On the Approximation of Operator-Valued Riccati Equations in Hilbert Spaces

Abstract

In this work, we present an abstract theory for the approximation of operator-valued Riccati equations posed on Hilbert spaces. It is demonstrated here that the error of the approximate solution to the operator-valued Riccati equation is bounded above by the approximation error of the governing semigroup, under the assumption of boundedness on the semigroup and compactness on the coefficient operators. One significant outcome of this result is the correct prediction of optimal convergence for finite element approximations of the operator-valued Riccati equations for when the governing semigroup involves parabolic, as well as hyperbolic processes. We derive the abstract theory for the time-dependent and time-independent operator-valued Riccati equations in the first part of this work. In the second part, we derive optimal error estimates for the finite element approximation of the functional gain associated with model weakly damped wave and thermal LQR control systems. These theoretical claims are then corroborated with computational evidence.

Figures (5)

• Figure 1: Numerical convergence plot with $\epsilon \in (0, 1]$ for the scalar-valued Riccati equation approximation \ref{['eqn: scalar Riccati equation']} with $a,f,g = 1$.
• Figure 2: Approximation error in $\left\| \kappa - \kappa_n \right\|_H$ for the finite element approximation of the thermal control system, where $H=L^2(\Omega)$.
• Figure 3: Approximation error in $\left\| \kappa - \kappa_n \right\|_H$ for the finite element approximation of the two-dimensional thermal control system, where $H = L^2(\Omega)$.
• Figure 4: Approximation error in $\left\| \kappa - \kappa_n \right\|_H$ for the finite element approximation of the weakly damped wave control system, where $H=H^1_0(\Omega) \times L^2(\Omega)$ in the left plot and $H=L^2(\Omega)\times L^2(\Omega)$ in the right figure.
• Figure 5: Left: Error of the functional gain approximation for the two-dimensional thermal model problem for when $b,q$ are Gaussian functions. Right: Error of the functional gain approximation for the 1D thermal model problem for when $b(x) = \delta(x)$.

Theorems & Definitions (51)

• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1
• Theorem 1: Trotter-Kato ito1998trotter
• Theorem 2: Burns-Rautenberg
• Proposition 3: Equivalence of Bochner Integral and Differential Form
• Remark 2
• proof