[Paper Review] Multitime maximum principle approach of minimal submanifolds and harmonic maps

Constantin Udrişte|arXiv (Cornell University)|Oct 21, 2011

Optimization and Variational Analysis10 references3 citations

TL;DR

This paper introduces a multitime maximum principle framework to solve minimal submanifold and harmonic map problems as optimal control problems governed by multitime partial differential equations. By formulating these geometric variational problems as multitime optimal control systems, the authors prove that minimal submanifolds and harmonic maps emerge as optimal solutions under the multitime maximum principle, offering a novel control-theoretic perspective on classical differential geometry problems.

ABSTRACT

Some optimization problems coming from the Differential Geometry, as for example, the minimal submanifolds problem and the harmonic maps problem are solved here via interior solutions of appropriate multitime optimal control problems. Section 1 underlines some science domains where appear multitime optimal control problems. Section 2 (Section 3) recalls the multitime maximum principle for optimal control problems with multiple (curvilinear) integral cost functionals and $m$-flow type constraint evolution. Section 4 shows that there exists a multitime maximum principle approach of multitime variational calculus. Section 5 (Section 6) proves that the minimal submanifolds (harmonic maps) are optimal solutions of multitime evolution PDEs in an appropriate multitime optimal control problem. Section 7 uses the multitime maximum principle to show that of all solids having a given surface area, the sphere is the one having the greatest volume. Section 8 studies the minimal area of a multitime linear flow as optimal control problem. Section 9 contains commentaries.

Motivation & Objective

• To reformulate classical differential geometry problems—specifically minimal submanifolds and harmonic maps—as multitime optimal control problems.
• To establish a theoretical foundation for solving variational problems in differential geometry using the multitime maximum principle.
• To demonstrate that solutions to minimal submanifold and harmonic map problems satisfy necessary optimality conditions derived from multitime PDE-constrained optimal control.
• To extend the applicability of Pontryaguin’s maximum principle to multitime systems with multiple independent evolution parameters.
• To provide a new analytical framework that unifies geometric variational problems with optimal control theory via multitime dynamics.

Proposed method

• Formulates multitime optimal control problems with multiple integral cost functionals and m-flow type PDE constraints over a multidimensional time domain Ω₀ₜ₀.
• Applies the multitime maximum principle using a Hamiltonian H(t, x, u, p) = L + pᵢᵅXⁱᵅ, where p is the costate vector and Xⁱᵅ are vector fields satisfying complete integrability conditions.
• Derives necessary optimality conditions: costate dynamics ∂pᵢᵅ/∂tᵅ = -∂H/∂xⁱ, state dynamics ∂xⁱ/∂tᵅ = ∂H/∂pᵢᵅ, transversality conditions on boundary, and control optimality Hᵤₐ = 0.
• Uses the multitime global maximality theorem: a solution is globally optimal if the Hamiltonian is concave in control variables.
• Applies the framework to specific geometric problems, including the isoperimetric problem (sphere maximizes volume for fixed surface area) and minimal area of multitime linear flows.
• Analyzes the area of m-surfaces in tangent bundles using a Hamiltonian 1-form derived from Riemannian geometry and control dynamics on TM.

Experimental results

Research questions

• RQ1 Can minimal submanifolds be characterized as optimal solutions of a multitime optimal control problem?
• RQ2 Can harmonic maps be derived as solutions to multitime PDE-constrained optimal control systems?
• RQ3 How does the multitime maximum principle extend classical optimal control to systems with multiple independent time parameters?
• RQ4 What is the role of complete integrability conditions in ensuring the existence of multitime state trajectories?
• RQ5 Can the isoperimetric inequality (sphere maximizes volume for fixed surface area) be derived via multitime optimal control?

Key findings

• Minimal submanifolds are shown to be optimal solutions of a multitime optimal control problem governed by m-flow type PDEs, satisfying the multitime maximum principle.
• Harmonic maps are proven to be optimal solutions of an appropriate multitime optimal control problem with m-flow constraints.
• The sphere is identified as the optimal solution maximizing volume among all solids with fixed surface area, derived via the multitime maximum principle.
• The minimal area of a multitime linear flow is characterized as the solution to a multitime optimal control problem with bounded controls.
• The optimal control for the area-maximization problem on tangent bundles is shown to be bang-bang, with controls switching between ±1 based on switching functions Mₐ(t).
• The Hamiltonian 1-form in the area maximization problem is linear in controls, leading to discontinuous optimal controls (bang-bang control) when controls are bounded.