Hamiltonian formalism for optimal control of nonlinear loaded integro-PDEs

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.