An embedding theorem for differential equations

The present paper establishes a general embedding theorem for differential equations, which is useful in establishing the maximum principle for optimal control problems.The preparation of this paper was sponsored by the Office of Naval Research and the US Army Research Office.     

Existence theorems for multidimensional Lagrange problems

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.This research was partially supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-942-65.     

Epi-convergent discretization of the generalized Bolza problem in dynamic optimization

The paper is devoted to well-posed discrete approximations of the so-called generalized Bolza problem of minimizing variational functionals defined via extended-real-valued functions. This problem covers more conventional Bolza-type problems in the calculus of variations and optimal control of differential inclusions as well of parameterized differential equations. Our main goal is find efficient conditions ensuring an appropriate epi-convergence of discrete approximations, which plays a significant role in both the qualitative theory and numerical algorithms of optimization and optimal control. The paper seems to be the first attempt to study epi-convergent discretizations of the generalized Bolza problem; it establishes several rather general results in this direction. Research of B. S. Mordukhovich was partially supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168. Research of T. Pennanen was supported by the Finnish Academy of Sciences under contract No. 3385.     

Differential riccati equation for the active control of a problem in structural acoustics

In this paper, we provide results concerning the optimal feedback control of a system of partial differential equations which arises within the context of modeling a particular fluid/structure interaction seen in structural acoustics, this application being the primary motivation for our work. This system consists of two coupled PDEs exhibiting hyperbolic and parabolic characteristics, respectively, with the control action being modeled by a highly unbounded operator. We rigorously justify an optimal control theory for this class of problems and further characterize the optimal control through a suitable Riccati equation. This is achieved in part by exploiting recent techniques in the area of optimization of analytic systems with unbounded inputs, along with a local microanalysis of the hyperbolic part of the dynamics, an analysis which considers the propagation of singularities and optimal trace behavior of the solutions.Research partially supported by National Science Foundation Grant DMS #9504822 and Army Research Office Grant #35170-MA.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Feedback exact null controllability for unbounded control problems in Hilbert space

Terminal constraint optimal control problems with unbounded control operators are considered. It is shown that the optimal solutions can be represented in a feedback form via a solution of an appropriate Riccati equation. In particular, it is proved that, for systems described by partial differential equations with infinite speed of propagation, boundary exact null controllability can be realized in feedback form.This work was partially supported by the National Science Foundation, Grant No. DMS-89-02811, and by the Air Force Office of Scientific Research, Grant No. AFOSR-89-0511 DEF.     

Multiobjective PDE-constrained optimization using the reduced-basis method

In this paper the reduced basis (RB) method is applied to solve quadratic multiobjective optimal control problems governed by linear parametrized variational equations. These problems often arise in applications, where the quality of the system behavior has to be measured by more than one criterium. The weighted sum method is exploited for defining scalar-valued linear-quadratic optimal control problems built by introducing additional optimization parameters. The optimal controls corresponding to specific choices of the optimization parameters are efficiently computed by the RB method. The accuracy is guaranteed by an a-posteriori error estimate. An effective sensitivity analysis allows to further reduce the computational times for identifying a suitable and representative set of optimal controls.     

Anchoring conditions for the sequential gradient-restoration algorithm and the modified quasilinearization algorithm for optimal control problems with bounded state

In Refs. 1–2, the sequential gradient-restoration algorithm and the modified quasilinearization algorithm were developed for optimal control problems with bounded state. These algorithms have a basic property: for a subarc lying on the state boundary, the state boundary equations are satisfied at every iteration, if they are satisfied at the beginning of the computational process. Thus, the subarc remains anchored on the state boundary. In this paper, the anchoring conditions employed in Refs. 1–2 are derived.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185.     

Spline approximations for nonlinear hereditary control systems

A spline-based approximation scheme is discussed for optimal control problems governed by nonlinear nonautonomous delay differential equations. The approximating framework reduces the original control problem to a sequence of optimization problems governed by ordinary differential equations. Convergence proofs, which appeal directly to dissipative-type estimates for the underlying nonlinear operator, are given and numerical findings are summarized.This work was supported in part by the Air Force Office of Scientific Research under Contract No. AFOSR-76-3092D, in part by the National Science Foundation under Grants Nos. NSF-MCS-79-05774-05 and NSF-MCS-82-00883, and in part by the US Army Research Office under Contract No. ARO-DAAG29-79-C-0161. The results reported here are a portion of the author's doctoral dissertation written under the supervision of Professor H. T. Banks, Brown University. The author is indebted to Professor Banks for his many valuable comments and suggestions during the course of this work.Part of this research was completed while the author was a visitor at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia.     

A finite element method for some integral equations of the first kind

This paper discusses a finite element approximation for a class of singular integral equations of the first kind. These integral equations are deduced from Dirichlet problems for strongly elliptic differential equations in two independent variables. By a variation of technique due to Aubin, it is shown that the Galerkin method with finite elements as trial functions leads to an optimal rate of convergence.     

Optimality,stability, and convergence in nonlinear control

Sufficient optimality conditions for infinite-dimensional optimization problems are derived in a setting that is applicable to optimal control with endpoint constraints and with equality and inequality constraints on the controls. These conditions involve controllability of the system dynamics, independence of the gradients of active control constraints, and a relatively weak coercivity assumption for the integral cost functional. Under these hypotheses, we show that the solution to an optimal control problem is Lipschitz stable relative to problem perturbations. As an application of this stability result, we establish convergence results for the sequential quadratic programming algorithm and for penalty and multiplier approximations applied to optimal control problems.This research was supported by the U.S. Army Research Office under Contract. Number DAAL03-89-G-0082, by the National Science Foundation under Grant Number DMS 9404431, and by Air Force Office of Scientific Research under Grant Number AFOSR-88-0059. A. L. Dontchev is on leave from the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

Parametric Sensitivity Analysis of Perturbed PDE Optimal Control Problems with State and Control Constraints

We study parametric optimal control problems governed by a system of time-dependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute the optimal control functions and the so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control.The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization then transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP), which can be solved efficiently by SQP methods.Second-order sufficient conditions can be checked numerically and we propose to apply an NLP-based approach for the robust computation of the sensitivity differentials of the optimal solutions with respect to the perturbation parameters. The numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation.Communicated by H. J. Pesch     

Approximating infinite horizon stochastic optimal control in discrete time with constraints

Traditional approaches to solving stochastic optimal control problems involve dynamic programming, and solving certain optimality equations. When recast as stochastic programming problems, structural aspects such as convexity are retained, and numerical solution procedures based on decomposition and duality may be exploited. This paper explores a class of stationary, infinite-horizon stochastic optimization problems with discounted cost criterion. Constraints on both states and controls are permitted, and modeled in the objective function by allowing it to take infinite values. Approximating techniques are developed using variational analysis, and intuitive lower bounds are obtained via averaging the future. These bounds could be used in a finite-time horizon stochastic programming setting to find solutions numerically. Research supported in part by a grant of the National Science Foundation. AMS Classification 46N10, 49N15, 65K10, 90C15, 90C46     

Stabilization and control of distributed systems with time-dependent spatial domains

This paper considers the problem of the stabilization and control of distributed systems with time-dependent spatial domains. The evolution of the spatial domains with time is described by a finite-dimensional system of ordinary differential equations, while the distributed systems are described by first-order or second-order linear evolution equations defined on appropriate Hilbert spaces. First, results pertaining to the existence and uniqueness of solutions of the system equations are presented. Then, various optimal control and stabilization problems are considered. The paper concludes with some examples which illustrate the application of the main results.This work was supported by the Air Force Office of Scientific Research, Grant No. AFOSR 86-0132, by the National Science Foundation, Grant No. 87-18473, and by the Jet Propulsion Laboratory, Pasadena, California.     

Analytic Solution for Fuel-Optimal Reconfiguration in Relative Motion

The current paper presents simple and general analytic solutions to the optimal reconfiguration of multiple satellites governed by a variety of linear dynamic equations. The calculus of variations is used to analytically find optimal trajectories and controls. Unlike what has been determined from previous research, the inverse of the fundamental matrix associated with the dynamic equations is not required for the general solution in the current study if a basic feature in the state equations is met. This feature is very common due to the fact that most relative motion equations are represented in the LVLH frame. The method suggested not only reduces the amount of calculations required, but also allows predicting the explicit form of optimal solutions in advance without having to solve the problem. It is illustrated that the optimal thrust vector is a function of the fundamental matrix of the given state equations, and other quantities, such as the cost function and the state vector during the reconfiguration, can be concisely represented as well. The analytic solutions developed in the current paper can be applied to most reconfiguration problems in linearized relative motions. Numerical simulations confirm the brevity and accuracy of the general analytic solutions developed in the current paper. This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the National Research Lab. Program funded by the Ministry of Education, Science and Technology (No. M10600000282-06J0000-28210).     

Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control

Semilinear parabolic differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. Applications to stochastic optimal control problems are studied by solving the associated Hamilton–Jacobi–Bellman equation. These results are applied to some controlled stochastic partial differential equations.     

Domain Decomposition,Optimal Control of Systems Governed by Partial Differential Equations,and Synthesis of Feedback Laws

We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The equations can be of elliptic, parabolic, or hyperbolic type. The space region supporting the partial differential equations is decomposed and the original global optimal control problem is reduced to a sequence of similar local optimal control problems set on the subdomains. The local problems communicate through transmission conditions, which take the form of carefully chosen boundary conditions on the interfaces between the subdomains. This domain decomposition method can be combined with any suitable numerical procedure to solve the local optimal control problems. We remark that it offers a good potential for using feedback laws (synthesis) in the case of time-dependent partial differential equations. A test problem for the wave equation is solved using this combination of synthesis and domain decomposition methods. Numerical results are presented and discussed. Details on discretization and implementation can be found in Ref. 1.     

Optimization of natural gas production by waterflooding

This paper discusses optimization of natural gas production using waterflooding of a gas reservoir. The functions governing the rate of gas withdrawal and the rate of water injection control the operation of the reservoir. These, together with volume and mass balance and the ideal gas law, give a simple system of ordinary differential equations modelling the reservoir. We use several physical and economic definitions of an optimal production rate. Under each definition, we establish the optimal controls. The relation between prices of water and natural gas and the optimal controls is discussed.Work performed under the auspices of the Energy Research and Development Administration.     

Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations

This work concerns analysis and error estimates for optimal control problems related to implicit parabolic equations. The minimization of the tracking functional subject to implicit parabolic equations is examined. Existence of an optimal solution is proved and an optimality system of equations is derived. Semi-discrete (in space) error estimates for the finite element approximations of the optimality system are presented. These estimates are symmetric and applicable for higher-order discretizations. Finally, fully-discrete error estimates of arbitrarily high-order are presented based on a discontinuous Galerkin (in time) and conforming (in space) scheme. Two examples related to the Lagrangian moving mesh Galerkin formulation for the convection-diffusion equation are described.