Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.      

On some optimal control problems and their finite difference approximations and regularization for quasilinear elliptic equations with controls in the coefficients

Mathematical statements of the optimal control problems for quasilinear elliptic equations with the controls in the variable coefficients of the equation of state are considered. Both local and integral constraints on the controls are considered. The objective functionals correspond to the optimization with respect to a certain number of quality indexes. Finite difference approximations of optimization problems are constructed, and estimates of the approximation error with respect to the state and to the objective functional are established. The weak convergence in control is proved. The approximations are regularized after Tikhonov. Interesting examples of some applied optimization problems that naturally lead to the nonlinear optimal control problems examined in this paper are considered.     

Regularity properties of optimal controls for problems with time-varying state and control constraints

In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Numerical Solution of Optimal Control Problems with Discrete-Valued System Parameters

In this paper, we propose a new approach to solve a class of optimal control problems involving discrete-valued system parameters. The basic idea is to formulate a problem of this type as a combination of a discrete global optimization problem and a standard optimal control problem, and then solve it using a two-level approach. Numerical results show that the proposed method is efficient and capable of finding optimal or near optimal solutions.     

Problem of optimal control by the natural frequency of oscillations of an orthotropic shell of revolution and its finite-dimensional approximation

The questions of optimization in problems of oscillations in orthotropic shells of revolution of variable thickness are studied for the case when the thickness and radius of curvature of the shell generatrix are used as the controls. Restrictions are imposed on the principal oscillation eigenfrequency, thickness, internal volume and other parameters. It is shown that a solution of the problem exists and, that the problem can be approximated by a sequence of the finite-dimensional problems. Certain questions of the optimal control in the problem concerning the oscillations of plates of variable thickness with the thickness serving as the control, were studied in /1–4/.     

Optimization of ship manoeuvring within the project GALILEOnautic

The goal of the project GALILEOnautic is to develop a system for autonomous navigation and optimal manoeuvring of cooperative ships within safety-critical areas. In this context, many challenges arise in the field of optimization and optimal control. The research presented here addresses one of them, namely, the calculation of optimal trajectories and controls for a group of cooperative ships navigating within the limits of a harbour. The adopted approach is to embed all aspects of the problem into a single optimal control problem, whose objective is to minimize the time to destination of each ship as well as their overall energy consumption. Path constraints are applied to maintain the ships at a safe distance from each other and from the infrastructure of the harbour. A two-ship scenario is implemented, for which optimal trajectories and controls are successfully computed. The numerical computation of this problem is performed using the software library WORHP, the official ESA NLP solver, and moreover the software TransWORHP, which transcribes optimal control problems into optimization problems. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameter-less algorithm for evolutionary-based optimization

The development of a simple, adaptive, parameter-less search algorithm was initiated by the need for an algorithm that is able to find optimal solutions relatively quick, and without the need for a control-parameter-setting specialist. Its control parameters are calculated during the optimization process, according to the progress of the search. The algorithm is intended for continuous and combinatorial problems. The efficiency of the proposed parameter-less algorithm was evaluated using one theoretical and three real-world industrial optimization problems. A comparison with other evolutionary approaches shows that the presented adaptive parameter-less algorithm has a competitive convergence with regards to the comparable algorithms. Also, it proves algorithm’s ability to finding the optimal solutions without the need for predefined control parameters.     

Analysis of a Two-Dimensional Thermal Cloaking Problem on the Basis of Optimization

For a two-dimensional model of thermal scattering, inverse problems arising in the development of tools for cloaking material bodies on the basis of a mixed thermal cloaking strategy are considered. By applying the optimization approach, these problems are reduced to optimization ones in which the role of controls is played by variable parameters of the medium occupying the cloaking shell and by the heat flux through a boundary segment of the basic domain. The solvability of the direct and optimization problems is proved, and an optimality system is derived. Based on its analysis, sufficient conditions on the input data are established that ensure the uniqueness and stability of optimal solutions.     

A filled function method for optimal discrete-valued control problems

In this paper, we present a new approach to solve a class of optimal discrete-valued control problems. This type of problem is first transformed into an equivalent two-level optimization problem involving a combination of a discrete optimization problem and a standard optimal control problem. The standard optimal control problem can be solved by existing optimal control software packages such as MISER 3.2. For the discrete optimization problem, a discrete filled function method is developed to solve it. A numerical example is solved to illustrate the efficiency of our method.     

A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems

We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann–Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz’s method. Examples show that the proposed approach is more accurate than recent methods available in the literature.     

Exploiting the Special Structure for Real‐Time Control of Optimal Control Problems with Linear Controls: Numerical Experience

Often optimal control problems possess control variables appearing linearly in the dynamics, the objective function and the constraints. The special bang‐bang and singular structure of the optimal control is exploited to formulate a nonlinear programming problem (NLP)w ith variables in the switching points and singular subarcs of the controls. This method has several advantages: The dimension of the resulting NLP problem is considerably reduced compared to usual direct optimization methods, several constraints can be neglected, and a parametric sensitivity analysis and real‐time control with respect to the switching points can be performed.     

Near-Optimal Controls of a Class of Volterra Integral Systems

In a recent paper by Zhou (Ref. 1), the concept of near-optimal controls was introduced for a class of optimal control problems involving ordinary differential equations. Necessary and sufficient conditions for near-optimal controls were derived. This paper extends the results obtained by Zhou to a class of optimal control problems involving Volterra integral equations. The results are applied to study near-optimal controls obtained by the control parametrization method.     

Constrained optimization problems using multiplier methods

A modified multiplier method for optimization problems with equality constraints is suggested and its application to constrained optimal control problems described. For optimal control problems with free terminal time, a gradient descent technique for updating control functions as well as the terminal time is developed. The modified multiplier method with the simplified conjugate gradient method is used to compute the solution of a time-optimal control problem for a V/STOL aircraft.     

Solving some optimal control problems using the barrier penalty function method

In this paper we present a new approach to solve a two-level optimization problem arising from an approximation by means of the finite element method of optimal control problems governed by unilateral boundary-value problems. The problem considered is to find a minimum of a functional with respect to the control variablesu. The minimized functional depends on control variables and state variablesx. The latter are the optimal solution of an auxiliary quadratic programming problem, whose parameters depend onu.Our main idea is to replace this QP problem by its dual and then apply the barrier penalty method to this dual QP problem or to the primal one if it is in an appropriate form. As a result we obtain a problem approximating the original one. Its good property is the differentiable dependence of state variables with respect to the control variables. Furthermore, we propose a method for finding an approximate solution of a penalized lower-level problem if the optimal solution of the original QP problem is known. We apply the result obtained to some optimal shape design problems governed by the Dirichlet-Signorini boundary-value problem.This research was supported by the Academy of Finland and the Systems Research Institute of the Polish Academy of Sciences.     

Instantaneous control for traffic flow

The solution methods for optimal control problems with coupled partial differential equations as constraints are computationally costly and memory intensive; in particular for problems stated on networks, this prevents the methods from being relevant. We present instantaneous control problems for the optimization of traffic flow problems on road networks. We derive the optimality conditions, investigate the relation to the full optimal control problem and prove that certain properties of the optimal control problem carry over to the instantaneous one. We propose a solution algorithm and compare quality of the computed controls and run‐times. Copyright © 2006 John Wiley & Sons, Ltd.     

An extension of Gilbert's algorithm for computing optimal controls

Gilbert's algorithm for computing optimal controls for systems with lumped parameters is extended to a class of control problems for systems with distributed parameters. It is also shown that the algorithm can be used to solve a larger class of problems in systems with lumped parameters than was originally considered by Gilbert.     

Ultraspherical Integral Method for Optimal Control Problems Governed by Ordinary Differential Equations

In this paper an ultraspherical integral method is proposed to solve optimal control problems governed by ordinary differential equations. Ultraspherical approximation method reduced the problem to a constrained optimization problem. Penalty leap frog method is presented to solve the resulting constrained optimization problem. Error estimates for the ultraspherical approximations are derived and a technique that gives an optimal approximation of the problems is introduced. Numerical results are included to confirm the efficiency and accuracy of the method.     

Trajectory Following Methods in Control System Design

A trajectory following method for solving optimization problems is based on the idea of solving ordinary differential equations whose equilibrium solutions satisfy the necessary conditions for a minimum. The method is `trajectory following' in the sense that an initial guess for the solution is moved along a trajectory generated by the differential equations to a solution point. With the advent of fast computers and efficient integration solvers, this relatively old idea is now an attractive alternative to traditional optimization methods. One area in control theory that the trajectory following method is particularly useful is in the design of Lyapunov optimizing feedback controls. Such a controller is one in which the control at each instant in time either minimizes the `steepest decent' or `quickest decent' as determined from the system dynamics and an appropriate (Lyapunov- like) decent function. The method is particularly appealing in that it allows the Lyapunov control system design method to be used `on-line'. That is, the controller is part of a normal feedback loop with no off-line calculations required. This approach eliminates the need to solve two-point boundary value problems associated with classical optimal control approaches. We demonstrate the method with two examples. The first example is a nonlinear system with no constraints on the control and the second example is a linear system subject to bounded control.     

Optimization method in problems of acoustic cloaking of material bodies

Optimization problems for a three-dimensional model of acoustic scattering are formulated and studied. These problems arise in designing tools for cloaking material bodies by applying the wave flow method. The cloaking effect is achieved due to an optimal choice of variable parameters of the inhomogeneous isotropic medium occupying the sought shell. The solvability of direct and optimization problems for the acoustic scattering model is proved, and sufficient conditions ensuring the uniqueness and stability of optimal solutions are established.