Pontryagin Maximum Principle in Contact Geometry

• Pontryagin-Type Maximum Principle is a framework providing necessary optimality conditions for control problems using a Hamiltonian formulation within a contact-geometric setting.
• It leverages the projectivized cotangent bundle and costate duality to distinguish normal and abnormal extremals in a coordinate-independent manner.
• The approach unifies the dynamic, adjoint, and transversality conditions into a single contact Hamiltonian flow, streamlining optimal control analysis.

The Pontryagin-Type Maximum Principle characterizes necessary optimality conditions for control problems governed by differential equations, leveraging a Hamiltonian framework involving state, costate, and control variables. Its geometric structure becomes especially transparent in the language of contact geometry, where key constructs such as costates, separating hyperplanes, and extremality conditions acquire concise differential-geometric meaning. The approach presented in (Ohsawa, 2015) provides a canonical, coordinate-independent formulation of the Maximum Principle, unifying classical ingredients, and yielding neat interpretations for normal/abnormal extremality and the transversality condition.

1. Projectivized Cotangent Bundle and Contact Structure

The fundamental geometric setting is the extended state space $\hat M = \mathbb{R} \times M$, with $\hat x = (x^0, x^1, ..., x^n)$, where $x^0$ represents accumulated cost. The relevant phase space for the Pontryagin principle is not the full cotangent bundle $T^*\hat M$, but rather its projectivization: $$\mathcal{C} = \mathbb{P}\left(T^*\hat M\right) = \{\, (\hat x, [\hat \nu])\,|\,\hat x \in \hat M,\, \hat \nu \in T^*_{\hat x}\hat M \setminus \{0\}\,\}$$ $\mathcal{C}$ is a $(2n+1)$-dimensional contact manifold, locally charted (in the "normal" region $\nu_0 \neq 0$) by $(x^0, x^i, \lambda_i)$ with $\lambda_i = -\nu_i/\nu_0$.

The canonical contact 1-form is

$\theta = -dx^0 + \sum_{i=1}^n \lambda_i\, dx^i$

whose kernel defines the contact distribution. Its exterior derivative, restricted to $\ker \theta$, yields a symplectic structure: $\omega = -d\theta = \sum_i dx^i \wedge d\lambda_i$ Nondegeneracy of $\omega|_{\ker\theta}$ ensures the geometric foundation for Hamiltonian flows in the contact bundle, rather than the full symplectic cotangent bundle.

2. Costate as Projective Object and Adjoint Dynamics

Classically, the adjoint (costate) is introduced as a covector $\hat \nu(t) \in T^*_{\hat x(t)} \hat M$, but contact geometry identifies only its projective class $[\hat\nu(t)]$ as essential. In the normal chart ($\nu_0 \neq 0$), one uses the coordinates $\lambda_i = -\nu_i/\nu_0$.

The contact Hamiltonian $h$ generates a vector field $X_h$ via

$$\iota_{X_h} \omega = h - (R_\theta h)\,\theta, \qquad \theta(X_h) = h$$

with $R_\theta$ the Reeb vector field ($\theta(R_\theta) = 1$, $\iota_{R_\theta} \omega = 0$). Explicitly,

$$\dot x^0 = h - \sum_i \lambda_i \dot x^i, \qquad \dot x^i = \frac{\partial h}{\partial \lambda_i}, \qquad \dot\lambda_i = \lambda_i \frac{\partial h}{\partial x^0} - \frac{\partial h}{\partial x^i}$$

For the optimal Hamiltonian $$h(x^0, x, \lambda) = \max_{u \in \mathcal{U}}\left[\lambda \cdot f(x, u) - L(x, u)\right]$$, the equations for $\lambda$ coincide with the classical adjoint system arising from Pontryagin's principle: $$\dot\lambda_i = -\sum_j \lambda_j \frac{\partial f^j}{\partial x^i}(x, u^*) + \frac{\partial L}{\partial x^i}(x, u^*)$$

3. Contact Hamiltonian and the Extremal Condition

The underlying dynamical system is defined by the control Hamiltonian

$$H_c(\hat x, \hat \nu, u) = \langle \hat \nu, \hat f(\hat x, u)\rangle = \nu \cdot f(x, u) + \nu_0 L(x, u)$$

Projectivizing to the contact manifold gives

$$h_c(\hat x, [\hat\nu], u) = H_c\left(\hat x, \frac{\hat\nu}{\nu_0}, u\right) = \lambda \cdot f(x, u) - L(x, u)$$

The maximum principle is then the extremality condition

$$h(\hat x, [\hat\nu]) = \max_{u \in \mathcal U} h_c(\hat x, [\hat\nu], u)$$

or, in standard notation,

$$H(q, u, \lambda) = \langle \lambda, f(q, u) \rangle + L(q, u), \qquad \max_{u \in \mathcal U} H(q, u, \lambda)$$

4. Normal and Abnormal Extremals: Geometric Interpretation

The normal/abnormal distinction is encoded as a geometric property of coordinates in projective space:

• Normal extremals: $\nu_0 \neq 0$; the contact coordinates $\lambda$ are regular.
• Abnormal extremals: $\nu_0 = 0$; singular in the standard chart, but regular in alternative charts (e.g., using $\nu_a \neq 0$). The projectivization encodes this as a geometric property, not an ad hoc case division.

This distinguishes abnormal extremals (which arise purely from the geometry of constraints, not from cost) as points where the cotangent vector lies in the "infinity" of the normal projective chart.

5. Separating Hyperplanes and Duality with Costates

At the terminal point, needle and endpoint variations define:

• Tangential cone $C_{\hat x_1^*} \subset T_{\hat x_1^*} \hat M$ of attainable perturbations.
• Forbidden subspace $R_{\leq 0} \times T_{x_1^*} S_1$ (for equality/inequality constraints).

Strict separation by a hyperplane $\mathcal{H}_{\hat x_1^*}$ is enforced by optimality. Each such hyperplane through the origin corresponds uniquely to a projective covector $[\hat\nu]$; the costate line $[\hat\nu^*(t_1)]$ is then exactly the separating hyperplane.

Transporting $[\hat\nu]$ via the (co)tangent lift of the flow yields a field of hyperplanes along the optimal curve, confirming that costates and separating conditions are equivalent in this contact-geometric framework. This removes redundancy present in symplectic ($T^*\hat M$) pictures.

6. Transversality Condition and Contact Transformations

For terminal cost $K(x(t_1))$, introduce the canonical contact transformation: $y^0 = x^0 - K(x), \qquad y = x$

$$T^*\Phi_K\colon (\hat x, \hat\nu) \mapsto (\hat y, T^*_{\hat x}\Phi_K\,\hat\nu)$$

and its projectivization $\Psi_K$. The transversality condition (normal case) imposes

$\lambda^*(t_1) \in (T_{x_1^*} S_1)^\perp$

Pushing forward by $\Psi_K$ gives the modified terminal condition for the transformed coordinates: $\mu^*(t_1) + dK(x_1^*) \in (T_{x_1^*} S_1)^\perp$ Crucially, $\Psi_K$ is a contactomorphism: it preserves the contact structure up to scaling, mapping separating hyperplanes to separating hyperplanes. Therefore, the transversality condition for problems with terminal cost is derived as a corollary of the underlying contact geometry.

7. Synthesis: Unification via Contact Geometry

The contact-geometric formulation of the Pontryagin Maximum Principle confers several structural advantages:

• Costate and hyperplane duality: Costate is fundamentally a projective class; hyperplanes dual to costate lines delineate permissible variations.
• Normal/abnormal distinction: Abnormality is the geometric phenomenon of being at the “boundary” of projective coordinates.
• Single vector field: The full dynamical-adjoint-maximum system arises as a single contact Hamiltonian vector field.
• Transversality: Push-forward under contact transformations provides the correct transversality for problems with terminal costs.
• Minimal description: The contact structure reduces the PMP to its minimal geometric components, dispensing with the irrelevant scaling degrees of freedom present in symplectic formalism.

This approach clarifies the geometric essence of Pontryagin-type necessary conditions, providing a canonical, coordinate-independent, and structurally economic account of optimal control extremality. The symplectic $T^*\hat M$ picture adds only redundant scaling; all essential features are already present in the contact-geometric framework (Ohsawa, 2015).