Optimal control of Gao beam contact with double Signorini conditions

State Transition Tensors for Continuous-Thrust Control of Three-Body Relative Motion

This work is concerned with the maximum principles for optimal control problems governed by 3-dimensional Navier--Stokes equations. Some types of state constraints (time variables) are considered.

In the last decade, pseudospectral methods have become popular for solving optimal control problems. Pseudospectral methods do not need prior knowledge about the optimal control structure and are thus very flexible for problems with complex path constraints, which are common in optimal train control, or train trajectory optimization. Practical optimal train control problems are nonsmooth with discontinuities in the dynamic equations and path constraints corresponding to gradients and speed limits varying along the track. Moreover, optimal train control problems typically include singular solutions with a vanishing Hessian of the associated Hamiltonian. These characteristics make these problems hard to solve and also lead to convergence issues in pseudospectral methods. We propose a computational framework that connects pseudospectral methods with Pontryagin’s Maximum Principle allowing flexible computations, verification and validation of the numerical approximations, and improvements of the continuous solution accuracy. We apply the framework to two basic problems in optimal train control: minimum-time train control and energy-efficient train control, and consider cases with short-distance regional trains and long-distance intercity trains for various scenarios including varying gradients, speed limits, and scheduled running time supplements. The framework confirms the flexibility of the pseudospectral method with regards to state, control and mixed algebraic inequality path constraints, and is able to identify conditions that lead to inconsistencies between the necessary optimality conditions and the numerical approximations of the states, costates, and controls. A new approach is proposed to correct the discrete approximations by incorporating implicit equations from the optimality conditions. In particular, the issue of oscillations in the singular solution for energy-efficient driving as computed by the pseudospectral method has been solved.

This paper is concerned with the optimal distributed control of the viscous weakly dispersive Degasperis–Procesi equation in nonlinear shallow water dynamics. It is well known that the Pontryagin maximum principle, which unifies calculus of variations and control theory of ordinary differential equations, sets up the theoretical basis of the modern optimal control theory along with the Bellman dynamic programming principle. In this paper, we commit ourselves to infinite dimensional generalizations of the maximum principle and aim at the optimal control theory of partial differential equations. In contrast to the finite dimensional setting, the maximum principle for the infinite dimensional system does not generally hold as a necessary condition for optimal control. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the controlled viscous weakly dispersive Degasperis–Procesi equation. The necessary optimality condition is established for the problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, optimal control is applied to study generator bidding in an oligopolistic electricity market. The repeated bidding process in (hourly-based) real-time electricity markets is modeled as a dynamic feedback system; an optimal control problem is then formulated to explore individual generator's long-term/multiperiod optimization behavior. Particularly in our formulation, the periodic property of the system demand is considered. Several lemmas are included for concerning system stability. Based on the necessary conditions for optimality from the Pontryagin maximum principle, a sweeping method is proposed, and an optimal state-feedback control rule is then obtained via backward induction. Numerical results suggest that the generator who unilaterally applies optimal control for generation decisions will obtain more profits. A sensitivity analysis is also performed, identifying these market factors that affect the performance of optimal control.

Practical Methods for Optimal Control using Nonlinear Programming

In this paper, we investigate the optimal control of the Burgers equation. For both optimal distributed and (Neumann) boundary control problems, the Dubovitskii and Milyutin functional analytical approach is adopted in investigation of the Pontryagin maximum principles of the systems. The necessary optimality conditions are, respectively, presented for two kinds of optimal control problems in both fixed and free final horizon cases, four extremum problems in all. Moreover, in free final horizon case, the assumptions of admissible control set on convexity and non-empty interior are removed so that it can be any set including an interesting case contains only finite many points. Finally, a remark on how to utilize the obtained results is also made for the illustration.

Matrix quadratic equation solution derivation applied in finding steady state solutions of Riccati differential equations with constant coefficients

In this paper, optimal control is applied to study generator bidding in an oligopolistic electricity market. The repeated bidding process in (hourly-based) real-time electricity markets is modeled as a dynamic feedback system; an optimal control problem is then formulated to explore individual generator's long-term/multiperiod optimization behavior. Particularly in our formulation, the periodic property of the system demand is considered. Several lemmas are included for concerning system stability. Based on the necessary conditions for optimality from the Pontryagin maximum principle, a sweeping method is proposed, and an optimal state-feedback control rule is then obtained via backward induction. Numerical results suggest that the generator who unilaterally applies optimal control for generation decisions will obtain more profits. A sensitivity analysis is also performed, identifying these market factors that affect the performance of optimal control

A neural network based optimal control synthesis is presented for solving optimal control problems with control and state constraints. The optimal control problem is transcribed into a nonlinear programming problem which is implemented with adaptive critic neural network. The proposed simulation method is illustrated by the optimal control problem of nitrogen transformation cycle model. Results show that adaptive critic based systematic approach holds promise for obtaining the optimal control with control and state constraints. INTRODUCTION Optimal control of nonlinear systems is one of the most active subjects in control theory. There is rarely an analytical solution although several numerical computation approaches have been proposed (for example, see (Polak, 1997), (Kirk, 1998)) for solving a optimal control problem. Most of the literature that deals with numerical methods for the solution of general optimal control problems focuses on the algorithms for solving discretized problems. The basic idea of these methods is to apply nonlinear programming techniques to the resulting finite dimensional optimization problem (Buskens at al., 2000). When Euler integration methods are used, the recursive structure of the resulting discrete time dynamic can be exploited in computing first-order necessary condition. In the recent years, the multi-layer feedforward neural networks have been used for obtaining numerical solutions to the optimal control problem. (Padhi at al., 2001), (Padhi et al., 2006). We have taken hyperbolic tangent sigmoid transfer function for the hidden layer and a linear transfer function for the output layer. The paper extends adaptive critic neural network architecture proposed by (Padhi at al., 2001) to the optimal control problems with control and state constraints. The paper is organized as follows. In Section 2, the optimal control problems with control and state constraints are introduced. We summarize necessary optimality conditions and give a short overview of basic result including the iterative numerical methods. Section 3 discusses discretization methods for the given optimal control problem. It also discusses a form of the resulting nonlinear programming problems. Section 4 presents a short description of adaptive critic neural network synthesis for optimal problem with state and control constraints. Section 5 consists of a nitrogen transformation model. In section 6, we apply the discussed methods to the nitrogen transformation cycle. The goal is to compare short-term and long-term strategies of assimilation of nitrogen compounds. Conclusions are presented in Section 7. OPTIMAL CONTROL PROBLEM We consider a nonlinear control problem subject to control and state constraints. Let x(t) ∈ R denote the state of a system and u(t) ∈ R the control in a given time interval [t0, tf ]. Optimal control problem is to minimize F (x, u) = g(x(tf )) + ∫ tf t0 f0(x(t), u(t))dt (1)

Special issue on “Optimal design and operation of energy systems”

This book is devoted to a consistent presentation of the recent results obtained by the authors related to controlled systems created based on the Lotka-Volterra competition model, as well as to theoretical and numerical study of the corresponding optimal control problems. These controlled systems describe various modern methods of treating blood cancers, and the optimal control problems stated for such systems, reflect the search for the optimal treatment strategies. The main tool of the theoretical analysis used in this book is the Pontryagin maximum principle - a necessary condition for optimality in optimal control problems. Possible types of the optimal blood cancer treatment - the optimal controls - are obtained as a result of analytical investigations and are confirmed by corresponding numerical calculations. This book can be used as a supplement text in courses of mathematical modeling for upper undergraduate and graduate students. It is our believe that this text will be of interest to all professors teaching such or similar courses as well as for everyone interested in modern optimal control theory and its biomedical applications.

This paper is devoted to applications of modern methods of variational analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main focus is on the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the “lack of compactness” in infinite dimensions, which leads to imposing certain “normal compactness” properties and developing their comprehensive calculus, together with appropriate calculus rules of generalized differentiation. On the other hand, one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems. In particular, we fully avoid the requirements of this type imposed on target endpoint sets in infinite-dimensional optimal control for discrete-time inclusions.

The optimal control problem is investigated for oscillation processes, described by integrodifferential equations with the Fredholm operator when functions of external and boundary sources nonlinearly depend on components of optimal vector controls. Optimality conditions having specific properties in the case of vector controls were found. A sufficient condition is established for unique solvability of the nonlinear optimization problem and its complete solution is constructed in the form of optimal control, an optimal process, and a minimum value of the functional.

Singular Optimal Control Problems