP extremal.

Summary Table: Key Characterizations

Context	Extremal description	Key technique
Classical finite-dimensional control	Maximum principle via needle variations	Lagrange multipliers, operator $P$
Geometric variational calculus	Covariant Hamiltonian systems, abnormality classification	Tensorial calculus, abnormality index
Measure-theoretic control (Wasserstein)	Measure-valued adjoints, subdifferential and continuity equations	Needle variations, chain rule
Lie groups, polyhedral Finsler	Vertical part via Euler–Arnold, maximizers on unit sphere	Asymptotic curvature, uniqueness criteria
Conservation laws	Algebraic minimizers for entropy solutions	Pointwise minimization, Chebfun algorithm
Nonsmooth/low regularity control	Gâteaux/Hadamard envelope theorems, minimal multipliers	Directional differentiability
Sub-Riemannian geometry, Carnot groups	Explicit control laws, conjugate/cut locus detection	Exponential map, determinant conditions

Pontryagin extremals thus serve as a unifying concept for the necessary conditions and analytic structure of minimizing or maximizing trajectories across a diverse array of control-theoretic, geometric, and analytic contexts.