Flow Control

E-mail to Max Gunzburger

Collaborators

The major portion of our efforts in analysis, in algorithmic development, and in computations connected with flow control and optimization has been directed at incompressible viscous flows as typified by the Navier-Stokes system.We have treated both steady state and unsteady problems. We have considered numerous objective functionalsdefined with respect to either a portion of the flow domain or a portion of the boundary; these include flow matching, drag minimization, rotation minimization, stress matching, temperature matching, temperature gradient minimization, etc. Our analyses, numerical analyses, and computations have addressed all three types of controls: distributed controls, e.g., body forces due to magnetic fields or heat sources, boundary value controls, e.g., injection or suction of fluid through orificies in the boundary, heat flux or temperature on portions of the boundary, etc., and shape controls, i.e., the shape of a portion of the boundary of the flow domain.

Our work in the optimal control of viscous incompressible flows includes the following aspects:

• the mathematical analyses of the optimal control problems;
• the definition and analysis of finite element methods for the approximate solution of the optimal control problems;
• the implementation of the finite element methods;
• the study, through extensive computational experiments, of issues and difficulties that arise in the practice of obtaining approximate solutions of optimal control problems;
• the application of the finite element codes to the solution of specific interesting and practical problems.

• the setup of precise mathematical models of optimal control problems; this includes the definition of objective functionals corresponding to physical objectives and, in a function space setting, of the sets of admissible states and controls;
• proving the existence of solutions of optimal control problems;
• rigorously justifyiing the use of the Lagrange multiplier rule to enforce the constraints, i.e., the Navier-Stokes system;
• applying the Lagrange multiplier rule to obtain optimality systems of partial differential equations whose solutions provide optimal states and controls;
• studying the regularity of solutions of the optimality systems;
• developing finite element methods for approximating solutions of the optimality systems;
• proving the convergence and deriving error estimates for the finite element discretizations;
• developing solutions algorithms, e.g., gradient methods, for the solution of the discrete optimality systems and rigorously examining the convergence properties of these algorithms.

Here, we give a very brief account of some of the analytical and computational studies we have done in connection with the optimal control of viscous incompressible flows.

Analysis

Boundary value control problems for the Navier-Stokes equations

The control of fluid flows has emerged as an important area of technological and scientific research. From a rigorous mathematical point of view, much of that study has been directed at stationary problems, or at time dependent problems with specially structured controls, mostly of the distributed type. We have addressed the interesting problems of distributed and boundary control problems for the two and three dimensional Navier-Stokes equations of unsteady, viscous, incompressible flow. The allowable controls are quite general. One thrust of out efforts has been to consider settings of minimal regularity, i.e., the state and control spaces are chosen to be as large as possible. We have carried out the analyses of the existence of optimal solutions and of the existence of Lagrange multipliers and have derived the optimality system for a drag minimization problem. In so doing, we have also derived some new results about the Navier-Stokes equations, especially related to the regularity of allowable boundary data. A second thrust of our efforts involves more practical settings, i.e., chosing state and control spaces that are more amenable to efficient and relatively simple computations. In these cases, we have treated both boundary value and distributed controls and we have not only carried out analyses similar to those for the first thrust, but we have also defined and analyzed finite element discretizations of the optimality systems and developed and analyzed gradient methods for the solution of the discrete optimality system. We have also analyzed optimal control problems and their approximation in the steady state setting. Here, we have treated numerous objective functionals and boundary value and distributed controls.

Shape control problems for the Navier-Stokes equations

Shape controls are of great interest in the control of fluid flows. For example, the problem of the optimal design of wings is a optimal shape control problem. There have been substantial analyses of linear shape control problems for partial differential equations. However, for the steady, nonlinear Navier-Stokes equations of incompressible flow, few rigorous analyses have been previously carried out. We have given such analyses in the context of a two-dimensional flow in a channel-like domain. Our rigorous analyses include showing, in the context of a drag minimization problem, that optimal solutions exist, showing that the Lagrange multiplier rule may be used to enforce constraints, deriving an optimality system from which optimal states and co-states may be deduced, deriving the shape gradient, and finally, deriving error estimates for finite element approximations of optimal states. We have also carried out extensive numerical experiments with shape control problems for incompressible viscous flows. In this study, we examined various issues related to local minima, functional regularization, non-feasible solutions, sensitivities, and the efficiency of computations. Our analyses and computations have involved both the sensitivity equation method and the adjoint or co-state method based optimization strategies.

Example applications

Flow matching in time-dependent, incompressible, viscous flows

As an example of the use of adjoint methods for optimal control problems for time-dependent, viscous, incompressible flows, we consider a simple flow matching problem. We have a given reference flow. We want to match, in an L² sense, the reference flow with a new flow having a different initial condition. We will force the new flow to match the reference flow through adjustments in the forcing function. (This setup is not very realistic, but serves to illustrate the effectiveness of optimal control strategies. Furthermore, we have done extensive computations with more realistic controls, e.g., heating and cooling and flow injection and suction.)

The mathematical problem at hand is to minimize the time integral over a specified time period of the L² norm of the difference between the velocity of the flow and that of the given reference flow. Candidate flows are required to satisfy the time-dependent Navier-Stokes system. Control is effected through the body force. The size of the control may be limited by either penalizing the cost functional or by imposing an explicit bound on an appropriate norm of the control.

We solve this problem by using an optimality system resulting from the application of the Lagrange mutiplier rule to enforce the Navier-Stokes system constraint. A gradient method is used to uncouple the forward in time state equation from the backward in time adjoint equation. As for may other problems, we have fully analyzed this problem. We have shown the existence of optimal solutions, justified the use of the Lagrange multiplier rule, derived the optimality system, and defined and analyzed semidiscrete in time and fully discrete finite element methods for determining approximate solutions.

Here, we present the results of one of the many computational experiments that we have performed.

The initial target and initial optimal velocity fields

The target and optimal velocity fields at t = 0.1

The target and optimal velocity fields at t = 0.3

The target and optimal velocity fields at t = 0.4

The target and optimal velocity fields at t = 0.5

From these figures we see that the optimal control strategy has indeed steered the flow towards the target flow. This can also be seen from a plot of the norm of the difference between the optimal and target flows.

The norm of the difference between the target and optimal velocities vs. time

Finally, we show the norm of the control.

The norm of the control vs. time

Instability suppression in transitional flows

As an another application of our theory for the control of time dependent viscous flows, we have developed a self-contained, automated methodology for flow control for the problem of boundary layer instability suppression. The objective of control is to match the stress vector along a portion of the boundary to a given vector; instability suppression is achieved by choosing the given vector to be that of a reference steady flow, e.g., Blasius flow for boundary layers. Control is effected through the injection or suction of fluid through a single orifice on the boundary. The present approach couples the time-dependent Navier-Stokes system with an adjoint Navier-Stokes system and optimality conditions from which optimal states, i.e., unsteady flow fields, and controls, e.g., actuators, may be determined. The results demonstrate that instability suppression can be achieved without any a priori knowledge of the disturbance, which is significant because other control techniques have required some knowledge of the flow unsteadiness such as frequencies, instability type, etc. The present methodology can be extended to three dimensions and may be directly applied to separation control, re-laminarization, and turbulence control applications using one to many sensors and actuators.

The specific problem setup is that of boundary layer flow at a high enough value of the Reynolds number so that Tollmien-Schlicting instability waves appear. We note at the outset that although we are treating a boundary layer flow, we use the full time-dependent Navier-Stokes system in our study.

The problem setup

Lagrange multipliers, or adjoint variables are introduced to enforce the constraints, i.e., the Navier-Stokes system. The optimality system then follows from the application of the Lagrange multiplier rule. The optimality system consists of:

• the Navier-Stokes system posed forward in time with initial conditions;
• the adjoint system, a system of partial differential equations posed backward in time with terminal conditions;
• the optimality condition that, due to the penalization of the functional, is a two-point boundary value problem in time for a second-order ordinary differential equation for the control with the adjoint variable appearing in the data.

Since the adjoint equations are well-posed backward in time, one cannot solve the coupled optimality system by marching all equations forward in time.

Our solution strategy for the optimality system is the obvious one. We make a guess for the control parameters and solve the Navier-Stokes equations forward in time. Using the flow solution so obtained, we then solve the adjoint system backwards in time. Using the adjoint variables, we can solve the optimality condition for a new guess of the control. This process is repeated until, hopefully, satisfactory convergence is achieved.

Convergence of the forward state-backward adjoint sweep strategy

It can be shown that this forward state-backward adjoint sweep is equivalent to a gradient method for the objective functional. There remain many difficulties, especially related to storage, connected to the forward state-backward adjoint sweep method; we have done many computational experiments testing different strategies for overcoming these difficulties.

We next see how effective the optimal control strategy is at suppressing the instabitility in the flow.

Comparison of the controlled and uncontrolled flows

We have many more computational results illustrating the effectiveness of both the optimal control strategy for suppressing instabilities and the forward state-backward adjoint sweep scheme for solving the optimality system.

We have devoted considerable effort to designing sensitivity equation algorithms for the optimal control and design of inviscid, compressible flows. Here, we only mention one simple example of an optimal design problem we have solved, i.e., flow matching through shape design, one example of a computational study of the important issue of flow sensitivities at surfaces in flows with discontinuities that arises in the optimal design of such flows, and one bit of analysis connected with the jump conditions for flow sensitivities.

Example application - Flow matching through optimal shape design

The problem is an "academic" version of a wind tunnel design problem. Given an obstacle and free stream flow conditions, we compute a reference flow, i.e., a target flow. We then try to match this flow along the computational outflow plane, in an L² sense, by a flow around a body that is half as long as the reference body.

The problem setup

The shortened body is described in terms of Bezier polynomials and is determined by choosing two Bezier parameters. (The positions of the left and right end points of the shortened body are fixed.)

Effect on the shortened body due to changes in the Bezier parameters

The inflow Mach number is a third design parameter.

This optimal design problem was solved using the sensitivity equation method.The flow simulation code used was the 2D, unstructured mesh version of the GASPcode of the Aerosoft Corp. This code was modified so that it computes flow sensitivities by differentiating the flow equations and then discretizing them using the same discretization method used for the flow itself. The computed flow sensitivities were used to determine an approximation the the gradient of the objective functional; the latter was chosen to be the L² norm, along the comparison plane, of the difference between the candidate flow solution and the reference flow solution. The optimization algorithm employed was the ACM TOMS 611 implementation of the BFGS quasi-Newton method, using the approximate gradients of the objective functional determined from the approximate sensitivities. The optimization algorithm converged after a handful of iterations.

The target flow

The initial flow used in the optimization

The optimal flow

The target, initial, and optimal flows side-by-side

One important observation about optimization for flows with discontinuities is that the optimizer finds the correct position for the shock wave almost immediately, i.e., in only one or two iterations. Trying to match the smooth part of the flow takes more iterations. The reason for this is that the flow sensitivities are very large at surfaces of discontinuity of the flow, and thus those discontinuities dominate the early part of the iteration.

Computational study - Sensitivities in flows with discontinuities

The flow sensitivity equations are derived by differentiating the flow equations with respect to a parameter. This can be done at the level of the partial differential equations or at the level of a discretization of those equations. Supersonic, inviscid, compressible flows usually contain discontinuities, e.g., shock waves and contact discontinuities. Strictly speaking, the flow equations cannot be differentiated across these discontinuities; we can only derive the sensitivity equations for regions in which the flow is smooth. In most optimization settings, the flow equations are differentiated without regard to the presence of discontinuities in the flow. Indeed, in the example discussed above, we ourselves have done so. Of course, differentiating across a discontinuity produces a delta function which accounts for the "large" sensitivities, e.g., at shock waves, faced by the optimizer. We can see this large sensitivity explicitly in the following plot related to the example discussed above.

A flow sensitivity along the comparison plane

We have performed extensive computational studies of the effects this crime, i.e., differentiating across flow discontinuties, has on sensitivity approximations and on the performance of optimization methods used in optimal design and control. Here, we discuss a portion of those studies in the context of the Reimann problem of gas dynamics.

The Reimann problem provides a very good setting for studying flow sensitivities because an exact solution for the flow variables is available.

The exact solution of the Reimann problem

Furthermore, the exact solution may be differentiated within smooth parts of the flows to determine the exact flow sensitivities. Then, we may compare the results of computations with these exact result.

Exact sensitivities for the Reimann problem

We have determined approximate sensitivities using three different flow approximation methods:

• Lax-Wendroff
• Godunov
• Roe,

three different methods for defining approximate sensitivities:

• use multiple flow solves in a finite difference quotientapproximation of the sensitivity derivatives
• use the ADIFORpackage (an automatic differentiation code) to differentiate the discrete flow equations
• use the sensitivity equation method, i.e., differentiate the flow equations and then discretize them by the same method used to determine the approximate flow,

and three different grids:

• relatively fine
• relatively medium
• relatively coarse.

Note that the finite difference quotient and the ADIFOR approaches to defining sensitivities are examples of discretize-then-differentiate approaches while the sensitivity equation method is an example of the differentiate-then-discretize approach.

We use the fine grid to compare the three ways of defining approximate sensitivities for each of the three numerical schemes:

Approximate sensitivities for the Lax-Wendroff method

Approximate sensitivities for the Godunov method

Approximate sensitivities for the Roe method.

Comparing the results for the different numerical schemes, we see that all three do a poor job of approximating sensitivites; this is true for all three methods of defining approximate sensitivities. It does seem that the Roe scheme, the most accurate one as far as resolving flow discontinuites, produces the largest spike-like approximations to delta functions and has less polution effects into the smooth regions of the flow. However, there seems to be no difference between using a discretize-then-differentiate or a differentiate-the-discretize approach to determining approximate sensitivities.

We use the Lax-Wendroff method to examine the effect of grid size on the three ways of defining approximate sensitivities:

Approximate sensitivities for the Lax-Wendroff method on the medium grid

Approximate sensitivities for the Lax-Wendroff method on the coarse grid.

Comparing the results for the different grids, we see that the finer the grid, the better is the approximation to the delta function and the better is the approximation in the smooth parts of the flow. However, for any case, the sensitivities are nowhere good enough to use for, e.g., calculating a nearby flow, i.e., flows for perturbed values of a parameter.

On the other hand, if we use the sensitivites determined by any of the above combination of approximate sensitivity definitions, discretization methods, and grid sizes in an optimization setting, one finds that the optimizer does quite well. A natural question to ask is

how can such awful approximate sensitivites be useful in the optimization setting?

The answer seems to be that the optimizer only cares about receiving enough information so that it can determine a step that "substantially" reduces the objective functional; evidently, even the inaccurate sensitivities are good enough for this. We have, in fact, used the exact sensitivities in an optimization setting and have observed that only modest gains in optimizer performance are achieved. One should also note that, at least in the early stages of optimization, the optimizer likes to see the spikes in the sensitivities. This allows the optimizer to quickly find the correct position of flow discontinuities. However, at the later stages of optimization during which the optimizer is trying to find the smooth parts of the optimal flow, the inaccurate sensitivities have a detrimental effect on the convergence.

We have examined numerous other issues related to sensitivities for discontinuous flows including how to find more accurate sensitivities in regions of smooth flow, how to more accurately determine perturbed flows using sensitivities, and other questions related to the use of approximate sensitivities within a flow optimization setting.

Analysis - Jump conditions for sensitivities at discontinuities in the flow

The sensitivity equations for invscid, compressible flows are a system of linear (in the sensitivities) hyperbolic partial differential equations. The flow variables appear in the coefficients of the sensitivity system. If the flow variables are discontinuous, e.g., across a shock, then the coefficients in the sensitivity equations are discontinuous. Linear hyperbolic partial differential equations have solutions for anyprescribed jump in the independent variables, in our case, the sensitivities, across surfaces of discontinuity in the coefficients. A natural question to ask is:

what jump conditions for the sensitivities should be imposed across shock waves and other surfaces of discontinuity in its coefficients?

The Reimann problem provides a good context for studying this question since the exact sensitivities may be determined from the known exact solution. We have shown, in this context, that

the correct jump conditions for the sensitivities across surfaces of discontinuity in the flow are found by differentiating the Rankine-Hugoniot conditions for the flow variables.

In particular, the exact sensitivities satisfy these differentiated Rankine-Hugoniot conditions. In addition,

the differentiated Rankine-Hugoniot conditions provide the shock position sensitivity,

i.e., the derivative of the shock position with respect to a parameter.

Analysis

The need to solve optimization and control problems arises in many settings. Although in some cases these problems may be easily solved, either analytically or computationally, in many other cases substantial difficulties are encountered. For example, candidate optimal states and controls may belong to infinite dimensional function spaces and one may have to minimize a nonlinear functional of the state and control variables subject to nonlinear constraints that take the form of a system of partial differential equations whose solutions are in general not unique. Our goal, which we have reached, was to construct, analyze, and apply a framework for the approximate solution of many such problems. The setting for our framework is a class of nonlinear control or optimization problems which is general enough to be of use in numerous applications. The major steps in the development and analysis of our framework are as follows:

1. define an abstract class of nonlinear control or optimization problems;
2. show that, under certain assumptions, optimal solutions exist;
3. show that, under certain additional assumptions, Lagrange multipliers exist that may be used to enforce the constraints;
4. use the Lagrange multiplier technique to derive an optimality system from which optimal states and controls may be deduced;
5. define algorithms for the finite dimensional approximation of optimal states and controls; and
6. derive estimates for the error in the approximations to the optimal states and controls.

Two of the key ingredients used to carry out the above plan are the Tikhomorov version of the Lagrange multiplier theory and the Brezzi-Rappaz-Raviart theory for the approximation of a class of nonlinear problems. In both of these theories, certain properties of compact operators on Banach spaces play a central role. We point out that the nonuniqueness of solutions of the nonlinear constraint equations makes it appropriate to employ Lagrange multiplier principles.

By applying the Lagrange multiplier rule, one obtains an optimality system of partial differential equations whose solution provides optimal states and controls. The optimality system consists of the given state equations, i.e., the given partial differential equations, a set of adjoint equations involving the formal adjoint operator of the linearized state equations, and the optimality condition that relates the optimal controls to the adjoint variables and perhaps also the state variables. These are all coupled and it is usually a formidable task to solve them simultaneously. For this reason, one instead invokes and iterative uncoupling method that involves the separate solution of each of the three components of the optimality system. An alternate iterative approach is to use sensitivities instead of adjoint variables within an optimzation setting. However, if it is possible to solve the optimality system in a coupled manner, then the optimal states and controls may be determined without any optimization iteration, i.e., only one solution of the nonlinear equations (which in itself may require and iteration) is needed. An example for which this "all-at-once" or "one-shot" approach may be used is given below.

Applications

The abstract framework described previously has been applied to some specific, concrete problems. In each case, we used the abstract framework to analyze the concrete problems by merely showing that the latter fit into the former. The particular applications we considered are:

1. control problems in structural mechanics having geometric nonlinearities that are governed by the von Karman equations;
2. control problems in superconductivity that are governed by the Ginzburg-Landau equations;
3. control problems for incompressible, viscous flows that are governed by the Navier-Stokes equations;
4. coupled solid/fluid heat control problem; and
5. control problems for a Ladyzhenskaya model for viscous flows.

In considering these applications, we purposely chose different types of controls in order to illustrate how these could be accounted for within the abstract framework. In all cases, approximations were effected through the use of finite element methods.

Computational example - A coupled solid/fluid heat control problem

We use a coupled coupled solid/fluid heat control problem as an example of a relatively simple problem that can be solved by solving the full optimality system all at once, i.e., as a coupled system. The configuration is given in the following sketch.

The problem setup

The control: the inflow boundary temperature

The objective: the temperature at the top of the solid

Although most of our efforts have been directed at optimal control and optimal design problems for fluid flows, we have also looked at some issues connected with both the analysis and computation of feedback control problems in that setting. Here, we briefly mention two such studies, one analytical and one computational.

Analysis - Linear feedback laws for Navier-Stokes flows

Some systematic approaches to the mathematical formulation and numerical resolution of the linear feedback control problem for tracking the velocity in Navier-Stokes flows in a bounded two-dimensional domain with bounded distributed control have been developed. Time semidiscretizations and full space-time discrete approximations have also been studied. Computational results have been compared with analogous results from optimal control theory.

Computational example - Feedback control of Karman vortex shedding

The control of the forces, e.g., lift and drag, exerted on a submerged obstacle by the fluid that surrounds it is important in many applications. We have focused on the problem of controlling the lift in the plane, unsteady, incompressible, viscous flow around a circular cylinder. Flows about a cylinder at even moderate values of the Reynolds number exhibit an unsymmetric, periodic shedding of vortices. As a result, the lift force exerted by the fluid on the cylinder is periodic in time. Our goal is to show the efficacy of a simple feedback control law for the reduction of the magnitude of the lift. The control mechanism used to attempt to reduce the size of the lift oscillations is the injection and suction of fluid through orifices on the surface of the cylinder. The amount of fluid injected or sucked through the orifices is determined, using a simple feedback law, from the pressure sensed at various stations on the cylinder. Thus the sensor determines the pressure at various stations on the cylinder and the actuator injects or sucks fluid through orifices also on the cylinder. Other sensing and actuating mechanisms could also be used, e.g., the vorticity or stress components on the cylinder for sensing and rotation or shape modification of the cylinder for actuating. The type of feedback laws used in our study are based on the observation that differences in the pressure on the top and bottom halves of the cylinder should give an indication of the asymmetric behavior of the lift. Such pressure differences are then used to determine how much fluid to inject or suck through the orifices. No attempt is made to systematically design an optimal feedback law, i.e., one that in some sense does the best possible job in meeting the objective. However, the computational results we have obtained indicate that the simple feedback law we have used is quite effective in reducing the size of the oscillations in the lift.

We have tested numerous different combinations for the number and location of the injection/suction orifices and the pressure sensing points. We only report on one especially efective combination. We place two sensors at symmetric (about the axis parallel to the flow at infinity) locations on the forward face of the cylinder and place three orifices on the backward face of the cylinder. Fluid is injected through the middle orifice at twice the rate that fluid is sucked through two off-axis orifices. In this way, the rate at which fluid is injected into the flow is equal to that rate at which it is sucked from the flow at other locations. The rate at which fluid is injected and sucked through the three orifices is determined from the difference in the pressures at the the two sensing locations through a simple feeback law. There is also an explicit upper bound placed on the rate at which fluid can be injected or sucked. The flow is impulsively started from rest.

Configuration of pressure sensors and injection/suction actuators.
Sensors and actuators are on the surface of the cylinder.

Figure of lift coefficient vs. time for the uncontrolled flow.
With no control applied, after a transient period, the lift coefficient oscillates periodically in time.

Figure of lift coefficient vs. time for the controlled flow.
With feedback control applied, the amplitude of the oscillations in the lift is greatly reduced.

We have re-examined the well-known and well-studied problem of the exact controllability of linear partial differential equations. Our main result in this direction is a new, weaker condition for the exact controllability of such equations. The result is derived through an examination of a related optimal control problem. It is shown that if the terminal state belongs to the Sobolev space of functions having two square integrable derivatives, then the parabolic equation is exactly controllable.

We have applied an optimization-based domain decomposition approach to optimization problems for partial differential equations, extending our domain decomposition methodologies developed for simulation problems. Here, the original problem involves the minimization of a functional with a partial differential equation playing the role of a contraint on the minimization problem. We subdivide the region into subdomains and pose the partial differential equation constraint in each subdomain. We have unknown data in the boundary conditions posed along the interfaces between subdomains. We then define a new objective functional that is a weighted sum of the original objective functional and a functional that measures the jumps in the solution across the interfaces between subdomains. Finally, we minimize the new objective functional with respect to both the control parameters of the original optimization problem and the unknown data in the subdomain problems boundary conditions. We have shown that, in appropriate limits of the weighting parameters in the new functional, that the solution of the new optimization problem converges to the solution of the original problem. We have also studied gradient methods for solving the optimization problem and demonstrated that the subdomain problems may be solved in parallel. We have also carried out some computational experiments that illustrate the implementation of our approach.

Numerical approximation of an optimal control problem associated with the Navier-Stokes equations; Appl. Math. Let. 2, 1989, 29-31; L. Hou, M. Gunzburger and T. Svobodny.

An algorithm for the boundary control of the wave equation; Appl. Math. Let. 2, 1989, 225-228; M. Gunzburger and R. Nicolaides.

Optimal boundary control of nonsteady incompressible flow with an application to viscous drag reduction; Proc. 29th IEEE Conf. Decision and Control IEEE, 1990, 377-378; L. Hou, M. Gunzburger and T. Svobodny.

A numerical method for drag minimization via the suction and injection of mass through the boundary; Stabilization of Flexible Structures Springer, 1990, 312-321; L. Hou, M. Gunzburger and T. Svobodny.

Finite element approximations of an optimal control problem associated with the scalar Ginzburg-Landau equation; Comput. Math. Appl. 21, 11991, 123-131; L. Hou, M. Gunzburger and T. Svobodny.

Control of temperature distributions along boundaries of engine components; Numerical Methods in Laminar and Turbulent Flow VII, Pineridge, 1991, 765-773; L. Hou, M. Gunzburger and T. Svobodny.

Approximation of boundary control and optimization problems for fluid flows; 4th International Symposium on Computational Fluid Dynamics U. California, Davis, 1991, 455-460; L. Hou, M. Gunzburger and T. Svobodny.

Analysis and finite element approximations of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls; Math. Comp. 57, 1991, 123-151; L. Hou, M. Gunzburger and T. Svobodny.

Analysis and finite element approximations of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls; Math. Model. Numer. Anal. 25, 1991, 711-748; with L. Hou, M. Gunzburger and T. Svobodny.

Boundary velocity control of incompressible flow with an application to viscous drag reduction; SIAM J. Control Optim. 30, 1992, 167-181; L. Hou, M. Gunzburger and T. Svobodny.

Issues in shape optimization for the Navier-Stokes equations; Proc. 31th IEEE Conf. Decision and Control IEEE, 1992, 3390-3392; M. Gunzburger and J. Peterson.

Heating and cooling control of temperature distributions along boundaries of flow domains; J. Math. Systems Estim. Control 3 1993, 147-172; M. Gunzburger, L. Hou and T. Svobodny.

Optimal control and optimization of viscous, incompressible flows; Incompressible Computational Fluid Dynamics: Trends and Advances Cambridge, 1992, 109-150; M. Gunzburger, L. Hou and T. Svobodny.

The approximation of boundary control problems for fluid flows an application to control by heating and cooling; Comput. Fluids 22 1993, 239-251; M. Gunzburger, L. Hou and T. Svobodny.

The reduced basis method in control problems; Computation and Control III Birkhauser Cambridge, 1993, 211-218; M. Gunzburger and J. Peterson.

Analysis and algorithms for flow control and optimization problems; Proc. Applied Math. Workshop on Numerical Analysis: Finite Element Methods KAIST, Taejon, Korea, 1993, 21-41; M. Gunzburger.

Sensitivity calculations for a 2D, inviscid, supersonic forebody problem; Identification and Control of Systems Governed by Partial Differential Equations SIAM, Philadelphia, 1993, 14-24; J. Borggaard, J. Burns, E. Cliff and M. Gunzburger.

Analysis, approximation, and computation of a coupled solid/fluid temperature control problem; Comput. Methods Appl. Mech. Engng. 118 1994, 133-152; M. Gunzburger and H.-C. Lee.

Boundary control of incompressible flows; Computational Fluid Dynamics Techniques Harwood, UK, 1994 839-846; M. Gunzburger, L. Hou and T. Svobodny.

Feedback control of fluid flows; Proc. 14th IMACS World Congress on Computational and Applied Mathematics Georgia Tech, Atlanta, 1994, 716-719; M. Gunzburger and H.-C. Lee.

Analysis and approximation of optimal control problems nonlinear constraints; Proc. 33th IEEE Conf. Decision and Control IEEE, Lake Buena Vista, 1994, 299-304; M. Gunzburger, L. Hou, S. Ravindran, and J. Turner.

Algorithms for flow control and optimization; Optimal Control and Design Birkhauser, Boston, 1995, 97-116; J. Borggaard, J. Burkardt, J. Burns, E. Cliff, M. Gunzburger, H. Kim, H. Lee, J. Peterson, A. Shenoy and X. Wu.

A prehistory of flow control and optimization; Flow Control Springer, New York, 1995, 185-195; M. Gunzburger

Active control of instabilities in laminar boundary-layer flow: Use of sensors and spectral controller; AIAA J. 33 1995, 1521-1523; G. Erlebacher, M. Gunzburger, M. Hussaini, R. Joslin, and R. Nicolaides.

Flow control and optimization; Computational Fluid Dynamics Review Wiley, West Sussex, 1995, 548-566; M. Gunzburger.

Sensitivity discrepancy for geometric parameters; CFD for Design and Optmization ASME, New York, 1995, 9-15; J. Burkardt and M. Gunzburger.

Optimization-based design in high-speed flows; CFD for Design and Optmization ASME, New York, 1995, 61-68; J. Appel, E. Cliff, A. Godfrey and M. Gunzburger.

Discretization of cost and sensitivities in shape optimization; Computation and Control IV Birkhauser, 1995, 43-56; J. Burkardt, M. Gunzburger and J. Peterson.

Analysis and finite element approximation of optimal control problems for a Ladyzhenskaya model for stationary, incompressible, viscous flows; J. Comp. Appl. Math. 61 1995, 323-343; Q. Du, M. Gunzburger and L. Hou.

Finite dimensional approximation of a class of constrained nonlinear optimal control problems; SIAM J. Cont. Opt. 34, 1996, 1001-1043; M. Gunzburger and L. Hou.

A methodology for the automated optimal control of flows including transitional flows; Proc. ASME Forum on Control of Transitional and Turbulent Flows FED-237, ASME, 1996 287-294; G. Erlebacher, M. Gunzburger, M. Hussaini, R. Joslin and R. Nicolaides.

Feedback control of Karman vortex shedding; J. Appl. Mech 63, 1996, 828-835; M. Gunzburger and H.-C. Lee.

An optimal design problem for a two dimensional flow in a duct; Opt. Control Appl. Meth. 17, 1996, 329-339; E. Cliff, M. Gunzburger and X. Wu.

Sensitivity calculation in flows discontinuitites; AIAA Paper 96-2471, Proc. 14th AIAA Applied Aerodynamics Conference, AIAA, 1996; J. Appel and M. Gunzburger.

An automated methodology for optimal flow control an application to transition delay; AIAA J. 35, 1997, 816-824; G. Erlebacher, M. Gunzburger, M. Hussaini, R. Joslin and R. Nicolaides.

Difficulties in sensitivity calculations for flows discontinuities; AIAA J. 35, 1997, 842-848; J. Appel and M. Gunzburger.

Introduction into mathematical aspects of flow control and optimization; Inverse Design and Optimisation Methods, von Karman Institute for Fluid Dynamics, Rhode-Saint-Gen\`ese, Belgium, 1997; M. Gunzburger.

Approximate solutions via sensitivities; Inverse Design and Optimisation Methods, von Karman Institute for Fluid Dynamics, Rhode-Saint-Gen\`ese, Belgium, 1997; J. Appel and M. Gunzburger.

Lagrange multiplier techniques; Inverse Design and Optimisation Methods, von Karman Institute for Fluid Dynamics, Rhode-Saint-Gen\`ese, Belgium, 1997; M. Gunzburger.

Analysis and approximation of optimal control problems for a simplified Ginzburg-Landau model of superconductivity; Numer. Math. 77, 1997, 243-268; M. Gunzburger, L. Hou and S. Ravindran.

The exact controllability of systems governed by linear parabolic equations; J. Math. Anal. Appl. 215, 1997, 174-189; Y. Cao, M. Gunzburger and J. Turner.

On exact controllability and convergence of optimal controls to exact controls of parabolic equations; Optimal Control: Theory, Algorithms, and Applications, Kluwer, 1998, 67-83; Y. Cao, M. Gunzburger and J. Turner.

Optimal control problems for a class of nonlinear equations an application to the control of fluids; Optimal Control of Viscous Flows, SIAM, Philadelphia, 1998, 43-62; M. Gunzburger, L. Hou and T. Svobodny.

Adjoint methods; Design Optimal et MDO, Centre de Recherche en Calcul Appliqu\'e, Montr\'eal, 1998, 1-25; M. Gunzburger.

Boundary value problems and optimal boundary control for the Navier-Stokes system: The two-dimensional case; SIAM J. Cont. Optim. 36, 1998, 852-894; A. Fursikov, M. Gunzburger and L. Hou.

Existence of an optimal solution of a shape control problem for the stationary Navier-Stokes equations; SIAM J. Cont. Optim. 36, 1998, 895-909; M. Gunzburger and H.-C. Kim.

Sensitivities in computational methods for optimal flow control; Computational Methods for Optimal Design and Control, Birkhauser, Boston, 1998, 197-236; M. Gunzburger.

Optimal Dirichlet control and inhomogeneous boundary value problems for the unsteady Navier-Stokes equations; to appear; A. Fursikov, M. Gunzburger and L. Hou.

Optimal boundary control of the Navier-Stokes equations bounds on the control; to appear in Proc. Korean Advanced Institute for Science and Technology Workshop on Finite Elements; A. Fursikov, M. Gunzburger and L. Hou.

Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions; Appl. Math. Comp.100 1999, 49-70; M. Gunzburger and H.-C. Lee.

Computations of optimal controls for incompressible flows; to appear in Int. J. Comp. Fluid Dyn.; M. Gunzburger, L. Hou, S. Manservisi and Y. Yan.

A penalty/least-squares method for optimal control problems for first-order elliptic systems; to appear in Appl. Math. Comp.; M. Gunzburger and H.-C. Lee.

Sensitivity analysis of a shape control problem for the stationary Navier-Stokes equations; to appear; M. Gunzburger and H.-C. Kim.

Analysis and approximation of the velocity tracking problem for Navier-Stokes flows distributed control; to appear; M. Gunzburger and S. Manservisi.

The velocity tracking problem for Navier-Stokes flows bounded distributed control; to appear; M. Gunzburger and S. Manservisi.

Analysis and approximation for linear feedback control for tracking the velocity in Navier-Stokes flows; to appear; M. Gunzburger and S. Manservisi.

Sensitivites, adjoints, and flow optimization; to appear; M. Gunzburger.

The velocity tracking problem for Navier-Stokes flows with boundary control; to appear; M. Gunzburger and S. Manservisi.

Domain decomposition methods for optimization problems for partial differential equations; in preparation; M. Gunzburger and J. Lee.

Selected abstracts

L. Hou, M. Gunzburger and T. Svobodny

We consider the problem of the encouragement, in a least squares sense, of a viscous incompressible flow towards a given vector field through the adjustment of the stress on the boundary. We discuss the question of existence of optimal solutions and the use of the Lagrange multiplier rule to enforce the contraints and derive an optimality system from which optimal states and controls may be determined. We also discuss the finite element approximation of solutions of the optimal control problem.

M. Gunzburger and R. Nicolaides

An algorithm for the approximate determination of boundary controls in controllability problems for the wave equation is presented. The algorithm and its variants apply in any number of space dimensions. Furthermore, the algorithm can be used in connection with more general controllability problems involving hyperbolic partial differential equations.

L. Hou, M. Gunzburger and T. Svobodny

We study the problem of minimizing the viscous drag on a body via the addition or removal of mass through the boundary. The control considered is the mass flux through all or part of the boundary; the functional to be minimized is the viscous dissipation. We use Lagrange multiplier techniques to derive a system of partial differential equations from which optimal, i.e., minimum drag, solutions may be determined. Then, finite element approximations of solutions of the optimality system are defined and optimal error estimates are derived.

L. Hou, M. Gunzburger and T. Svobodny

We consider finite element approximations of an optimal control problem associated with a scalar version of the Ginzburg-Landau equations of superconductivity. The control is the Neumann data on the boundary and the optimization goal is to obtain a best approximation, in the least squares sense, to some desired state. The existence of optimal solutions is proved. The use of Lagrange multipliers is justified and an optimality system of equations is derived. Then, the regularity of solutions of the optimality system is studied, and finally, finite element algorithms are defined and optimal error estimates are obtained.

L. Hou, M. Gunzburger and T. Svobodny

The problem considered is the control of the temperature along boundaries of flow domains by adjustments in the heat flux at those boundaries. An optimization problem is formulated in which the objective is to remove any hot spots, i.e., temperature peaks, along boundary surfaces. Among the issues discussed are the existence and uniqueness of the solution of the optimization problem and the use of Lagrange multipliers to derive a system of partial differential equations whose solution provide the optimal state and control. Then, finite element discretizations of the optimality system are defined; error estimates are provided along with an efficient solution algorithm for the discrete equations. Finally, a simple numerical example is provided to show what kind of results can be achieved through the use of our algorithm.

L. Hou, M. Gunzburger and T. Svobodny

We consider finite element approximations of various optimal control problems associated with incompressible flows. The goal is to achieve some desired objective by minimizing an associated cost functional through the application of a control mechanism along a portion of the boundary of the flow domain. Included in the objectives we consider are some related to drag reduction, to the avoidance of high temperatures along boundary surfaces, and to conformation to a desired flow pattern. The control mechanisms considered are injection or suction of fluid through boundary orifices and heating or cooling along boundary surfaces. The method of Lagrange multipliers is used to derive a system of partial differential equations whose solutions provide the optimal states and controls. No simplifications concerning the flow are invoked, i.e., the full Navier-Stokes equations are employed. Finite element algorithms for approximating the optimal states and controls are discussed, as is the accuracy of approximations resulting from these algorithms.

L. Hou, M. Gunzburger and T. Svobodny

We examine certain analytic and numerical aspects of optimal control problems for the stationary Navier-Stokes equations. The controls considered may be of either the distributed or Neumann type; the functionals minimized are either the viscous dissipation or the L⁴--distance of candidate flows to some desired flow. We show the existence of optimal solutions and justify the use of Lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. Then, we consider the approximation, by finite element methods, of solutions of the optimality system and derive optimal error estimates.

L. Hou, M. Gunzburger and T. Svobodny

Optimal control problems for the stationary Navier-Stokes equations are examined from analytical and numerical points of view. The controls considered are of Dirichlet type, that is, control is effected through the velocity field on or the mass flux through the boundary; the functionals minimized are either the viscous dissipation or the L⁴--distance of candidate flows to some desired flow. We show that optimal solutions exist and justify the use of Lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. Then, finite element approximations of solutions of the optimality system are defined and optimal error estimates are derived.

L. Hou, M. Gunzburger and T. Svobodny

An optimal boundary control problem for the Navier-Stokes equations is presented. The control is the velocity on the boundary, which is constrained to lie in a closed, convex subset of H^1/2 of the boundary. A necessary condition for optimality is derived. Computations are done when the control set is actually finite-dimensional, resulting in an application to viscous drag reduction.

M. Gunzburger and J. Peterson

We consider sensitivity based methods for shape optimization of fluid flow governed by the Navier-Stokes equations. The crucial components of an algorithm for this task are the proper paarmetrization of the boundary, the efficient calculation of the sensitivities with respect to the boundary parameters, the selection of a good optimization algorithm, and an efficient scheme for state calculations. We discuss some of these issues, pointing out pitfalls and potential solutions.

M. Gunzburger, L. Hou and T. Svobodny

An optimization problem is formulated motivated by the desire to remove temperature peaks, i.e., hot spots, along bounding surfaces of fluid flows. Control is effected by adjustments to the heat flux along those boundaries. The existence and uniqueness of the solution of the optimization problem are shown, and an optimality system of partial differential equations is derived from which optimal controls and states may be determined. Then, finite element discretizations of the optimality system are defined and optimal error estimates are provided along with an efficient solution algorithm for the discrete equations. Finally, computational results are given ilustrating various features of the discretization and solution algorithms.

M. Gunzburger, L. Hou and T. Svobodny

The control of fluid motions for the purpose of achieving some desired objective is crucial to many technological applications. In the past, these control problems have been addressed either through expensive experimental processes or through the introduction of significant simplifications into the analyses used in the development of control mechanisms. Only recently have flow control problems been addressed, by scientists and mathematicians, in a systematic, rigorous manner. Here, we discuss the constraints that the state and control variables are required to satisfy. Then, we provide a small sampling of flow control and optimization objectives of practical interest and describe some of the possible control mechanisms that can be used to achieve the desired objectives. We describe some specific flow control problems which serve to illustrate some of the possible combinations of constraints, objectives, and control mechanisms. We also describe, in general terms, the issues that arise when one attempts to analyse and computationally simulate flow control and optimization problems and then we treat, in more detail, a particular problem, illustrating the resolution of some of these issues.

M. Gunzburger, L. Hou and T. Svobodny

The goal of a boundary control or optimization problem for a fluid flow is to achieve some desired objective by the application of a control mechanism along a portion of the boundary of the flow domain. Among the objectives of interest are some related to drag reduction, to the avoidance of high temperatures along boundary surfaces, and to conformation to a desired flow pattern. Possible control mechanisms are injection or suction of fluid through boundary orifices and heating or cooling along boundary surfaces. The objective is met through the minimization of an associated cost functional. The method of Lagrange multipliers is used to derive a system of partial differential equations whose solutions provide the optimal states and controls. No simplifications concerning the flow are invoked, \ie, the full Navier-Stokes equations are employed. Finite element algorithms for approximating the optimal states and controls are discussed, as is the accuracy of approximations resulting from these algorithms. Details are provided for the example of using heating and cooling controls in order to avoid hot spots along bounding surfaces.

M. Gunzburger and J. Peterson

The standard methods for determining solutions of partial differential equations are only loosely tied to the particular problem being solved. For example, in the context of finite element methods, one uses the same type of local basis functions, e.g., piecewise polynomials, for solving Poisson's equation, the heat equation, the Navier-Stokes equations, etc. What if one were to use basis functions that still belong to one's favorite finite element space and are global in nature, but which are intimately related to the solution of the particular partial differential equation of interest? This is the rationale behind the reduced basis method. The reduced basis method was developed for structural engineering applications, and has met with considerable success in that setting. Its potential usefulness for fluid calculations has also been demonstrated. In this paper we examine the possibilities for its application to the computational resolution of control problems. Although our focus is on flow control problems, it is clear that our discussion can also be applied to other settings having partial differential equations playing the role of state contraints.

J. Borggaard, J. Burns, E. Cliff and M. Gunzburger

In this paper, we discuss the use of a sensitivity equation to compute derivatives for optimization based design algorithms. The problem of designing an optimal forebody simulator is used to motivate the algorithm and to illustrate the basic ideas. Finally, we indicate how and existing CFD code can be modified to compute sensitivities and a numerical example is presented.

M. Gunzburger and H.-C. Lee

An optimization problem is formulated motivated by the desire to remove temperature peaks, i.e., hot spots, along the bounding surfaces of containers of fluid flows. The heat equation of the solid container is coupled to the energy equation for the fluid. Heat sources can be located on the solid body, the fluid, or both. Control is effected by adjustments to the temprature of the fluid at the inflow boundary. Both mathematical analyses and computational experiments are given.

M. Gunzburger, L. Hou, S. Ravindran, and J. Turner

A general framework for treating nonlinearly constrained optimal control and optimization problems is given. Based on a set of hypotheses, optimal solutions are shown to exist and the use of Lagrange multipliers to enforce the constraints is justified. An optimality system is derived whose solutions provide the optimal states and controls. Finite dimensional approximations are then considered; an approximate problem is defined and optimal error estimates are derived. The general framework has been applied to numerous concrete settings; here we illustrate its use in the context of a magentohydrodynamics control problem.

J. Borggaard, J. Burkardt, J. Burns, E. Cliff, M. Gunzburger, H. Kim, H. Lee, J. Peterson, A. Shenoy and X. Wu

The work on flow control and optimization at the Air Force Center for Optimal Design and Control (CODAC) has focused on new tools for improved aerodynamic design and the {optimal and feedback control of fluid flows. The aerodynamic performance of aerospace systems (or of their components) and of test facilities can often be enhanced by the judicious tailoring of their shape or of other parameters that determine the flows. Computational tools that combine modern computational fluid dynamics and optimization methods can aid in the design of these objects and result in enhanced performance and reduced development costs. Efficient optimization schemes require information about both the flow and its sensitivity to design changes. The use of standard finite-difference quotient approximations to the sensitivities requires the expenditure of far too large amounts of computational resources. As an effective alternative, we use the continuum model to derive sensitivity equations which are a system of linear partial differential equations that may be solved, in an approximate manner, for the flow sensitivities. We have successfully applied this method to several problems. This approach is now the basis for several joint ventures with Air Force Laboratories and industrial collaborators.The development of convergent computational schemes for optimal control of fluid flows is an extremely complex undertaking. The problems are highly nonlinear and often the control is applied by using wall shapes, temperature, or mass fluxes as control mechanisms. A general framework for constructing finite dimensional approximating optimal control problems has been developed. This framework is applicable not only to flow control, but to nonlinear problems in many other settings as well. Our efforts in this direction have focused on building mathematical models of the physical problems, invoking a minimum of assumptions about the physical phenomena, and then rigorously analyzing these mathematical models. After this first step, we develop and analyze discretization methods for determining approximate solutions of these problems, develop computer codes implementing the algorithms, first for the purpose of showing the efficacy of these methods, and ultimately, to solve problems of practical interest. We have also considered various algorithmic issues in the feedback control of fluid flows. In this paper, we first give a brief summary of the various projects that CODAC personnel have been involved in. The references cited may be consulted for details about these projects and the results that have been obtained. We then give a more detailed discussion of two of these projects.

M. Gunzburger

Flow control and optimization is an ancient practice o man. For example, any dam, sluice, canal, levee, irrigation ditch, valve, duct, pipe, pump, hose, vane, etc., is an exercise in flow control or optimization, i.e., an attempt to control the mechanical and/or thermodynamical state of a fluid in order to achieve a desired purpose. In this paper, we describe the history of flow control and optimization before the recent introduction of sophisticated computational fluid dynamics simulations and sophisticated optimization strategies into flow control methodologies. We also discuss why this recent introduction is a timely one.

G. Erlebacher, M. Gunzburger, M. Hussaini, R. Joslin and R. Nicolaides

This paper focuses on the suppression of instability growth using an automated active-control technique and is a next logical step based on previous experimental and computational studies in which control was in the form of wave-cancellation based on analytical information. Here, for boundary layer flows, control is effected by determining spectral information about the instability through a Fourier analysis of sensed data, namely shear or pressure on a portion of the wall. The control mechanism is injection of suction of fluid through and orifice in the wall. The amount of control is determined through a simple relation involving the spectral information obtained from the sensed data. It is shown that such an approach is effective in suppressing instabilities.

M. Gunzburger

We review the modeling, analysis, and approximation of flow control and optimization problems. We discuss some objectives and control mechanisms that are of practical interest. We then discuss two possible approaches for the approximate solution of optimal flow control problems, and give a computational example of the application of each method. The two methods are based on sensitivity derivatives and on Lagrange multiplier techniques, respectively. We then examine in more detail the mathematical analysis and approximation of Lagrange multiplier methods. We close by considering a problem in feeback control of fluid flows.

J. Burkardt and M. Gunzburger

When discretized sensitivities are used to approximate the sensitivities of a discretized variable, a discrepancy may occur, particularly when the underlying parameter respresents a geometric or implicit quantity. For a model problem, the primary source of error is found to be in the boundary conditions, which in turn are affected by errors in approximate spatial derivatives. A related optimization problem shows that the discretized sensitvities provide superior approximation of the behavior of the continous variables.

J. Appel, E. Cliff, A. Godfrey and M. Gunzburger

The Sensitivity-Equation approach is implemented to determine sensitivity derivatives in the framework of aerodynamic optimization. Two shape-optimization problems are studied on both structured and unstructured meshes using an inflow design parameter and two shape-design variables. The first is a perfect-gas, forebody simulator and the second is a finite-rate chemistry optimization maximizing the mass fraction of a species throughout the flow field.

J. Burkardt, M. Gunzburger and J. Peterson

We consider a number of computational difficulties arising in the discretization of costs and sensitivities, and ultimately in optimal design strategies, in shape optimization problems. We show that one can encounter spurious local minima and how these can be eliminated through smoothing of the functional. We also demonstrate sensitivity failure, i.e., discretized sensitivities can yield incorrect information about the gradient of the objective functional. We discuss means for avoiding this difficulty as well.

Q. Du, M. Gunzburger and L. Hou

We examine certain analytic and numerical aspects of optimal control problems for a Ladyzhenskaya model for stationary, incompressible, viscous flows. The control considered is of the distributed type; the functionals minimized are the L²-distance of candidate flows to some desired flow and the viscous drag on bounding surfaces. We show the existence of optimal solutions and justify the use of Lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. Then, we consider approximations, by finite element methods, of solutions of the optimality system and examine their convergence properties.

M. Gunzburger and L. Hou

An abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed. Nonlinearities occur in both the objective functional and in the constraints. The framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces, an approximate problem posed on finite dimensional spaces, together with a number of hypotheses concerning the two problems. The framework is used to show that optimal solutions exist, to show that Lagrange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal states and controls may be deduced, and to derive existence results and error estimates for solutions of the approximate problem. The abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods. The first involves the von Karman plate equations of nonlinear elasticity, the second the Ginzburg-Landau equations of superconductivity, and the third the Navier-Stokes equations for incompressible, viscous flows.

M. Gunzburger and H.-C. Lee

A computational study of the feedback control of the magnitude of the lift in flow around a cylinder is presented. The uncontrolled flow exhibits an unsymmetric Karman vortex street and a periodic lift coefficient. The size of the oscillations in the lift is reduced through an active feedback control system. The control used is the injection and suction of fluid through orifices on the cylinder; the amount of fluid injected or sucked is determined, through a simple feedback law, from pressure measurements at stations along the surface of the cylinder. The results of some computational experiments are given; these indicate that the simple feedback law used is effective in reducing the size of the oscillations in the lift.

E. Cliff, M. Gunzburger and X. Wu

An inverse design problem in which we seek a flow solution that matches a prescribed target flow is formulated as an optimization problem. Some numerical results are presented which show that the optimization procedure converges quickly and results in a very good match between the design and target flows.

J. Appel and M. Gunzburger

The use of flow sensitivities has become an integral part of flow optimization-based aerodynamic design and is useful for the efficient calculation of perturbed flows. However, there are still many issues to be resolved, especially for flows having discontinuities. In this paper, flow sensitivities for a variety of numerical methods are calculated using finite differences, automatic differentiation and the sensitivity equation method. The flow sensitivities are calculated for the Riemann problem of gas dynamics. An attempt is made to explain and understand difficulties that arise in the calculation of flow sensitivities in flows with discontinuities.

G. Erlebacher, M. Gunzburger, M. Hussaini, R. Joslin and R. Nicolaides

This paper describes a self-contained, automated methodology for active flow control which couples the time-dependent Navier-Stokes system with an adjoint Navier-Stokes system and optimality conditions from which optimal states, i.e., unsteady flow fields and controls (e.g., actuators), may be determined. The problem of boundary layer instability suppression through wave cancellation is used as the initial validation case to test the methodology. Here, the objective of control is to match the stress vector along a portion of the boundary to a given vector; instability suppression is achieved by choosing the given vector to be that of a steady base flow. Control is effected through the injection or suction of fluid through a single orifice on the boundary. The results demonstrate that instability suppression can be achieved without any a priori knowledge of the disturbance, which is significant because other control techniques have required some knowledge of the flow unsteadiness such as frequencies, instability type, etc. The present methodology has been extended to three dimensions and may potentially be applied to separation control, re-laminarization, and turbulence control applications using one to many sensors and actuators.

J. Appel and M. Gunzburger

The use of flow sensitivities has become and integral part of flow optimization-based aerodynamic design and is useful for the efficient calculation of perturbed flows. However, there are still many issues to be resolved, especially for flows having discontinuities. Flow sensitivities for several numerical methods are calculated using finite difference quotients, automatic differentiation, and the sensitivity equation method. The flow sensitivities are calculated for the Riemann problem of gas dynamics. An attempt is made to expose, explain, and understand difficulties that arise in the calculation of flow sensitivities in flows with discontinuiuties.

M. Gunzburger, L. Hou and S. Ravindran

This paper is concerned with optimal control problems for a Ginzburg-Landau model of superconductivity that is valid for high values of the Ginzburg-Landau parameter and high external fields. The control is of Neumann type. We first show that optimal solutions exist. We then show that Lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced. Then we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations. Finally, we report on some numerical results.

Y. Cao, M. Gunzburger and J. Turner

The main result of this paper is a new, weaker condition for the exact controllability of linear parablic partial differential equations. The result is derived through an examination of a related optimal control problem. It is shown that if the terminal state belongs to the Sobolev space of functions having two square integrable derivatives, then the parabolic equation is exactly controllable.

Y. Cao, M. Gunzburger and J. Turner

The main result of this paper is the convergence of optimal controls to the exact controls of linear parablic partial differential equations. The result is derived through a representaion of the terminal state by a operator that is similar to the one used by Lions in Hilbert Uniqueness Method.

M. Gunzburger, L. Hou and T. Svobodny

In the first part of this paper we state and analyze an abstract constrained optimization problem. The problem considered, although having a certain structure, is general enough to allow for a variety of applications including fluid control problems associated with the Navier-Stokes equations of incompressible flow. We consider this specific application in the second part of this paper.

A. Fursikov, M. Gunzburger and L. Hou

We study optimal boundary control problems for the two-dimensional Navier-Stokes equations in an unbounded domain. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates. An objective of interest is the drag functional. We first establish three important results for inhomogeneous boundary value problems for the Navier-Stokes equations, namely, we identify the trace space for the velocity fields possessing finite energy, we prove the existence of a solution for the Navier-Stokes equations with boundary data belonging to the trace space, and we identify the space in which the stress vector (along the boundary) of admissible solutions is well-defined. Then, we prove the existence of an optimal solution over the control set. Finally, we justify the use of Lagrange multiplier principles, derive an optimality system of equations in the weak sense from which optimal states and controls may be determined, and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values.

M. Gunzburger and H.-C. Kim

This paper is concerned with an optimal shape control problem for the stationary Navier--Stokes system. A two--dimensional channel flow of an incompressible, viscous fluid is examined to determine the shape of a bump on a part of the boundary that minimizes the viscous drag. After giving a precise formulation of the extremal problem in a function analytic setting, it is shown that optimal solutions exist.

M. Gunzburger

There are several different approaches to solving the optimization problem. One of these uses Lagrange multipliers to enforce the constraints, e.g., the flow equations, and results in a large coupled system from which optimal states and controls may be determined. Another approach is based on the use of an optimization algorithm that require (at least an approximation) to the gradient of the objective functional. One can then use flow sensitivities, i.e., the derivatives of the flow variables with respect to the design parameters, to aid in the determination of the functional gradient. Alternately, one could use solutions of adjoint problems for the same purpose. Due to space limitations, this paper will be limited to a discussion of sensitivity-based method for determining functional gradients; much of what we have to say also applies to adjoint equation-based gradient determinations, and even to the Lagrange multiplier approach. We address numerous questions and difficulties that arise in sensitivity computaions within an optimization setting. Included in our discussions are the relative merits of the discretize-then-differentiate and differentiate-then-discretize approaches to determining sensitivities and difficulties in sensitivity calculations for flows with discontinuities.

M. Gunzburger and H.-C. Lee

We examine analytical and numerical aspects of optimal control problems for first-order elliptic systems in three dimensions. The particular setting we use is that of div-curl systems. After formulating some optimization problems, we prove the existence and uniqueness of the optimal solution. We then demonstrate the existence of Lagrange multipliers and derive an optimality system of partial differential equations from which optimal controls and states may be deduced. We then define least-squares finite element approximations of the solution of the optimality system and derive optimal estimates for the error in these approximations.

M. Gunzburger, L. Hou, S. Manservisi and Y. Yan

This paper is concerned with numerical solutions of optimal control problems for unsteady, viscous, incompressible flows. In general, controls can be of the distributed type (external body force) or Dirichlet type (e.g., boundary velocity). Here, we only consider the former case, although most of what we present is also applicable to the latter. Two different optimization objectives and associated solution methodologies are described. One involves a global-in-time functional, the other a local-in-time functional. Which method is preferred depends on the specific application. Some test computational results are presented.

M. Gunzburger and H.-C. Lee

A penalty/least-squares method for optimal control problems for first-order elliptic systems is considered wherein the constraint equations are enforced via penalization. The convergence, as the penalty parameter tends to zero, of the solution to the penalized optimal control problem to that unpenalized one is demonstrated as is the convergence of a gradient method for determining solutions of the penalized optimal control problem. Finally, finite element approximations of the penalized optimal control problem are studied and optimal error estimates are obtained.

M. Gunzburger and H.-C. Kim

This paper is concerned with a sensitivity analysis of an optimal shape control problem for the stationary Navier--Stokes system. A two--dimensional channel flow of an incompressible, viscous fluid is examined to determine the shape of a bump on a part of the boundary that minimizes the viscous drag. The shape sensitivity analysis uses the material derivative method and adjoint variables to determine the shape gradient of the design functional.

M. Gunzburger and S. Manservisi

We consider the mathematical formulation, analysis and the numerical solution of a time-dependent optimal control problem associated with the tracking of the velocity of a Navier-Stokes flow in a bounded two-dimensional domain through the adjustment of a distributed control. The existence of optimal solutions is proved and the first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown. A gradient method for the solution of the fully discrete equations is examined as are its convergence properties. Finally, the results of some illustrative computational experiments are presented.

M. Gunzburger and S. Manservisi

We present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for Navier-Stokes flows in bounded two-dimensional domains with bounded distributed controls. We study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined. We also define and analyse semidiscrete-in-time and fully space-time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system. The results of some computational experiments are provided.

M. Gunzburger and S. Manservisi

Some systematic approaches to the mathematical formulation and numerical resolution of the linear feedback control problem for tracking the velocity in Navier-Stokes flows in a bounded two-dimensional domain with bounded distributed control are presented. Time semidiscretizations and full space-time discrete approximations are also studied. Some computational results are presented and compared with analogous results from optimal control theory.

M. Gunzburger

Several issues related to the calculation of flow sensitivities and the solution of flow optimization problems are considered. For the latter, one-shot Lagrange multiplier methods are presented, as well as sensitivity-based and adjoint-based iterative algorithms. A sample application of each method to a specific flow optimization problem is provided. Then, some difficulties asssociated with the practical implementation of the methods are discussed. Particular emphasis is placed on the effect of flow discontinuities on approximate sensitivities and adjoints. A discussion of these issues is given in the context of the Reimann problem for which exact information is known.

M. Gunzburger and S. Manservisi

We present some systematic approaches to the mathematical formulation and numerical resolution of the time-dependent optimal control problem of tracking the velocity for Navier-Stokes flows in a bounded two-dimensional domain with boundary control. We study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined. We also define and analyse semidiscrete-in-time and fully space-time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system. The results of some computational experiments are provided. problem.

M. Gunzburger and J. Lee

We consider an optimization-based domain decomposition approach for optimization problems for partial differential equations. Here, the given optimization problem involves the minimization of a functional that depends on the solution of a boundary value problem for a partial differential equation. The minimization is with respect to some unknown data in the boundary value problem. We subdivide the region into subdomains and pose the partial differential equation constraints in each subdomain. The subdomain boundary value problems have boundary conditions along the interfaces between the subdomains that contain unknown data. We also define a new objective functional that is a weighted sum of the original objective functional and a functional that measures the jumps in the solution across the interfaces between subdomains. Finally, we minimize the new objective functional with respect to both the unknown parameters in the given optimization problem and the unknown data introduced in the definition of the subdomain problems. We show, in an appropriate limit for the weighting parameters in the new functional, that the solution of the new optimization problem converges to the solution of the original problem. We study a gradient method for solving the new optimization problem and demonstrate that the subdomain problems may be solved in parallel. We also provide the results of some computational experiments that illustrate the implementation of our approach.

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Last updated: 3/6/99 by Max Gunzburger