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Extension of Switch Point Algorithm to Boundary-Value Problems

William W. Hager

TL;DR

This work extends the Switch Point Algorithm from initial-value optimal control problems to boundary-value problems with both initial and terminal constraints. By deriving a derivative formula for the objective with respect to switch points that equals the jump in the Hamiltonian at each switch, and establishing existence and stability results under invertibility of key state-transition and costate-transition matrices, the authors enable gradient-based optimization in boundary-value settings. The analysis accommodates singular controls and includes an extension to cases where singular controls depend on both state and costate via a generalized Hamiltonian. Practical algorithms are outlined, including Newton-type updates for switch points and Euler-based initialization with total variation regularization, with applications relevant to boundary-value problems such as fish harvesting.

Abstract

In an earlier paper (https://doi.org/10.1137/21M1393315), the Switch Point Algorithm was developed for solving optimal control problems whose solutions are either singular or bang-bang or both singular and bang-bang, and which possess a finite number of jump discontinuities in an optimal control at the points in time where the solution structure changes. The class of control problems that were considered had a given initial condition, but no terminal constraint. The theory is now extended to include problems with both initial and terminal constraints, a structure that often arises in boundary-value problems. Substantial changes to the theory are needed to handle this more general setting. Nonetheless, the derivative of the cost with respect to a switch point is again the jump in the Hamiltonian at the switch point.

Extension of Switch Point Algorithm to Boundary-Value Problems

TL;DR

This work extends the Switch Point Algorithm from initial-value optimal control problems to boundary-value problems with both initial and terminal constraints. By deriving a derivative formula for the objective with respect to switch points that equals the jump in the Hamiltonian at each switch, and establishing existence and stability results under invertibility of key state-transition and costate-transition matrices, the authors enable gradient-based optimization in boundary-value settings. The analysis accommodates singular controls and includes an extension to cases where singular controls depend on both state and costate via a generalized Hamiltonian. Practical algorithms are outlined, including Newton-type updates for switch points and Euler-based initialization with total variation regularization, with applications relevant to boundary-value problems such as fish harvesting.

Abstract

In an earlier paper (https://doi.org/10.1137/21M1393315), the Switch Point Algorithm was developed for solving optimal control problems whose solutions are either singular or bang-bang or both singular and bang-bang, and which possess a finite number of jump discontinuities in an optimal control at the points in time where the solution structure changes. The class of control problems that were considered had a given initial condition, but no terminal constraint. The theory is now extended to include problems with both initial and terminal constraints, a structure that often arises in boundary-value problems. Substantial changes to the theory are needed to handle this more general setting. Nonetheless, the derivative of the cost with respect to a switch point is again the jump in the Hamiltonian at the switch point.
Paper Structure (9 sections, 4 theorems, 64 equations)

This paper contains 9 sections, 4 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. An Existence Result
  3. Stability with Respect to Terminal Constraint
  4. Stability with respect to the Switch Point
  5. Objective Derivative with Respect to Switch Point
  6. Singular Control Depending on Both State and Costate
  7. Algorithms
  8. Conclusions
  9. Acknowledgments

Key Result

proposition 1

Suppose that ${\bf{g}} : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously differentiable in ${\cal {B}}_r({\bf{0}})$ for some $r > 0$, and define $\delta =$$\|{\bf{g}}({\bf{0}})\|$. Let ${\cal {L}} \in \mathbb{R}^{n \times n}$ be an invertible matrix with $\gamma := \|{\cal {L}}^{-1}\|$ and wi If $\epsilon \gamma < 1$ and $\delta \le r(1-\gamma \epsilon)/\gamma$, then there exists a unique $

Theorems & Definitions (7)

  • proposition 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • theorem 1
  • proof