Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint (Q698855)

This paper deals with an abstract model for controlled fluid flow. The system is \[ y'(t) + \gamma Ay(t) + B(y(t)) = Du(t) + f(t) \] and the solution \(y(t)\) lives in a Hilbert space \(H,\) while the control \(u(t)\) takes values in a subset \(U\) of a second Hilbert space \(F.\) The assumptions on the linear operator \(A\) and the nonlinear operator \(B\) are designed to accommodate the Navier-Stokes equations. The functional undergoing minimization is \[ L(y, u) = \int_0^T \ell(t, y(t), u(t)) dt \] and the state constraint is of the form \((y(0), y(T)) \in S.\) The author proves a version of Pontryagin's maximum principle for this problem.