Final-state, Open-loop Control of Parabolic PDEs with Dirichlet Boundary Conditions

TL;DR

The paper tackles final-state open-loop control for parabolic PDEs with Dirichlet BC using time-dependent point controls. It reduces the infinite-dimensional problem to finite-dimensional Galerkin approximations, yielding an explicit Lyapunov-based solution for the unconstrained case and a Lagrangian-duality approach for peak-value constrained problems. The authors prove convergence of the approximations to the true optimum and validate the method with 1D and 2D heat equation examples, showing that larger penalty weights $ ho_F$ improve final-state accuracy and that dual-based solutions are nearly optimal. This yields computationally tractable open-loop control schemes with practical applicability to heat-like processes.

Abstract

In this paper, a quadratic optimal control problem is considered for second-order parabolic PDEs with homogeneous Dirichlet boundary conditions, in which the "point" control function (depending only on time) constitutes a source term. These problems involve choosing a control function (with or without "peak-value" constraints) to approximately steer the solution of the PDE in question to a desired function at the end of a prescribed (finite) time-interval. To compute approximations to the desired optimal control functions, semi-discrete, Galerkin approximations to the equation involved are introduced and the corresponding (approximating) control problems are tackled. It is shown that the sequences of solutions to both the constrained and unconstrained approximating (finite-dimensional) control problems converge, respectively, to the optimal solutions of the control problems involving the original initial/boundary value problem. The solution to the unconstrained approximating problem can be quite explicitly characterized, with the main numerical step for its computation requiring only the solution of a Lyapunov equation. Whereas approximate solutions to the constrained control problems can be obtained on the basis of Lagrangian duality and piecewise linear multipliers. These points are worked out in detail and illustrated by numerical examples involving the heat equation (HEq).

Figures (11)

• Figure 1: $\theta_{r}$: target final state.
• Figure 2: $\boldsymbol{\beta}_{_{\boldsymbol{S}}}$: control-to-state actuator.
• Figure 3: Control signals $\boldsymbol{u}_{_{K}}$ (blue dashed), $\boldsymbol{u}_{_{R}}^{^{K}}$ (red solid) for $\rho_F=2000$.
• Figure 4: Approximations to target final state for $\rho_{_{F}}=2000$.
• Figure 5: Control signals $\boldsymbol{u}_{_{K}}$ (blue dashed), $\boldsymbol{u}_{_{R}}^{^{K}}$ (red solid) for $\rho_{_{F}}=4000$.

Theorems & Definitions (14)

• Proposition 3.1
• Remark 3.1
• Remark 4.1
• Proposition 4.1
• Proposition 4.2
• Remark 4.2
• Remark 4.3
• Proposition 4.3
• Proposition 4.4
• Remark 4.4