[Paper Review] Penalization via global functionals of optimal-control problems for dissipative evolution

Lorenzo Portinale, Ulisse Stefanelli|arXiv (Cornell University)|Oct 22, 2019

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TL;DR

This paper proposes a penalization method for optimal control problems governed by dissipative evolution equations using global-in-time functionals (De Giorgi and Brezis-Ekeland-Nayroles) to enforce the differential constraint. It establishes Γ-convergence of the penalized functional to the constrained problem across gradient, doubly nonlinear, and GENERIC systems, ensuring convergence of minimizers as the penalty parameter vanishes.

ABSTRACT

We consider an optimal control problem for an abstract nonlinear dissipative evolution equation. The differential constraint is penalized by augmenting the target functional by a nonnegative global-in-time functional which is null-minimized in the evolution equation is satisfied. Different variational settings are presented, leading to the convergence of the penalization method for gradient flows, noncyclic and semimonotone flows, doubly nonlinear evolutions, and GENERIC systems.

Motivation & Objective

• To develop a variational penalization framework for optimal control of abstract dissipative evolution equations.
• To ensure convergence of penalized minimizers to solutions of the constrained problem as the penalty parameter ε → 0.
• To extend the method to diverse dissipative systems, including gradient flows, doubly nonlinear, and GENERIC systems.
• To establish the validity of the penalization approach under different variational structures, such as convex and nonconvex potentials.
• To provide a unified theoretical foundation for numerical implementation via alternating minimization or energy-based optimization.

Proposed method

• Augment the target functional F(u,y) with a nonnegative global functional G(u,y), which vanishes if and only if the evolution equation is satisfied.
• Use the De Giorgi functional G_DG = ∫₀ᵀ [½‖y′‖² + ½‖∂ϕ(y)−u‖² − (u,y′)] dt + ϕ(y(T))−ϕ(y₀) to penalize the residual in a variational form.
• Employ the Brezis-Ekeland-Nayroles functional G BEN = ∫₀ᵀ [ϕ(y) + ϕ*(u−y′) − (u,y)] dt + ½‖y(T)‖² − ½‖y₀‖² for convex cases.
• Prove Γ-convergence of the penalized functional E_ε = F + (1/ε)G to the constrained limit E₀, defined as F(u,y) if G(u,y)=0 and ∞ otherwise.
• Leverage weak and strong compactness, lower semicontinuity of G, and coercivity of F to establish existence and convergence of minimizers.
• Apply the framework to specific systems: gradient flows, noncyclic/semimonotone flows, doubly nonlinear evolutions, and GENERIC systems with energy and entropy structure.

Experimental results

Research questions

• RQ1 Can the penalization of differential constraints in optimal control problems for dissipative evolution equations be achieved via global-in-time functionals that vanish iff the equation is satisfied?
• RQ2 Does the penalized functional E_ε = F + (1/ε)G converge to the constrained problem as ε → 0, ensuring convergence of minimizers?
• RQ3 How does the choice of the global functional (De Giorgi vs. Brezis-Ekeland-Nayroles) affect the variational structure and numerical implementability?
• RQ4 Can the penalization method be extended to nonconvex potentials and complex systems like GENERIC?
• RQ5 What conditions ensure the Γ-convergence of E_ε to E₀ under different topologies and functional settings?

Key findings

• Γ-convergence of the penalized functional E_ε to the constrained limit E₀ is established under weak and strong topologies for various dissipative systems.
• For the De Giorgi functional, convergence holds even for nonconvex potentials, enabling broader applicability beyond convex energy functionals.
• The Brezis-Ekeland-Nayroles functional enables separate convexity in u and y, supporting alternating minimization algorithms.
• Lower semicontinuity of G_DG is proven under strong × weak convergence in L²(0,T;H) × H¹(0,T;H), ensuring existence of minimizers.
• The method applies to GENERIC systems, as demonstrated on a thermalized oscillator model with energy and entropy balance.
• Coercivity of F on sublevels of ϕ ensures strong compactness of y_ε in C([0,T];H), crucial for convergence.