Abstract
A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics, and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the Successive Galerkin Approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear Generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.
Highlights
Optimal feedback controls for evolutionary control systems are of significant practical importance
In the very special but important case of a linear control system with quadratic cost without constraints on the control or the state variables, the Hamilton–Jacobi– Bellman (HJB) equation reduces to a Riccati equation which has received a tremendous amount of attention, both for the cases when the control system is related to ordinary or to partial differential equations
It is very important to note that semidiscretization in space of a wide class of time-dependent partial differential equation (PDE) will lead to finite-dimensional state space representations of this type, and the applicability of the presented framework is only limited by the dimensionality of the associated HJB equation
Summary
Optimal feedback controls for evolutionary control systems are of significant practical importance. A related approach to numerical optimal feedback control of PDEs is to semidiscretize the dynamics and to add a model order reduction step, either with balanced truncation or proper orthogonal decomposition, in order to reduce the dimension of the dynamics to a number that is tractable for grid-based, semi-Lagrangian schemes. This approach has been successfully explored, for instance, in [1, 31, 34] and references therein. To solve the resulting HJB we utilize a Newton method based on the GHJB equa-
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