We consider the data-driven approximation of Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is estimation error if data are collected from long-term ergodic simulations. derive both an exact expression variance cross-covariance operator, measured in Hilbert-Schmidt norm, and probabilistic bounds finite-data error. Moreover, we a bound prediction observables RKHS using finite Mercer series expansion. Further, assuming...

Abstract The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic the and prediction depending on number training data points, both ordinary stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these by...

Abstract Adaptive structures are equipped with sensors and actuators to actively counteract external loads such as wind. This can significantly reduce resource consumption emissions during the life cycle compared conventional structures. A common approach for active damping is derive a port-Hamiltonian model employ linear-quadratic control. However, quadratic control penalization lacks physical interpretation merely serves regularization term. Rather, we propose controller, which achieves...

We analyze the sensitivity of extremal equations that arise from first order optimality conditions for time dependent optimization problems. More specifically, we consider parabolic PDEs with distributed or boundary control and a linear quadratic performance criterion. prove solution's boundedness respect to right-hand side condition which includes initial data. If system fulfills particular stabilizability detectability assumption, bound is independent horizon. As consequence, influence...

While Koopman-based techniques like extended Dynamic Mode Decomposition are nowadays ubiquitous in the data-driven approximation of dynamical systems, quantitative error estimates were only recently established. To this end, both sources resulting from a finite dictionary and finitely-many data points generation surrogate model have to be taken into account. We generalize rigorous analysis control setting while simultaneously reducing impact curse dimensionality by using proposed bilinear...

The Koopman operator serves as the theoretical backbone for machine learning of dynamical control systems, where is heuristically approximated by extended dynamic mode decomposition (EDMD). In this paper, we propose Stability- and certificate-oriented EDMD (SafEDMD): a novel EDMD-based architecture which comes along with rigorous certificates, resulting in reliable surrogate model generated data-driven fashion. To ensure trustworthiness SafEDMD, derive proportional error bounds, vanish at...

We present a method to design state-feedback controller ensuring exponential stability for nonlinear systems using only measurement data.Our approach relies on Koopmanoperator theory and uses robust control explicitly account approximation errors due finitely many data samples.To simplify practical usage across various applications, we provide tutorial-style exposition of the feedback its guarantees single-input systems.Moreover, extend this multi-input more flexible controllers...

Extended dynamic mode decomposition (EDMD) is a popular data-driven method to predict the action of Koopman operator, i.e., evolution an observable function along flow dynamical system. In this paper, we leverage recently-introduced kernel EDMD for control systems model predictive control. Building upon pointwise error bounds proportional in state, rigorously show practical asymptotic stability origin w.r.t. MPC closed loop without stabilizing terminal conditions. The key novelty that avoid...

We study the problem of state transition on a finite time interval with minimal energy supply for linear port-Hamiltonian systems. While cost functional is intrinsic to structure, necessary conditions optimality resulting from Pontryagin's maximum principle may yield singular arcs. The underlying reason dependence control, which makes determining optimal control as function and adjoint more complicated or even impossible. To resolve this issue, we fully characterize regularity...

We analyze the robustness of optimally controlled evolution equations with respect to spatially localized perturbations. prove that if involved operators are domain-uniformly stabilizable and detectable, then these perturbations only have a local effect on optimal solution. characterize this domain-uniform stabilizability detectability for transport equation constant velocity, showing even unitary semigroups, optimality implies exponential damping. Finally, we extend our result case...

For linear-quadratic optimal control problems (OCPs) governed by elliptic and parabolic partial differential equations, we investigate the impact of perturbations on solutions. Local may occur, e.g., due to discretization optimality system or disturbed problem data. Whereas these exhibit global effects in uncontrolled case, prove that ramifications are exponentially damped space under stabilizability detectability conditions. To this end, a bound condition’s solution operator is uniform...

Recently, domain-uniform stabilizability and detectability has been the central assumption %in order robustness results on to ensure in sense of exponential decay spatially localized perturbations optimally controlled evolution equations. In present paper we analyze a chain transport equations with boundary point controls regard this property. Both for Dirichlet Neumann coupling conditions, show necessary sufficient criterion control domains which allow stabilization equation. We illustrate...

We examine the minimization of a quadratic cost functional composed output and final state abstract infinite-dimensional evolution equations in view existence solutions optimality conditions. While initial value is prescribed, we are minimizing over all inputs within specified convex subset square integrable controls with values Hilbert space. The considered class systems based on system node formulation. Thus, our developed approach includes optimal control wide variety linear partial...

We consider the singular optimal control problem of minimizing energy supply linear dissipative port-Hamiltonian descriptor systems. study reachability properties system and prove that states exhibit a turnpike behavior with respect to conservative subspace. Further, we derive input-state toward subspace for ordinary differential equations feed-through term property corresponding adjoint zero. In an appendix characterize class Hamiltonian matrices pencils.

Abstract Classical turnpikes correspond to optimal steady states which are attractors of infinite-horizon control problems. In this paper, motivated by mechanical systems with symmetries, we generalize concept manifold turnpikes. Specifically, the necessary optimality conditions projected onto a symmetry-induced coincide those reduced-order problem defined on under certain conditions. We also propose sufficient for existence based tailored notion dissipativity respect manifolds. Furthermore,...