Solving 1D Conservation Laws Using Pontryagin's Minimum Principle

This paper discusses a connection between scalar convex conservation laws and Pontryagin's minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar space-independent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the Lax--Oleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin's minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both space-dependent and space-independent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with space-independent flux functions

Optimal control problems with delays in state and control and mixed control-state constraints

Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control-state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods for the delayed control problem are discussed which amount to solving a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and two numerical examples from chemical engineering and economics illustrate the results

Pontryagin's Minimum Principle and Forward-Backward Sweep Method for the System of HJB-FP Equations in Memory-Limited Partially Observable Stochastic Control

Memory-limited partially observable stochastic control (ML-POSC) is the stochastic optimal control problem under incomplete information and memory limitation. In order to obtain the optimal control function of ML-POSC, a system of the forward Fokker-Planck (FP) equation and the backward Hamilton-Jacobi-Bellman (HJB) equation needs to be solved. In this work, we firstly show that the system of HJB-FP equations can be interpreted via the Pontryagin's minimum principle on the probability density function space. Based on this interpretation, we then propose the forward-backward sweep method (FBSM) to ML-POSC, which has been used in the Pontryagin's minimum principle. FBSM is an algorithm to compute the forward FP equation and the backward HJB equation alternately. Although the convergence of FBSM is generally not guaranteed, it is guaranteed in ML-POSC because the coupling of HJB-FP equations is limited to the optimal control function in ML-POSC

Shoot-1.1 Package - User Guide

This package implements a shooting method for solving boundary value problems, for instance resulting of the application of Pontryagin's Minimum Principle to an optimal control problem. The software is mostly Fortran90, with some third party Fortran77 codes for the numerical integration and non-linear equations system. Its features include the handling of right hand side discontinuities (such as caused by a bang-bang control) for the integration of the trajectory and the computation of Jacobians for the shooting method. The particular case of singular arcs for optimal control problems is also addressed