Pontryagin’s Maximum Principle for a State-Constrained System of Douglis-Nirenberg Type

This article is concerned with optimal control problems for plants described by systems of high order nonlinear PDE’s (whose linear approximation is elliptic in the sense of Douglis-Nirenberg), with a special attention being given to their particular case: the standard stationary system of non-linear Navier–Stokes equations. The objective is to establish an analog of the Pontryagin’s maximum principle. The major challenge stems from the presence of infinitely many point-wise constraints on the system’s state, which are imposed at any point from a given continuum set of independent variables. Necessary conditions for optimality in the form of an “abstract” maximum principle are first obtained for a general optimal control problem couched in the language of functional analysis. This result is targeted at a wide class of problems, with an idea to absorb, in its proof, a great deal of technical work needed for derivation of optimality conditions so that only an interpretation of the discussed result would be basically needed to handle a particular problem. The applicability of this approach is demonstrated via obtaining the afore-mentioned analog of the Pontryagin’s maximum principle for a state-constrained system of high-order elliptic equations and the Navier–Stokes equations.