Control problems for evolution PDEs are most often considered in finite time intervals, without paying attention to the length of the control horizon and how it affects the optimal trajectories and controls. However, the effective available time horizon is one of the critical factors in applications, as it occurs in the design of medical therapies, or in sonic boom minimisation for supersonic aircrafts. Often it is assumed that, if the uncontrolled free dynamics tends to a steady configuration, optimal dynamic control strategies will also converge to the optimal steady state ones. However, this fact has rarely been proved to hold rigorously.
Inspired by recent work on coupled forward-backward parabolic systems in mean field game theory, we proved that the turnpike property is fulfilled (optimal controls in long time intervals tend to be steady), if the underlying control system is controllable, a key assumption that is often ignored in applications.
These relevant contributions have so far been limited, mostly, to linear PDEs, and the problem is wide open in the nonlinear frame, where the existing results require smallness conditions on the objectives, possibly of a purely technical nature, which restrict the scope of applications. Achieving optimal turnpike results for nonlinear PDEs without smallness conditions on the targets will be our first objective in this context.
This theory will also have important consequences for quasi-static models and for the development of efficient numerical solvers and software. Indeed, in practice, when the control problem is formulated in long time intervals, standard iterative algorithms based on adjoint methodologies are computationally expensive, relying on the recurrent resolution of the forward state dynamics and the backward adjoint one. The use of the turnpike property, starting from the optimal steady control and state, can to achieve a fast approximation procedure in the long time horizon.
We shall also develop the interface between turnpike properties and optimal design problems.
We will also develop specific algorithms to long time horizons.