Boundary control of linear one-dimensional parabolic PDE using neuro-dynamic programming

This paper develops a neuro-dynamic programming (NDP) based near optimal boundary control of distributed parameter systems (DPS) governed by linear one-dimensional parabolic partial differential equations (PDE) under Dirichlet boundary control condition. The structure of the optimal cost functional is defined as an extension of its definition from lumped parameter systems (LPS) but for the infinite dimensional state space. Subsequently, the Hamilton-Jacobi-Bellman (HJB) equation is formulated in the infinite dimensional domain without using any model reduction. Since solving the HJB equation for the exact optimal value functional is burdensome, a radial basis function network (RBF) is subsequently proposed to achieve a computationally feasible solution online and in a forward-in time manner. The optimal value functional is tuned by using conventional adaptive laws such that the HJB equation error is minimized and accordingly the optimal control policy is derived. Ultimate boundedness (UB) of the closed-loop system is verified by using the Lyapunov theory. The performance of proposed controller is successfully verified on an unstable diffusion reaction process.