Advancements in Functional Interpolation for Solving Optimal Control Problems

Abstract

This dissertation intends to advance functional interpolation (FI) for solving optimal control problems (OCPs). FI is a mathematical framework that derives functionals containing a free function and, no matter what the free function is, always satisfies a set of user-defined constraints. The FI frameworks considered in this work are the Theory of Functional Connections (TFC) and the Extreme Theory of Functional Connections (X-TFC). Both frameworks approximate linearly constrained variables in any mathematical problem as \textit{constrained expressions} (CEs), an arbitrary free function plus a summation of functionals that analytically satisfies the problem's linear constraints. In TFC, the free function is a linear combination of any orthogonal polynomial, e.g., Chebyshev and Legendre polynomials. By contrast, X-TFC's free function is a single-layer feed-forward neural network (SLFNN) with randomized input weights and biases, i.e., an extreme learning machine expansion. Using a constrained expression to analytically satisfy linear constraints effectively reduces the entire solution space of the problem in question to just the space of admissible solutions, i.e., those fully complying with all constraints. Hence, OCPs involving multiple linear constraints can be solved quicker and more accurately with TFC and X-TFC than state-of-the-art methods. In this work, indirect methods encompass the branch of optimal control theory used to solve OCPs with FI. Indirect methods analytically construct the first-order necessary conditions for an optimal solution via the Euler-Lagrange Theorem or Pontryagin's Minimum Principle. The result is a Hamiltonian boundary-value problem (HBVP) that, when solved, gives the original OCP solution. FI has been used to solve HBVPs many times in the literature. This work revisits the approach by using it to solve spacecraft energy-optimal rendezvous OCPs in relative motion. Furthermore, FI is advanced for solving OCPs indirectly by demonstrating how it can be incorporated with sample-based motion planners to handle inequality path constraints, solve the famous differential Riccati equation, coupled with receding horizon control to track trajectories, and utilized with an \textit{hp}-adaptive mesh refinement algorithm to solve more complex HBVPs. An in-depth overview of the FI frameworks used, how they relate to physics-informed neural networks, and optimal control theory is also provided.