Numerical computation of singular control problems with application to optimal heating and cooling by solar energy

The method presented here is an extension of the multiple shooting algorithm in order to handle multipoint boundary-value problems and problems of optimal control in the special situation of singular controls or constraints on the state variables. This generalization allows a direct treatment of (nonlinear) conditions at switching points. As an example a model of optimal heating and cooling by solar energy is considered. The model is given in the form of an optimal control problem with three control functions appearing linearly and a first order constraint on the state variables. Numerical solutions of this problem by multiple shooting techniques are presented.     

The finite topology and variational calculus on nontopological vector spaces

Certain aspects of the calculus of variations are presented in the setting of nontopological vector spaces, and the results are shown to have certain advantages in the investigation of various optimization problems of economics that seem more directly accessible by these techniques than by the maximum principle of optimal control theory.     

Multiple shooting methods for parabolic optimal control problems with control constraints

In the context of ordinary differential equations, shooting techniques are a state-of-the-art solver component, whereas their application in the framework of partial differential equations (PDE) is still at an early stage. We present two multiple shooting approaches for optimal control problems (OCP) governed by parabolic PDE. Direct and indirect shooting for PDE optimal control stem from the same extended problem formulation. Our approach reveals that they are structurally similar but show major differences in their algorithmic realizations. In the presented numerical examples we cover a nonlinear parabolic optimal control problem with additional control constraints. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Seismic Disturbance Rejection of Structures*

Using results of disturbance rejection in optimal control and computational mechanics' techniques, a new approach for the design of robust structural control systems for aseismic design applications is presented. A shear-type frame structure is used to outline the proposed methodology.     

Optimal Guidance of the Isotropic Rocket in the Presence of Wind

We address the minimum-time guidance problem for the so-called isotropic rocket in the presence of wind under an explicit constraint on the acceleration norm. We consider the guidance problem to a prescribed terminal position and a circular target set with a free terminal velocity in both cases. We employ standard techniques from optimal control theory to characterize the structure of the optimal guidance law as well as the corresponding minimum time-to-go function. It turns out that the complete characterization of the solution to the optimal control problem reduces to the solution of a system of nonlinear equations in triangular form. Numerical simulations, that illustrate the theoretical developments, are presented.     

Discrete Adjoint based Sensitivity Analysis for Optimal Active Flow Control of High-Lift Configurations

Blowing and suction type of active flow control techniques can be used to delay the flow separation on the flap and to enhance the aerodynamic performance of high-lift configurations. Effective separation control and maximum enhancement in the mean lift coefficient are achieved by finding the optimal actuation parameters. The optimal set of actuation parameters can be obtained by combining the gradient based algorithms with discrete adjoints. In the present work, an unsteady discrete adjoint incompressible RANS solver is developed for the optimal active separation control. The adjoint solver is applied to the test case of active flow control on the flap of a 2D high-lift configuration. Sensitivity gradients are presented to demonstrate the accuracy of unsteady adjoint RANS solver. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Successive approximation technique for a class of large-scale NLP problems and its application to dynamic programming

In this paper, two successive approximation techniques are presented for a class of large-scale nonlinear programming problems with decomposable constraints and a class of high-dimensional discrete optimal control problems, respectively. It is shown that: (a) the accumulation point of the sequence produced by the first method is a Kuhn-Tucker point if the constraint functions are decomposable and if the uniqueness condition holds; (b) the sequence converges to an optimum solution if the objective function is strictly pseudoconvex and if the constraint functions are decomposable and quasiconcave; and (c) similar conclusions for the second method hold also for a class of discrete optimal control problems under some assumptions.     

Improved Optimal Control Methods Based Upon the Adjoining Cell Mapping Technique

In this paper, cell mapping methods are studied and refined for the optimal control of autonomous dynamical systems. First, the method proposed by Hsu (Ref. 1) is analyzed and some improvements are presented. Second, adjoining cell mapping (ACM), based on an adaptive time of integration (Refs. 2–3), is formulated as an alternative technique for computing optimal control laws of nonlinear systems, employing the cellular state-space approximation. This technique overcomes the problem of determining an appropriate duration of the integration time for the simple cell mapping method and provides a suitable mapping for the search procedures. Artificial intelligence techniques, together with some improvements on the original formulation lead to a very efficient algorithm for computing optimal control laws with ACM (CACM). Several examples illustrate the performance of the CACM algorithm.     

On a Class of Direct Optimization Problems

Based on an earlier publication (Ref. 1), a coordinate transformation is proposed, which allows the direct global extremization of a class of integrals without the use of comparison methods such as variational or field techniques. This direct method is shown to be applicable to a class of unconstrained optimal control problems. A motivation for the proposed method as well as applications are presented.     

An approach for an algorithmic solution of discrete optimal control problems and their game-theoretical extension

We consider time discrete systems which are described by a system of difference equations. The related discrete optimal control problems are introduced. Additionally, a gametheoretic extension is derived, which leads to general multicriteria decision problems. The characterization of their optimal behavior is studied. Given starting and final states define the decision process; applying dynamic programming techniques suitable optimal solutions can be gained. We generalize that approach to a special gametheoretic decision procedure on networks. We characterize Nash equilibria and present sufficient conditions for their existence. A constructive algorithm is derived. The sufficient conditions are exploited to get the algorithmic solution. Its complexity analysis is presented and at the end we conclude with an extension to the complementary case of Pareto optima.Dmitrii Lozovanu was Supported by BGP CRDF-MRDA MOM2-3049-CS-03.     

Stochastic stability and robust control for sampled-data systems with Markovian jump parameters

In this paper, the problems of stochastic stability and robust control for a class of uncertain sampled-data systems are studied. The systems consist of random jumping parameters described by finite-state semi-Markov process. Sufficient conditions for stochastic stability or exponential mean square stability of the systems are presented. The conditions for the existence of a sampled-data feedback control and a multirate sampled-data optimal control for the continuous-time uncertain Markovian jump systems are also obtained. The design procedure for robust multirate sampled-data control is formulated as linear matrix inequalities (LMIs), which can be solved efficiently by available software toolboxes. Finally, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed techniques.     

Application of chaos-based chaotic invasive weed optimization techniques for environmental OPF problems in the power system

This paper presents efficient chaotic invasive weed optimization (CIWO) techniques based on chaos for solving optimal power flow (OPF) problems with non-smooth generator fuel cost functions (non-smooth OPF) with the minimum pollution level (environmental OPF) in electric power systems. OPF problem is used for developing corrective strategies and to perform least cost dispatches. However, cost based OPF problem solutions usually result in unattractive system gaze emission issue (environmental OPF). In the present paper, the OPF problem is formulated by considering the emission issue. The total emission can be expressed as a non-linear function of power generation, as a multi-objective optimization problem, where optimal control settings for simultaneous minimization of fuel cost and gaze emission issue are obtained. The IEEE 30-bus test power system is presented to illustrate the application of the environmental OPF problem using CIWO techniques. Our experimental results suggest that CIWO techniques hold immense promise to appear as efficient and powerful algorithm for optimization in the power systems.     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Optimal strategies for water quality control

In this paper, the use of optimal control theory to obtain optimal strategies for the control of aquatic models is illustrated. Several types of control variables are used including the rate of nutrient application and the rates of change of nutrient concentration in both the phytoplankton and zooplankton populations. Techniques are given to show how optimal control theory can be applied to several models with different states and control variables constraints. Explicit expressions and optimality conditions are given for singular controls whenever they exist. Some numerical techniques are suggested to couple the optimal control parts in the proper sequence.     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Numerical solution of minimax problems of optimal control,part 2

In a previous paper (Part 1), we presented general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We considered two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).In this paper, the transformation techniques presented in Part 1 are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer. Both the single-subarc approach and the multiple-subarc approach are employed. Three test problems characterized by known analytical solutions are solved numerically.It is found that the combination of transformation techniques and sequential gradient-restoration algorithm yields numerical solutions which are quite close to the analytical solutions from the point of view of the minimax performance index. The relative differences between the numerical values and the analytical values of the minimax performance index are of order 10^–3 if the single-subarc approach is employed. These relative differences are of order 10^–4 or better if the multiple-subarc approach is employed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

Dinkelbach Approach to Solving a Class of Fractional Optimal Control Problems

We consider optimal control problems with functional given by the ratio of two integrals (fractional optimal control problems). In particular, we focus on a special case with affine integrands and linear dynamics with respect to state and control. Since the standard optimal control theory cannot be used directly to solve a problem of this kind, we apply Dinkelbach’s approach to linearize it. Indeed, the fractional optimal control problem can be transformed into an equivalent monoparametric family {P_q} of linear optimal control problems. The special structure of the class of problems considered allows solving the fractional problem either explicitly or requiring straightforward classical numerical techniques to solve a single equation. An application to advertising efficiency maximization is presented. This work was partially supported by the Università Ca’ Foscari, Venezia, Italy, the MIUR (PRIN cofinancing 2005), the Council for Grants (under RF President) and State Aid to Fundamental Science Schools (Grant NSh-4113.2008.6). We thank Angelo Miele, Panos Pardalos and the anonymous referees for comments and suggestions.     

Optimal reinsurance–investment problem in a constant elasticity of variance stock market for jump‐diffusion risk model

In this paper, we consider the jump‐diffusion risk model with proportional reinsurance and stock price process following the constant elasticity of variance model. Compared with the geometric Brownian motion model, the advantage of the constant elasticity of variance model is that the volatility has correlation with the risky asset price, and thus, it can explain the empirical bias exhibited by the Black and Scholes model, such as volatility smile. Here, we study the optimal investment–reinsurance problem of maximizing the expected exponential utility of terminal wealth. By using techniques of stochastic control theory, we are able to derive the explicit expressions for the optimal strategy and value function. Numerical examples are presented to show the impact of model parameters on the optimal strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

Regularized state-constrained boundary optimal control of the Navier-Stokes equations

The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau-Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.