“…As an application example we consider the driven cavity as described in Sect. 6.1 and the task of finding inputs that cause a prescribed output y * in the domain of observation via the solution of (27). Figure 6 shows the result for y * ≡ [1 0] T .…”
Section: Application To Optimal Flow Controlmentioning
“…Our new approach is based on a recently introduced general semigroup approach [27,43] for linear time invariant (LTI) systems of abstract ordinary differential equations. Here, we will extend this approach to abstract differential-algebraic systems and with this the applicability to flow control problems.…”
The construction of reduced order models for flow control via a direct discretization of the input/output behavior of the system is discussed. The spatially discretized equations are linearized such that an explicit formula for the corresponding input/output map can be used to generate a matrix representation of the input/output map. Estimates for the approximation error are derived and the applicability is illustrated via a numerical example for the control of a driven cavity flow.
“…The presented framework can also be used for linearized flow systems. For the Stokes equation, results similar to (3) and (6) are obtained by working with appropriate subspaces of divergence-free functions  and for the spatially discretized Oseen equations, which arise as linearizations of the Navier-Stokes equations, it has been shown in [19,20] how the framework can be extended to linear time invariant descriptor systems.…”
Section: I/o Maps Of ∞-Dimensional Lti State Space Systemsmentioning
We present a framework for the direct discretization of the input/output map of dynamical systems governed by linear partial differential equations with distributed inputs and outputs. The approximation consists of two steps. First, the input and output signals are discretized in space and time, resulting in finite dimensional spaces for the input and output signals. These are then used to approximate the dynamics of the system. The approximation errors in both steps are balanced and a matrix representation of an approximate input/output map is constructed which can be further reduced using singular value decompositions. We present the discretization framework, corresponding error estimates, and the SVD-based system reduction method. The theoretical results are illustrated with some applications in the optimal control of partial differential equations. Keywords input/output maps, discretization, control of partial differential equations AMS subject classification. 39C20, 35B37.
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