Boundary Control of the Wave Equation via Linear Quadatic Regulation

TL;DR

This paper develops explicit solutions for boundary control of the 1D wave equation using Linear Quadratic Regulation, deriving Riccati PDEs for Dirichlet and Neumann controls through a straightforward completing-the-square method.

Contribution

It provides explicit Riccati PDE solutions for boundary control of the wave equation under both Dirichlet and Neumann conditions, advancing control design methods.

Findings

Explicit Riccati PDE solutions for boundary control

Unified approach for Dirichlet and Neumann controls

Simplified derivation via completing the square

Abstract

We consider the Linear Quadratic Regulation for the boundary control of the one dimensional linear wave equation under both Dirichlet and Neumann activation. For each activation we present a Riccati partial differential equation that we explicitly solve. The derivation the Riccati partial differential equations is by the simple and explicit technique of completing the square.