Hamiltonian formalism for optimal control of nonlinear loaded integro-PDEs

TL;DR

This paper develops a Hamiltonian formalism for nonlinear, nonlocal integro-PDEs with memory, providing necessary optimality conditions relevant to environmental control problems such as flood management and wildfire extinguishing.

Contribution

It introduces a novel Hamiltonian approach for complex loaded integro-PDEs, including new differential operators, advancing optimal control theory for nonlocal PDEs.

Findings

Derived necessary optimality conditions in Hamilton-Euler-Lagrange form.

Formulated new differential operators for loaded integro-PDEs.

Applied results to environmental control scenarios.

Abstract

We formulate nonlinear nonlocal integro-PDE with memory, biloaded (boundary integrals load the ambient space, and the ambient space loads the boundary), and the associated optimal control problems. We derive part of the necessary conditions for optimality in the form of Hamilton-Euler-Lagrange loaded integro-PDEs. In the process, we introduce an agglomeration of new differential operators. Our results have relevance to optimal amelioration of flooded areas, remediation of sites of contaminated groundwater, and active control methods for optimally extinguishing forest fires.