The paper considers some control problems for systems described on infinite-dimensional spaces. A mathematical framework in terms of semigroups is developed which enables the generalisation of the finite-dimensional results to infinite dimensions, and which includes partial differential equations and delay equations as special cases. After first surveying some finite-dimensional results, abstract dynamical systems are introduced and the systems theory concepts of stability, controllability, observabiity and the linear quadratic problem are analysed. Throughout the paper the abstract theory is applied to a number of examples to obtain specific results.

Infinite-Dimensional Linear Systems Theory

[1] A new planetary atmospheric numerical model, “planetWRF,” has been developed by modifying the Weather Research and Forecasting (WRF) model. The model has generalized map projection, multiscale, and nesting capabilities, blurring the distinction between global and mesoscale models and enabling investigation of coupling between processes on all scales, including global. The model can also be run in one, two, or three dimensions. The conversion of the dynamical core for global application by altering the map projection grid and the boundary conditions as well as conversion of the physics parameterizations and constants for planetary application are described. Validation of the global dynamical core through use of standard forcing scenarios is presented. Example results from a series of simulations for Mars, Titan, and Venus are shown to demonstrate that the model performs well for a variety of planets and operating modes (microscale, mesoscale, and global scale).

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2006JE002825

PlanetWRF: A general purpose, local to global numerical model for planetary atmospheric and climate dynamics

Full von Karman system accounting for in-plane accelerations and describing the transient deformations of a thin, elastic plate subject to edge loading is considered. The energy dissipation is introduced via the nonlinear velocity feedback acting on a part of the edge of the plate. It is known [J. Puel and M. Tucsnak, SIAM J. Control Optim., 33 (1995), pp. 255--273] that in the case of linear dissipation and "star-shaped" domains, boundary velocity feedback with the tangential derivatives of horizontal displacements leads to the exponential decay rates for the energy of the resulting closed loop system. The main goal of the paper is to derive the uniform energy decay rates valid for the model without the above-mentioned restrictions. In particular, it is shown that simple, monotone nonlinear feedback (without the tangential derivatives of the horizontal displacements) provides the uniform decay rates for the energy in the absence of geometric hypotheses imposed on the controlled part of the boundary. This is accomplished by establishing, among other things, "sharp" regularity results valid for the boundary traces of solutions corresponding to this nonlinear model and by employing a Holmgren-type uniqueness result proved recently in [V. Isakov, J. Differential Equations, 97 (1997), pp. 134--147] for the dynamical systems of elasticity which are overdetermined on the boundary.

Uniform Stabilizability of a Full Von Karman System with Nonlinear Boundary Feedback

Bilinear Optimal Control of the Velocity Term in a Kirchhoff Plate Equation

The boundary stabilization of a nonlinear plate model is studied. The equations used take in consideration the in-plane accelerations and the rotary inertia of the cross sections. Applying linear feedbacks, the authors obtain the exponential decay of the energy.

Boundary Stabilization for the von Karman Equations

The EPIC atmospheric model with an isentropic/terrain-following hybrid vertical coordinate

Bilinear optimal control of a Kirchhoff plate

Abstract We consider a bilinear optimal control problem where the state equation is a Kirchhoff plate equation. The control acts as a multiplier of the velocity term. We prove the existence of an optimal control in a class h ∈ UM = {h ∈ L∞(0, T); −M ≤ h(t) ≤ M} and uniqueness of this optimal control for T sufficiently small.

Bilinear spatial control of the velocity term in a Kirchhoff plate equation.

Global Decay Rates for the Solutions to a Von Kármán Plate without Geometric Conditions

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