Core Concepts
The core message of this article is that the optimal control problem governed by parabolic equations with measure-valued controls over time can be well-posed, and the optimal control exhibits a sparsity structure independent of space.
Abstract
The article investigates an optimal control problem governed by parabolic equations with measure-valued controls over time. The key highlights and insights are:
The authors establish the well-posedness of the optimal control problem and derive the first-order optimality condition using Clarke's subgradients, revealing a sparsity structure in time for the optimal control.
The optimal control problems represent a generalization of impulse control for evolution equations, where the control can be optimized in both time and magnitude.
The authors employ the space-time finite element method to discretize the optimal control problem, using piecewise linear and continuous finite elements in space and a Petrov-Galerkin method in time.
Error estimates are derived for the finite element approximations of the state and adjoint equations, and weak-* convergence is established for the control under the norm M(Ic̄; L2(ω)), with a convergence order of O(h^(1/2) + τ^(1/4)) for the state.