Constrained LQR problems governed by the potential equation on Lipschitz domains with point observations
Using layer potential theory and boundary element methods, the authors study linear quadratic regulator problems governed by the potential equation on Lipschitz domains, with point observations on the boundary. First, the linear quadratic regular problems without control constraints are studied. It is proved that in \(\mathbb{R}^3\), the linear quadratic regulator problem generally does not have any nontrivial optimal control, whereas in \(\mathbb{R}^2\), it has a unique optimal control. The explicit characterization and regularity of the optimal control are derived. Then the linear quadratic regulator problems with control constraints are discussed. In both \(\mathbb{R}^2\) and \(\mathbb{R}^3\), it is obtained the existence, uniqueness, regularity and an explicit characterization of the optimal control. Finally a decomposition of the optimal control of linear quadratic regulator problems on polygonal domains is established, which is very useful in the regularity analysis and numerical computation. Based upon the decomposition formula, a numerical algorithm insensitive of the boundary discretization is developed. Several numerical examples are included to illustrate the results.
scientific article; zbMATH DE number 891044