Section: New Results

Optimal control of PDEs

Optimal Control of an Age-Structured System with State Constraints

In [10] we study an optimal control problem with state constraints where the state is given by an age-structured, abstract parabolic differential equation. We prove the existence and uniqueness of solution for the state equation and provide first and second parabolic estimates. We analyze the differentiability of the cost function and, based on the general theory of Lagrange multipliers, we give a first order optimality condition. We also define and analyze the regularity of the costate. Finally, we present a pregnancy model, where two coupled age-structured equations are involved, and we apply the obtained results to this case.

Feedback laws

The articles [4], [5], [6], co-written by L. Pfeiffer in the framework of his former position at the University of Graz, deal with the computation of feedback laws for stabilization problems of PDE systems. These problems are formulated as infinite-horizon optimal control problems.

In [5], we prove that the value function associated with bilinear stabilization problems (including some control problems of the Fokker-Planck equation) can be expanded as a Taylor expansion, where the second-order term is the solution to an algebraic Riccati equation and where the terms of order three and more are solutions to well-posed linear equations. These equations are obtained by successive differentiation of the HJB equation. A polynomial feedback law can be deduced from the Taylor approximation and its efficiency is analyzed. This approach generalizes the classical LQR-stabilization method.

In [4], we apply the methodology previously described to a stabilization problem of the 2D Navier-Stokes equation. Numerical results are provided.

In [6], we analyze an implementation of the Receding-Horizon Control method utilizing the Taylor expansion of the value function as a terminal cost. More precisely, we show that the method converges at an exponential rate with respect to the prediction horizon and the degree of the Taylor approximation.