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Abstract. In this paper we consider the convergence of the infinite dimensional version of the Kleinman-Newton algorithm for solving the algebraic Riccati operator equation associated with the linear quadratic regulator problem in a Hilbert space. We establish mesh independence for this algorithm and apply the result to systems governed by delay equations. Numerical examples are presented to illustrate the results. There are two basic issues that need to be addressed in developing practical numerical approximations for control. First, it is essential that the approximation scheme leads to finite dimensional approximating Riccati equations that converge (under mesh refinement) to the solution of the infinite dimensional Riccati equation. This is a well-studied problem (see [7], [14], [26], [33], and [43]). It is now well known that to obtain norm convergence for the Riccati equation, the approximation scheme must satisfy some form of convergence, dual convergence, and uniform preservation of stabilizability and detectability under mesh refinement (see [7] and [33]). These concepts will be made more precise in section 7.1. The important point here is that many "standard" convergent approximation schemes do not satisfy all the conditions necessary for norm convergence of the Riccati operators (see [16]). If this issue is ignored when one develops an approximation scheme for control design and optimization, then the resulting numerical algorithm can fail to produce accurate and useful results (see the numerical examples in section 9). In this paper we show that these properties are also key ingredients in establishing mesh independence of Newton-type algorithms.

The problem of active feedback control of fluid flows falls into a class of problems in the area of distributed parameter control. Distributed parameter systems are typically defined by partial differential equations that model the time and spatial evolution of the process. We consider the problem of locating sensors for effective feedback control of a fluid flow problem described by the Navier-Stokes equations. In this setting, the state of the system is the velocity field v(t, x), and hence all feedback laws are a function of this velocity field or, in most practical settings, a function of sensor outputs. In many designs, the feedback control law can be represented as a linear function of the state defined by an integral operator with a kernel function called the functional gain. In this paper we show that these functional gains can be used to determine effective sensor placement in complex flow control applications. The approach is to choose measurements of the state that would provide good quadrature points for the integral operator. We provide a computational validation of this approach by controlling the vortex shedding in a two-dimensional cylinder flow using a pair of fluid actuators on the cylinder surface. This model is linearized about the mean flow and a feedback control is designed by pole placement. Distributed parameter control theory yields the existence and form of the functional gains which are used to locate sensors. In particular, we use the location of the supports of the functional gains to determine two sets of four sensor locations in the wake. One of these measurement sets coincides with large magnitudes of the gain and the other set coincides with small magnitudes. Numerical experiments with a reduced-order model confirm superior performance of the closed-loop (CL) system using the former sensor set. We also show that choosing sensor locations associated with small magnitudes of the functional gains actually destabilizes the CL system.

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