Abstract
In this work, we introduce Pontryagin Neural Networks (PoNNs) and employ them to learn the optimal control actions for unconstrained and constrained optimal intercept problems. PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed for tackling optimal control problems via the Pontryagin Minimum Principle (PMP) application (e.g., indirect method). The PMP provides first-order necessary optimality conditions, which result in a Two-Point Boundary Value Problem (TPBVP). More precisely, PoNNs learn the optimal control actions from the unknown solutions of the arising TPBVP, modeling them with Neural Networks (NNs). The characteristic feature of PoNNs is the use of PINNs combined with a functional interpolation technique, named the Theory of Functional Connections (TFC), which forms the so-called PINN-TFC based frameworks. According to these frameworks, the unknown solutions are modeled via the TFC’s constrained expressions using NNs as free functions. The results show that PoNNs can be successfully applied to learn optimal controls for the class of optimal intercept problems considered in this paper.
Highlights
The results show that Pontryagin Neural Networks (PoNNs) can be successfully applied to learn optimal controls for the class of optimal intercept problems considered in this paper
PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) designed to tackle optimal control problems via the Pontryagin Minimum Principle (PMP) application
The results that are presented in this paper show that PoNNs can effectively learn the optimal control for the class of optimal intercept problems with the integral quadratic cost
Summary
Optimal control theory has garnered interest in many fields of study. Solution methods for optimal control problems fall into two categories: direct methods and indirect methods [1]. The system’s state and control are discretized and, the problem is transcribed into a NonLinear Optimization Problem or Non-Linear Programming Problem (NLP). These NLPs can be solved using well-known optimization techniques, such as Trust Region Method, Nelder–Mead Method, or Interior Point Methods [2]. The optimal solution is found by transcribing an optimization problem from infinite-dimensional to finite-dimensional, according to these techniques. Indirect methods are based on the first-order necessary conditions retrieved by direct application of Pontryagin Minimum Principle (PMP)
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