Related papers: On the Approximation of Operator-Valued Riccati Eq…
For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple…
We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a…
In this chapter, the Hilbert space framework in the mathematical theory of composite materials is introduced for studying the properties of effective operators. The goal is to introduce some of the key concepts and fundamental theorems in…
In this paper, we extend a classical approach to linear quadratic (LQ) optimal control via Popov operators to abstract linear differential-algebraic equations (ADAEs) in Hilbert spaces. To ensure existence of solutions, we assume that the…
In this study, we provide an interpretation of the dual differential Riccati equation of Linear-Quadratic (LQ) optimal control problems. Adopting a novel viewpoint, we show that LQ optimal control can be seen as a regression problem over…
We consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The control will be considered in the energy norm of the anisotropic Sobolev space $[H_{0;,0}^{1,1/2}(Q)]^\ast$, such that the state…
Linear fixed point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random…
We examine the minimization of a quadratic cost functional composed of the output and the final state of abstract infinite-dimensional evolution equations in view of existence of solutions and optimality conditions. While the initial value…
We study decay rates for bounded $C_0$-semigroups from the perspective of $L^p$-infinite-time admissibility and related resolvent estimates. In the Hilbert space setting, polynomial decay of semigroup orbits is characterized by the…
We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$…
An abstract nonautonomous parabolic linear-quadratic regulator problem with very general final cost operator P_T is considered, subject to the same assumptions under which a classical solution of the associated differential Riccati equation…
We consider continuous structures which are obtained from finite dimensional Hilbert spaces over $\mathbb{C}$ by adding some unitary operators. Quantum automata and circuits are naturally interpretable in such structures. We consider…
Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is a sum of operators…
The long time behavior and detailed convergence analysis of Langevin equations has received increased attention over the last years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic…
Discrete algebraic Riccati equations and their fixed points are well understood and arise in a variety of applications, however, the time-varying equations have not yet been fully explored in the literature. In this article we provide a…
An important class of physical systems that are of interest in practice are input-output open quantum systems that can be described by quantum stochastic differential equations and defined on an infinite-dimensional underlying Hilbert…
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…
Regularization methods have been recently developed to construct stable approximate solutions to classical partial differential equations considered as final value problems. In this paper, we investigate the backward parabolic problem with…
We consider a family of second-order parabolic operators $\partial_t+\mathcal{L}_\varepsilon$ in divergence form with rapidly oscillating, time-dependent and almost-periodic coefficients. We establish uniform interior and boundary H\"older…
A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati…