G. Lance, E. Trélat, E. Zuazua Turnpike in optimal shape design for heat equation
Abstract: After a short presentation of the turnpike’s problem in optimal control, we focus on the example of the heat equation with a source term as a shape. We search the path and the shape of an optimal source which let us to warm a given domain at a given temperature’s function.
Under certain assumptions we establish the existence of solutions. Then, thanks to strict dissipativity, we establish measure-turnpike result for both state and adjoint, i.e. that the optimal solution (state and adjoint) essentially remains close to an optimal solution of an associated static problem.
We illustrate the turnpike phenomenon in shape design with several numerical simulations.