Optimal load scheduling of hydroelectric power stations.

SUMMARYThis paper presents a motion planning method for a simple wheeled robot in two cases: (i) where translational and rotational speeds are arbitrary, and (ii) where the robot is constrained to move forwards at unit speed. The motions are generated by formulating a constrained optimal control problem on the Special Euclidean group SE(2). An application of Pontryagin's maximum principle for arbitrary speeds yields an optimal Hamiltonian which is completely integrable in terms of Jacobi elliptic functions. In the unit speed case, the rotational velocity is described in terms of elliptic integrals, and the expression for the position is reduced to quadratures. Reachable sets are defined in the arbitrary speed case, and a numerical plot of the time-limited reachable sets is presented for the unit speed case. The resulting analytical functions for the position and orientation of the robot can be parametrically optimised to match prescribed target states within the reachable sets. The method is shown to be easily adapted to obstacle avoidance for static obstacles in a known environment.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

Pontryagin's maximum principle is only applicable when right hand side functions in a differential equation of a controlled system and an integrand in criterion function as well as their partial derivatives with respect to x are all continuous for both x and u. The optimal control problem in which any above condition is not satisfied is called a problem with discontinuity. In this case the state space is divided by boundary into some regions in which the maximum principle is applicable. In this paper one approach to find out the optimal control law for the problem is discussed. That is, d optimal backward time trajectories are sought in a whole state space by solving one after another the newly established optimal problem with the maximum principle for each region. Usually, costate vector experiences a jump at the discontinuity point but in our method it arises as the result of the solution and any account for jump condition is not required. The mechanism of jump, however, is also examined in this paper.

SUMMARYA mathematical model for leukemia therapy based on the Gompertzian law of cell growth is investigated. The effect of the therapy is described in terms of therapy functions. Two types of therapy functions are considered: monotone and non‐monotone. In the first case, the effect of the chemotherapy on the normal cells directly depends on the quantity of the chemotherapeutic agent. In the second case, we consider a therapy function with a threshold effect, which influences the leukemia cells. This effect can describe resistance to the therapy. The optimal control problem has the purpose of minimizing the number of the leukemia cells subject to the constraints of the normal cells, whose number must not fall below a prescribed critical limit and of the amount of the chemotherapeutic agents in the body of the patient (i.e. constraint on the toxicity). Furthermore, a third phase constraint on the cumulative amount of the medicine over the whole therapy process is considered, which can be interpreted as an economic factor of the therapy. This optimization problem is solved analytically with the help of the Pontryagin's Maximum Principle, extension of penalty functions and additional analysis. Copyright © 2013 John Wiley & Sons, Ltd.

This paper treats an optimization problem of time-varying network topology for continuous-time linear network systems. Although the topology from controllers to nodes has been widely investigated in the literature, this paper alternatively focuses on the topology among nodes. For the cost function, we employ the trace of the controllability Gramian to reduce the required control energy. For the constraint, we introduce an upper bound of the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> norm of the data transmission among nodes to find an effective communication schedule. Since the formulated optimization problem is combinatorial, we introduce a convex relaxation for its computational tractability. After reformulating the problem into an optimal control problem where Pontryagin's maximum principle can be applied, we give a sufficient condition under which the relaxation problem gives a solution of the original problem. We finally provide a numerical algorithm based on a derived optimality condition and illustrate the proposed method with an example.

In this article, the optimal boundary control for a generalised Korteweg–de Vries equation is considered. The Dubovitskii and Milyutin functional analytical approach is adopted in the investigation of Pontryagin's maximum principles of the system in both fixed and free final horizon cases. The necessary conditions are, respectively, presented for the optimal boundary control problems in these two cases.

Application of Pontryagin's Maximum Principle to the Distributed Parameter System

In this paper, we consider the problem of controlling the intensity of a point process in order to maximize the probability that the number of points in a fixed interval equals a given integer, under the constraint that the intensity belong to some closed interval of R +. The problem is stated as a problem of optimization on the set of probabilities over the basic measurable space of point processes, and shown to be equivalent to a problem of deterministic control. Structural results concerning the set of optimal solutions are given. The existence of the latter is proven; the control is shown to be bang-bang and a complete solution can be obtained by application of Pontryagin's Maximum Principle.

We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

Some optimization problems concerning a substrate in a fluid are considered. The concentration of the substrate is affected by diffusion, convection, and elimination by enzymes, and the problem is to find the optimal distribution of enzymes. In this paper, the rate of elimination and the transmission coefficient are optimized. Mathematically, these problems are optimal control problems, and they are analyzed by means of Pontryagin's maximum principle.

&lt;p style='text-indent:20px;'&gt;In this work, we have proposed a new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex. In this approach, we transform the continuous optimal control problem into a quadratic optimization problem using the Cauchy discretization technique, then we solve it with the active-set method. In order to study the efficiency and the accuracy of the proposed approach, we developed an implementation with MATLAB, and we performed numerical experiments on several convex and non-convex linear-quadratic optimal control problems. The obtained simulation results show that our method is more accurate and more efficient than the method using the classical Euler discretization technique. Furthermore, it was shown that our method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.&lt;/p&gt;

In this paper we define a class of optimal control problems which we denote “embedded optimal control problems”. These are not true optimal control problems since the control system is not locally controllable on the manifold on which it is defined. Despite this, they allow for a well defined associated optimal control problem which does not admit abnormal extremals. We apply Pontryagin's maximum principle to the embedded optimal control problem to derive the generating differential equations for the normal and abnormal extremals. We show that the normal extremal generating equations in a sense contain the extremal generating equations for the associated optimal control problem. We show that this is not the case for the abnormal extremal generating equations. This has applications to the study of the optimal control of systems constrained to a given submanifold of a configuration space, for example the sphere or hypersphere. We apply the theory to three examples in order to illustrate its applicability and to show how it relates to well known results.

A Variational Approach to Determine the Optimal Power Distribution for Cycling in a Time Trial

Under some circumstances, dependent on a ship's velocity, the wave period and wave direction, certain courses induce heavy rolling and must be avoided. This paper proposes a computational method for the solution of optimal control problems in ship routing for ships with such limited manoeuvrability. Known results for the control problem of Bolza with additional constraints are interpreted in terms of this new problem. This approach is equivalent to the application of Pontryagin's maximum principle. The method is an extension of an earlier method dealing with the meteorological navigation of ships with unrestricted manoeuvrability and gives a more realistic picture of what really could happen in practice.

&lt;p style='text-indent:20px;'&gt;The optimal harvesting of biological resources, which is directly relevant to sustainable development, has attracted more attention. In this paper, we first prove the existence and uniqueness of generalized solution of a size-stage-structured population model while the optimal harvesting effort is discontinuous. Next, we demonstrate the existence of the optimal harvesting policy. Further, based on the idea of the Pontryagin's maximum principle of the optimal control problem in ordinary differential equations, we derive the maximum principle describing the optimal control. Finally, the dynamical behavior of the population is simulated by solving the corresponding optimality system numerically with an algorithm based on the method of backward Euler implicit finite-difference approximation. The numerical simulations indicate harvesting activity will reduce the quantity of the population and that increasing harvesting cost will result in less adult harvested. This provides guideline of implementing harvesting tactic to guarantee the persistence of the population.&lt;/p&gt;