Infinite-dimensional Control Problem Research Articles

A steepest descent algorithm for the optimal control of a cascaded hydropower system

Optimal power generation along the cascaded Kainji-Jebba hydroelectric power system had been very difficult to achieve. The reservoirs operating heads are being affected by possible variation in impoundments upstream, stochastic factors that are weather-related, availability of the turbo-alternators and power generated at any time. Proposed in this paper, is an algorithm for solving the optimal release of water on the cascaded hydropower system based on steepest descent method. The uniqueness of this work is the conversion of the infinite dimensional control problem to a finite one, the introduction of clever techniques for choosing the steepest descent step size in each iteration and the nonlinear penalty embedded in the procedure. The control algorithm was implemented in an Excel VBA® environment to solve the ormulated Lagrange problem within an accuracy of 0.03%. It is recommended for use in system studies and control design for the optimal power generation in the cascaded hydropower system.

Koopman operator-based model reduction for switched-system control of PDEs

We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.

Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

We establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode control problem in a number of concrete nonlinear partial differential equations. We also develop applications concerning associated classes of stochastic partial differential equations (SPDEs). In particular, we study the support properties of probability laws corresponding to these SPDEs as well as provide applications concerning the ergodic and mixing properties of invariant measures for these stochastic systems.

Education

This issue contains two articles in the Education section. The first one is “An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation,” by Matthew Kelly. The author presents a tutorial on discretization ideas for optimal control problems with ordinary differential equations. The article starts with a simple example, which is used to introduce some basic notions and a general formulation of an optimal control problem, where the dynamics is described by an ordinary differential equation. That formulation includes general state and control constraints. The main focus is put on methods, which construct a polynomial approximation of the state and control functions. The state and control spaces are discretized, and the infinite-dimensional control problem becomes a finite-dimensional nonlinear optimization problem. Piecewise linear and quadratic approximations of the state and the control are explained and illustrated by figures. Attention is brought to the error accumulated in the numerical calculation of integrals according to several methods. The author offers some practical suggestions for the initialization of the numerical optimization algorithm, as well as for the interpretation of the solution provided by the optimization method. The optimization models obtained as a result of the discretization procedure typically lack convexity properties; this entails that the optimization procedure would provide a local solution, which would not be globally optimal, in general. Some ideas about how to look for a better solution or confirm the quality of the one at hand are provided. Further discussion includes suggestions for smoothing of nondifferentiable functions that may occur in the problem and a comparison of open loop vs. closed loop solutions. The last section of the paper briefly surveys methods based on discretization and numerical treatment of the conditions of optimality. The author has made his software---a general-purpose MATLAB library for solving trajectory optimization problems, complete with documentation---publicly available at https://GitHub.com/MatthewPeterKelly/OptimTraj. The electronic supplement is described in Appendix A. In addition to the optimization software, it includes all example problems, which are presented in the paper. The second article is “How to Mitigate Sloshing,” authored by H. Ockendon and J. R. Ockendon. The authors start with an example from everyday life familiar to most of us: carrying a mug of coffee can often lead to spillage. How should we reduce this danger? The authors devise a model of this dynamical system under certain simplifying assumptions. A point (representing the hand carrying the coffee) is oscillating horizontally with a given frequency and amplitude (representing the walk) about a fixed mean position. A spring connects the hand to the coffee mug, which can slide freely on a smooth table. The effect of that spring on the motion of the liquid is the focus of attention. It is assumed additionally that the mug is rectangular and that no motion perpendicular to the direction of the action of the spring is present. The goal becomes to model and analyze the two-dimensional motion of liquid in a rectangular container of given width and depth. To this end, the authors introduce and analyze a mathematical model based on partial differential equations. Although the article is compact, the authors make a number of suggestions for future investigation and extensions.

A Pontryagin Maximum Principle for Infinite-Dimensional Problems

A basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable “needle-like control variations.” We use this idea to prove the main result of the present paper—a Pontryagin maximum principle for infinite-dimensional optimal control problems with pointwise terminal constraints in arbitrary Banach state space. By refining the classical variational technique we are able to replace the differentiability of the norm of the state space (guaranteed by the strict convexity of its dual norm, which is assumed in the known results) by a separation argument. We also drop another key assumption which is common in the existing literature on infinite-dimensional control problems—that the set of variations (in the state space) of the state trajectory's endpoint (resulting from the control variations) be finite-codimensional. Instead, we require only that it has nonempty interior in its closed affine hull. As an application of the abstract result we present an illustrative example—an optimal control problem for an age-structured system with pointwise terminal state constraints.

A Paradox in the Approximation of Dirichlet Control Problems in Curved Domains.

In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain $\Omega$. To solve this problem numerically, it is usually necessary to approximate $\Omega$ by a (typically polygonal) new domain $\Omega_h$. The difference between the solutions of both infinite-dimensional control problems, one formulated in $\Omega$ and the second in $\Omega_h$, was studied in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746–3780], where an error of order $O(h)$ was proved. In [K. Deckelnick, A. Günther, and M. Hinze, SIAM J. Control Optim., 48 (2009), pp. 2798–2819], the numerical approximation of the problem defined in $\Omega$ was considered. The authors used a finite element method such that $\Omega_h$ was the polygon formed by the union of all triangles of the mesh of parameter h. They proved an error of order $O(h^{3/2})$ for the difference between continuous and discrete optimal controls. Here we show that the estimate obtained in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746–3780] cannot be improved, which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from $\Omega$ to $\Omega_h$.

Approximation of Boundary Control Problems on Curved Domains

In this paper we consider boundary control problems associated to a semilinear elliptic equation defined in a curved domain $\Omega$. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of $\Omega$ by an appropriate domain $\Omega_h$ (typically polygonal) is required. Here we do not consider the numerical approximation of the control problems. Instead, we formulate the corresponding infinite dimensional control problems in $\Omega_h$, and we study the influence of the replacement of $\Omega$ by $\Omega_h$ on the solutions of the control problems. Our goal is to compare the optimal controls defined on $\Gamma=\partial\Omega$ with those defined on $\Gamma_h=\partial\Omega_h$ and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates.

Fast iterative solution of elliptic control problems in wavelet discretization

We investigate wavelet methods for the efficient numerical solution of a class of control problems constrained by a linear elliptic boundary value problem where the cost functional may contain fractional Sobolev norms of the control and the state. Starting point is the formulation of the infinite-dimensional control problem in terms of (boundary-adapted biorthogonal spline-) wavelets, involving only ℓ 2 norms of wavelet expansion coefficients (where different norms are realized by a diagonal scaling together with a Riesz map) and constraints in form of an ℓ 2 isomorphism. The coupled system of equations resulting from optimization is solved by an inexact conjugate gradient (CG) method for the control, which involves the approximate inversion of the primal and the adjoint operator using again CG iterations. Starting from a coarse discretization level, we use nested iteration to solve the coupled system on successively finer uniform discretizations up to discretization error accuracy on each level. The resulting inexact CG scheme is a ‘fast solver’: it is of asymptotic optimal complexity in the sense that the overall computational effort to compute the solution up to discretization error on the finest grid is proportional to the number of unknowns on that grid, a consequence of grid-independent condition numbers of the linear operators in wavelet coordinates. In the numerical examples we study the choice of different norms and the regularization parameter in the cost functional and their effect on the solution. Moreover, for different situations the performance of the fully iterative inexact CG scheme is investigated, confirming the theoretical results.

The problem of measurement feedback control

The deterministic problem of constructing synthesized control strategies from the results of available observations under unknown but bounded disturbances subjected in advance to hard bounds on the uncertain parameters is considered. The suggested solutions rely on methods of dynamic programming and set-valued analysis and are formulated in terms of the Hamiltonian formalism. It is shown that the problem can be separated into two — a finite-dimensional problem of estimation and an infinite-dimensional control problem.

Existence and Structure of Optimal Solutions of Infinite-Dimensional Control Problems

In this work we analyze the structure of optimal solutions for a class of infinite-dimensional control systems. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson, Haurie, and Jabrane to a situation where the trajectories are not necessarily bounded. Also, we show that an optimal trajectory defined on an interval [0,τ] is contained in a small neighborhood of the optimal steady-state in the weak topology for all t ∈ [0,τ] \backslash E , where E \subset [0,τ] is a measurable set such that the Lebesgue measure of E does not exceed a constant which depends only on the neighborhood of the optimal steady-state and does not depend on τ .

A Remark on Existence of Solutions of Infinite-Dimensional Noncompact Optimal Control Problems

Without compactness of the semigroup, optimal controls for semilinear infinite dimensional control problems may not exist even under the usual convexity--lower semicontinuity assumptions. This is shown with two counterexamples, and two fixes for nonexistence are discussed.

Necessary conditions for infinite-dimensional control problems

We consider infinite-dimensional nonlinear programming problems which consist of minimizing a functionf 0(u) under a target set constraint. We obtain necessary conditions for minima that reduce to the Kuhn-Tucker conditions in the finite-dimensional case. Among other applications of these necessary conditions and related results, we derive Pontryagin’s maximum principle for a class of control systems described by semilinear equations in Hilbert space and study convergence properties of sequences of near-optimal controls for these systems.

Constructive control of the motion of oscillating systems with discrete and distributed parameters

The problem of controlling the motion of mechanical systems over an asymptotically large but fixed time interval is considered. The controlled objects may include elements with discrete parameters (material points, rigid bodies, weightless springs, etc.) and oscillating components with distributed parameters (strings, rods, elastic beams and shafts, membranes, plates, cavities with stratified liquid, etc.), whose frequency spectrum is denumerably infinite. The controlling actions, whether kinematic or dynamic in nature, are assumed to be concentrated with respect to the space variables. They may be movable, applied to the absolutely rigid parts of the system and / or fixed at the boundaries of the distributed elements (boundary control). This kind of control is of value in applications. On the assumption that techniques of mathematical physics /1, 2/ or the method of moments /3, 4/ yield an infinite-dimensional control problem for the Fourier coefficients of the solution relative to a set of basis functions for the boundary-value problem, an asymptotic approach is proposed for constructing approximate controls and the resulting scheme for the approximate solution of the problem is shown to be legitimate. A specific problem is examined as an illustration - rotation of an elastic rod in the plane by a torque applied at its end.

Constrained Controllability in Banach Spaces

The aim of this paper is to study null-controllability of the linear infinite dimensional control problem $\dot x = Ax + Bu$ where the control u is constrained to lie in a convex, weakly compact subset $\Omega $ of the control space with $0 \in \Omega $. A necessary and sufficient condition for a particular initial state to be $\Omega $-null-controllable within a fixed, finite time T is given. The result is extended to the case $\Omega = $ convex cone with vertex at 0. Applications to the one-dimensional heat- and wave-equation are given.

Some Results on Bellman Equation in Hilbert Spaces

We give an existence result on the Bellman equation related to an infinite dimensional control problem.

A Decoupling Approach to the Control of Large Spaceborne Antenna Systems

A simple practical method for designing antenna-feed positioning control systems for large deployable spaceborne antenna systems with flexible booms is proposed. The approach is based on the mechanical decoupling of the antenna-feed from the boom so that the positioning control system can be designed without taking boom dynamics into consideration, thus avoiding a complex infinite dimensional control problem. The basic idea is illustrated by a simple angular positional control system attached to a flexible boom restricted to torsional motion only. The application of this approach to more complex situations is discussed briefly.

Suboptimal control of a nuclear reactor using functional analysis

A functional optimization technique based on the abstract minimum norm problem in Hilbert spaces is applied to obtain a suboptimal control policy for the problem of adjusting the flux distribution, in finite time.Restricting the control action to be expanded in terms of a finite set of arbitrary known functions of time, the infinite dimensional control problem is reduced to a finite dimensional one. The control action is subject to linear constraints. The method is described for a general, linearized, distributed reactor model. An example is presented.

Infinite dimensional control problems I: On the closure of the set of attainable states for linear systems

Infinite dimensional control problems I: On the closure of the set of attainable states for linear systems