[Paper Review] A simple proof of the discrete time geometric Pontryagin maximum principle.

P K Mishal Assif, Debasish Chatterjee+1 more|arXiv (Cornell University)|Jun 29, 2018

Stability and Controllability of Differential Equations12 references3 citations

TL;DR

This paper presents a geometric proof of the discrete-time Pontryagin maximum principle for optimal control problems on finite-dimensional smooth manifolds, incorporating state, control, and frequency spectrum constraints. It extends the continuous-time geometric approach of [Cha11] to discrete time, establishing necessary optimality conditions via differential-geometric techniques on manifolds.

ABSTRACT

We establish a geometric Pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, c) constraints on the frequency spectrum of the optimal control trajectories. Our proof follows, in spirit, the path to establish geometric versions of the Pontryagin maximum principle on smooth manifolds indicated in [Cha11] in the context of continuous-time optimal control.

Motivation & Objective

• To extend the geometric Pontryagin maximum principle to discrete-time optimal control problems on finite-dimensional smooth manifolds.
• To incorporate three types of constraints: pointwise state constraints, pointwise control constraints, and frequency spectrum constraints on control trajectories.
• To provide a rigorous geometric framework for necessary optimality conditions in discrete-time settings, mirroring continuous-time results.
• To generalize existing discrete-time maximum principles by embedding them within a differential-geometric structure on manifolds.

Proposed method

• Adapts the geometric approach from continuous-time optimal control (as in [Cha11]) to discrete-time systems on smooth manifolds.
• Uses the calculus of variations and first-order optimality conditions on tangent bundles of manifolds to derive necessary conditions.
• Imposes constraints via embedded submanifolds and Lagrange multipliers in the tangent space framework.
• Applies spectral constraints by modeling control trajectories in the frequency domain using Fourier analysis on discrete sequences.
• Derives the maximum principle through the Hamiltonian formalism on the cotangent bundle, ensuring consistency with geometric control theory.
• Establishes the optimality condition as a maximization of the discrete Hamiltonian over admissible controls at each time step.

Experimental results

Research questions

• RQ1 How can the geometric Pontryagin maximum principle be extended to discrete-time optimal control problems on smooth manifolds?
• RQ2 What is the role of state and control constraints in shaping the necessary optimality conditions in discrete time?
• RQ3 How can frequency spectrum constraints on control trajectories be formally incorporated into the discrete-time maximum principle?
• RQ4 To what extent does the discrete-time geometric maximum principle mirror its continuous-time counterpart in structure and derivation?

Key findings

• A complete geometric discrete-time Pontryagin maximum principle is established for systems on finite-dimensional smooth manifolds.
• The necessary optimality conditions are derived using differential-geometric tools, including tangent and cotangent bundles.
• The principle incorporates pointwise state and control constraints through embedded submanifold constraints.
• Frequency spectrum constraints are formally integrated by constraining the discrete Fourier transform of control trajectories.
• The resulting optimality condition maximizes the discrete Hamiltonian at each time step, consistent with geometric control theory.
• The proof maintains the structural integrity of the continuous-time geometric approach, adapted to discrete dynamics.