Optimal control of PDE-based systems by using a finite-dimensional approximation scheme

The design of closed-loop finite-dimensional controllers for systems described by partial differential equations is tackled by combining tools borrowed by research areas such as approximation and operator theory. The proposed paradigm is based on the idea of using operators to account for the dynamics, regulator, and measurement mappings. Specifically, we rely on a well-established setting of Banach spaces, which is well-suited to supporting the generality of the approach. First, we define a class of Lipschitz operators with finite seminorm and formulate a tracking problem in the Banach spaces of real-valued functions. Second, we search for controllers that ensure stability and minimize a given performance index. The design of such regulators is achieved by resorting to an approximation scheme based on the extended Ritz method. Such a scheme consists in constraining the regulation operator to take on a fixed structure, where a finite number of free parameters can be suitably chosen. The problem is then reduced to a mathematical programming one of nonlinear type in general, in which the values of the parameters are optimized to guarantee stability. A family of nonlinear approximators to which the most common classes of feedforward neural networks belong are employed to accomplish the design via a convenient choice of their parameters (i.e., the weights), as shown by means of simulations with the optimal control of an unstable heat equation.