Schrödinger and Euler-Bernoulli beam equations: structure-preserving discretization and equivalence as port-Hamiltonian systems

In this work, the classical equivalence between Schrödinger and Euler-Bernoulli beam partial differential equations (PDEs) as closed physical systems is extended to open physical systems using the port-Hamiltonian framework: first, a damped version of both these models is presented; second, the possible collocated inputs and outputs of the two models are parameterized in the most general way; and third, a structure-preserving discretization method for a class of one-dimensional Boundary-Controlled Port-Hamiltonian System (BC-PHS) with collocated boundary actuation and sensing is developed. Unlike the classical Partitioned Finite Element Method (PFEM) approach, the proposed discretization method makes it possible to obtain an ordinary differential equation regardless of the boundary conditions, preventing the emergence of a differential-algebraic equation in the case of mixed boundary conditions. This novelty, recently validated for the wave and Timoshenko beam equations, is now extended to PDEs with a second-order differential operator. On the Schrödinger and Euler-Bernoulli equations, it is shown that the proposed numerical scheme can deal with a large type of boundary inputs and outputs.