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We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.

In the context of Galerkin discretizations of a partial differential equation (PDE), the modes of the classical method of Proper Orthogonal Decomposition (POD) can be interpreted as the ansatz and trial functions of a low-dimensional Galerkin scheme. If one also considers a Galerkin method for the time integration, one can similarly define a POD reduction of the temporal component. This has been described earlier but not expanded upon -- probably because the reduced time discretization globalizes time which is computationally inefficient. However, in finite-time optimal control systems, time \textit{is} a global variable and there is no disadvantage from using a POD reduced Galerkin scheme in time. In this paper, we provide a newly developed generalized theory for space-time Galerkin POD, prove its optimality in the relevant function spaces, show its application for the optimal control of nonlinear PDEs, and, by means of a numerical example with Burgers' equation, discuss the competitiveness by comparing to standard approaches.

We shall consider a stochastic maximum principle of optimal control for a control problem associated with a stochastic partial differential equations of the following type: d x(t) = (A(t) x(t) + a (t, u(t)) x(t) + b(t, u(t)) dt + [ _K + g (t, u(t))] dM(t), x(0) = x_0 \in K, with some given predictable mappings $a, b, \sigma, g$ and a continuous martingale $M$ taking its values in a Hilbert space $K,$ while $u(\cdot)$ represents a control. The equation is also driven by a random unbounded linear operator $A(t,w), \; t \in [0,T ], $ on $K .$ We shall derive necessary conditions of optimality for this control problem without a convexity assumption on the control domain, where $u(\cdot)$ lives, and also when this control variable is allowed to enter in the martingale part of the equation.

A convex penalty for promoting switching controls for partial differential equations is introduced; such controls consist of an arbitrary number of components of which at most one should be simultaneously active. Using a Moreau-Yosida approximation, a family of approximating problems is obtained that is amenable to solution by a semismooth Newton method. The efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples.

 This paper presents a novel approach, based on a partial differential equation (PDE) model, for analogical modeling and digital simulation of biomedical processes with distributed parameters. Such processes include penetration of radiation during radiation therapy or radiological imaging techniques. The present study simulates the propagation of radiation through different media. A valuable feature of the proposed method is its ability to simulate propagation along inhomogeneous media such as human tissues. It offers a starting point for automated control of radiological devices. 

This project was focused on fundamental issues of modeling and control of flexible structures. Particular problems considered were: patterns of transmission zeroes, sensor placement, robust control, and high-performance control using iterative learning approach. During the grant period a flexible beam control devise for experimental purpose was also developed. With most of the work carried out to the extent of laboratory implementation, the following results were concluded: 1) For flexible structure control the location of transmission zeroes in the mathematical models critically depend on the location of sensors. For flexible beams the movement of zeroes were mapped out for control design; 2) For flexible structures a class of new compensators called generalized lead/lag compensators were developed. Their implementation is very simple with delay elements; and 3) Optimal sensor placement can be based on the robustness of the dynamic observer to parameter uncertainty. A design procedure was developed for this purpose.

This project was focused on fundamental issues of modeling and control of flexible structures. Particular problems considered were: patterns of transmission zeroes, sensor placement, robust control, and high-performance control using iterative learning approach. During the grant period a flexible beam control devise for experimental purpose was also developed. With most of the work carried out to the extent of laboratory implementation, the following results were concluded: 1) For flexible structure control the location of transmission zeroes in the mathematical models critically depend on the location of sensors. For flexible beams the movement of zeroes were mapped out for control design; 2) For flexible structures a class of new compensators called generalized lead/lag compensators were developed. Their implementation is very simple with delay elements; and 3) Optimal sensor placement can be based on the robustness of the dynamic observer to parameter uncertainty. A design procedure was developed for this purpose.

This project was focused on fundamental issues of modeling and control of flexible structures. Particular problems considered were: patterns of transmission zeroes, sensor placement, robust control, and high-performance control using iterative learning approach. During the grant period a flexible beam control devise for experimental purpose was also developed. With most of the work carried out to the extent of laboratory implementation, the following results were concluded: 1) For flexible structure control the location of transmission zeroes in the mathematical models critically depend on the location of sensors. For flexible beams the movement of zeroes were mapped out for control design; 2) For flexible structures a class of new compensators called generalized lead/lag compensators were developed. Their implementation is very simple with delay elements; and 3) Optimal sensor placement can be based on the robustness of the dynamic observer to parameter uncertainty. A design procedure was developed for this purpose.

I development a Conjugate Gradient Method for solving a partial differential system with multiply controls. Some numerical results are depicted. Also, I present an explication of why the control over a partial differential equations system is necessary.

We generalize the concept "well-posed linear system" to stochastic linear control systems and study some basic properties of such kind systems. Under our generalized definition, we show the well-posedness of the stochastic heat equation and the stochastic Schr\"odinger equation with suitable boundary control and observation operators, respectively.

We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations arising from the finite horizon control of McKean-Vlasov dynamics. We give examples of existence and uniqueness results. Finally, we propose some simple models for the motion of pedestrians and report about numerical simulations in which we compare mean filed games and mean field type control.

We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets.

We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets.

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

We study the problem of optimal inside control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: (i) The controller has access to inside information, i.e. access to information about a future state of the system, (ii) The integro-differential operator of the SPDE might depend on the control. In the first part of the paper, we formulate a sufficient and a necessary maximum principle for this type of control problem, in two cases: (1) When the control is allowed to depend both on time t and on the space variable x. (2) When the control is not allowed to depend on x. In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from nonlinear filtering, we transform this noisy observation SDE inside control problem into a full observation SPDE insider control problem. The results are illustrated by explicit examples.

We study the problem of optimal inside control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: (i) The controller has access to inside information, i.e. access to information about a future state of the system, (ii) The integro-differential operator of the SPDE might depend on the control. In the first part of the paper, we formulate a sufficient and a necessary maximum principle for this type of control problem, in two cases: (1) When the control is allowed to depend both on time t and on the space variable x. (2) When the control is not allowed to depend on x. In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from nonlinear filtering, we transform this noisy observation SDE inside control problem into a full observation SPDE insider control problem. The results are illustrated by explicit examples.

The purpose of this grant has been to undertake research in the general area of adaptive-grid finite element methods in estimation and control problems. The overall goal has been to investigate adaptive gridding numerical algorithms and to develop a solid theory to handle the mathematical questions of convergence and stability which arise in the use of such algorithms. The P.I. has met this goal through the study of a number of adaptive-grid finite element methods for parameter estimation problems governed by distributed parameter systems. In addition, stability problems encountered in the implementation of such algorithm have led her to make extensive studies of the ill-posed nature of these problems. In particular, theoretical and numerical studies were undertaken concerning regularization schemes for a number of ill-posed estimation problems. Specific applications considered in this latter study include the inverse heat conduction problem and regularizing aspects of descent methods as used to minimize fit-to-data functions associated with linear estimation problems.