Analysis of optimal controls for a mathematical model of tumour anti‐angiogenesis

Anti‐angiogenic therapy is a novel treatment approach for cancer that aims at preventing a tumour from developing its own blood supply system that it needs for growth. In this paper we consider a mathematical model where the endogenous stimulation term in the dynamics is taken proportional to the number of endothelial cells. This system is an example from a class of mathematical models for anti‐angiogenic treatment that were derived from a biologically validated model by Hahnfeldt, Panigrahy, Folkman and Hlatky. The problem how to schedule a given amount of angiogenic inhibitors to achieve a maximum reduction in the primary cancer volume is considered as an optimal control problem and it is shown that optimal controls are bang‐bang of the type 0a0 with 0 denoting a trajectory corresponding to no treatment and a a trajectory with treatment at maximum dose along that all inhibitors are being exhausted. Copyright © 2007 John Wiley & Sons, Ltd.     

Bang‐bang optimal control for differentially flat systems using mapped pseudospectral method and analytic homotopic approach

The bang‐bang type optimal control problems arising from time‐optimal or fuel‐optimal trajectory planning in aerospace engineering are computationally intractable. This paper suggests a hybrid computational framework that utilizes differential flatness and mapped Chebyshev pseudospectral method to generate a related but smooth trajectory, from which the original non‐smooth solutions are achieved continuously by the analytic homotopic algorithm. The flatness allows for transcribing the original problem into an integration‐free flat outputs optimization problem with reduced number of decision variables. Chebyshev pseudospectral method is applied to parameterizing the flat outputs, and the numerical accuracy for the derivatives of flat outputs at collocation nodes, which are readily computed using differentiation matrices, is greatly enhanced by conformal map and barycentric rational interpolation techniques. Based on the obtained smooth trajectory, the analytic homotopic approach constructs an auxiliary optimal control problem whose costates are simply zero, avoiding the estimation of initial costates. The hybrid framework successfully addresses the difficulties of pseudospectral method and homotopic approach when they are applied separately. Numerical simulations of time‐optimal trajectory planning for spacecraft relative motion and attitude maneuver are presented, validating the performance of the hybrid computational framework. Copyright © 2016 John Wiley & Sons, Ltd.     

2,α‐norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations

This paper deals with a numerical solution method for optimal control problems subject to parabolic and hyperbolic evolution equations. Firstly, the problem is semi‐discretized in space with the boundary or distributed controls as input and those parts of the discretized state appearing in the cost functional as output variables. The corresponding transfer function is then approximated optimally with respect to the _2,α‐norm providing an optimally reduced optimal control problem, which is finally solved by a first‐discretize‐then‐optimize approach. To enable the application of this reduction method, a new constrained optimal model reduction problem subject to reduced systems with real system matrices is considered. Necessary optimality conditions and a transformation procedure for the reduced system to a canonical form of real matrices are presented. The method is illustrated with numerical examples where also complicated controls with many bang‐bang arcs are investigated. The approximation quality of the optimal control and its correlation to the decay rate of the Hankel singular values of the system are numerically studied. A comparison to the approach of using Balanced Truncation for model reduction is applied. Copyright © 2013 John Wiley & Sons, Ltd.     

Computations for bang–bang constrained optimal control using a mathematical programming formulation

An algorithm is proposed to solve the problem of bang–bang constrained optimal control of non‐linear systems with free terminal time. The initial and terminal states are prescribed. The problem is reduced to minimizing a Lagrangian subject to equality constraints defined by the terminal state. A solution is obtained by solving a system of non‐linear equations. Since the terminal time is free, time‐optimal control is given a special emphasis. Second‐order sufficient conditions of optimality are also stated. The algorithm is demonstrated by a detailed study of the switching structure for stabilizing the F–8 aircraft in minimum time, and other examples. Copyright © 2005 John Wiley & Sons, Ltd.     

On L1‐minimization in optimal control and applications to robotics

In this paper, we analyze optimal control problems with control variables appearing linearly in the dynamics. We discuss different cost functionals involving the L^p‐norm of the control. The case p = 0 represents the time‐optimal control, the case p > 1 yields a standard smooth optimal control problem, whereas the case p = 1 leads to a nonsmooth cost functional. Several techniques are developed to deal with the nonsmooth case p = 1. We present a thorough theoretical discussion of the necessary conditions. Two types of numerical methods are developed: either a regularization technique is used or an augmentation approach is applied in which the number of control variables is doubled. We show the precise relations between the L¹‐minimal control and the bang–bang or singular controls in the augmented problem. Using second‐order sufficient conditions (SSC) for bang–bang controls, we obtain SSC for L¹‐minimal controls. The different techniques and results are illustrated with an example of the optimal control for a free‐flying robot which is taken from Sakawa. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal control of a magnetic bearing without bias flux using finite voltage

Conventional active magnetic bearings (AMB) are operated using a bias current (or flux) to achieve greater linearity and dynamic capability. Bias, however, results in undesirable rotating losses and consequent rotor heating. While control without bias flux is an attractive alternative, it is considerably more complex due to both force slew rate limitations and actuator non-linearity. In this paper, optimal control of a magnetic bearing without bias is investigated. A single-degree-of-freedom system consisting of a mass and two opposing electromagnets is considered. The optimal control problem is examined for a cost function that penalizes both poor regulation and rotational energy lost. Though a standard optimization procedure does not directly yield an analytical solution, it does show that the optimal control is always bang–bang including possibly a singular arc. First, the minimum time problem is solved for a simple switching law in three dimensional state space. A non-standard, physics-based approach is then employed to obtain an optimal solution for the general problem. The final result is an optimal variable structure feedback controller. This result provides a benchmark which can be used for evaluation of the performance of a practical feedback controller designed via other methods. The practical controller will be designed to support a flexible rotor and achieve robustness and optimally reject disturbance. This result may also be applied to many other applications which contain opposing quadratic actuators. © 1998 John Wiley & Sons Ltd.     

Linear‐quadratic control problems with L1‐control cost

We analyze a class of linear‐quadratic optimal control problems with an additional L¹‐control cost depending on a parameter β. To deal with this nonsmooth problem, we use an augmentation approach known from linear programming in which the number of control variables is doubled. It is shown that if the optimal control for a given is bang‐zero‐bang and the switching function has a stable structure, the solutions are Lipschitz continuous functions of the parameter β. We also show that in this case the optimal controls for β ^* and a with | β ? β ^* | sufficiently small coincide except on a set of measure . Finally, we use the augmentation approach to derive error estimates for Euler discretizations. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimization methods for the verification of second order sufficient conditions for bang–bang controls

It has been common practice to find controls satisfying only necessary conditions for optimality, and then to use these controls assuming that they are (locally) optimal. However, sufficient conditions need to be used to ascertain that the control rule is optimal. Second order sufficient conditions (SSC) which have recently been derived by Agrachev, Stefani, and Zezza, and by Maurer and Osmolovskii, are a special form of sufficient conditions which are particularly suited for numerical verification. In this paper we present optimization methods and describe a numerical scheme for finding optimal bang–bang controls and verifying SSC. A straightforward transformation of the bang–bang arc durations allows one to use standard optimal control software to find the optimal arc durations as well as to check SSC. The proposed computational verification technique is illustrated on three example applications. Copyright © 2005 John Wiley & Sons, Ltd.     

Using switching detection and variational equations for the shooting method

We study in this paper the resolution by single shooting of an optimal control problem with a bang‐bang control involving a large number of commutations. We focus on the handling of these commutations regarding the precise computation of the shooting function and its Jacobian. We first observe the impact of a switching detection algorithm on the shooting method results. Then, we study the computation of the Jacobian of the shooting function, by comparing classical finite differences to a formulation using the variational equations. We consider as an application a low thrust orbital transfer with payload maximization. This kind of problem presents a discontinuous optimal control, and involves up to 1800 commutations for the lowest thrust. Copyright © 2007 John Wiley & Sons, Ltd.     

On the computation of optimal singular and bang–bang controls

To avoid difficulties associated with the computation of optimal singular/bang–bang controls, a common approach is to add a perturbed energy term. The efficacy of this perturbation method is assessed here via a direct search iterative dynamic programming procedure. A potential limitation of the strategy is shown from a computational point of view, and some guidelines for selecting the perturbation parameter are provided using numerical examples. It is demonstrated that many gradient-based methods may not be well suited for computing singular/bang–bang controls when perturbation methods are used to solve optimal control problems in chemical process control. © 1998 John Wiley & Sons, Ltd.     

New smoothing techniques for solving bang–bang optimal control problems—numerical results and statistical interpretation

In this paper, we investigate the solution of bang‐bang optimal control problems by shooting methods. We will show how modifying the performance index by a term depending on a small parameter ε yields more regular controls and shooting functions. A continuation procedure on ε will lead us to a good approximation of the initial solution. Then, a statistical interpretation of the method is given, providing us with a general framework for building new regular controls. Finally, two numerical examples are solved illustrating the interest of our method. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimal energy management for grid‐connected storage systems

This paper proposes three near‐optimal (to a desired degree) deterministic charge and discharge policies for the maximization of profit in a grid‐connected storage system. The changing price of electricity is assumed to be known in advance. Three near‐optimal algorithms are developed for the following three versions of this optimization problem: (1) The system has supercapacitor type storage, controlled in continuous time. (2) The system has supercapacitor or battery type storage, and it is controlled in discrete time (i.e., it must give constant power during each sampling period). A battery type storage model takes into account the diffusion of charges. (3) The system has battery type storage, controlled in continuous time. We give algorithms for the approximate solution of these problems using dynamic programming, and we compare the resulting optimal charge/discharge policies. We have proved that in case 1 a bang off bang type policy is optimal. This new result allows the use of more efficient optimal control algorithms in case 1. We discuss the advantages of using a battery model and give simulation and experimental results. Copyright © 2014 John Wiley & Sons, Ltd.     

Redundancy implies robustness for bang‐bang strategies

We develop in this paper a method ensuring robustness properties of bang‐bang strategies, for general nonlinear control systems. Our main idea is to add bang arcs in the form of needle‐like variations of the control. With such bang‐bang controls having additional degrees of freedom, steering the control system to some given target amounts to solving an overdetermined nonlinear shooting problem, what we do by developing a least‐square approach. In turn, we design a criterion to measure the quality of robustness of the bang‐bang strategy, based on the singular values of the end‐point mapping, and which we optimize. Our approach thus shows that redundancy implies robustness, and we show how to achieve some compromises in practice, by applying it to the attitude control of a 3d rigid body.     

Optimal control of steer‐braking systems: Non‐existence of minimizing trajectories

In this paper, we investigate an optimal control problem in which the objective is to decelerate a simplified vehicle model, subject to input constraints, from a given initial velocity down to zero by minimizing a quadratic cost functional. The problem is of interest because, although it involves apparently simple drift‐less dynamics, a minimizing trajectory does not exist over the admissible input trajectories. This problem is motivated by a minimum‐time problem for a fairly complex car vehicle model on a race track. Numerical computations run on the car trajectory optimization problem provide evidence of convergence issues and of an apparently unmotivated ripple in the steer angle. Characterizing this ripple behavior is important to fully understand and exploit minimizing vehicle trajectories. We are able to isolate the key features of this chattering behavior in a very simple dynamics/objective setting. We show that the cost functional has an infimum, but an admissible minimizing input trajectory does not exist. We also show that the infimum can be arbitrarily approximated by bang‐bang inputs with a sufficiently large number of switches. We reproduce this phenomenon in numerical computations and characterize it by means of non‐existence of admissible minimizing trajectories. Copyright © 2015 John Wiley & Sons, Ltd.     

dsoa: The implementation of a dynamic system optimization algorithm

This paper describes the ANSI C/C++ computer program dsoa , which implements an algorithm for the approximate solution of dynamics system optimization problems. The algorithm is a direct method that can be applied to the optimization of dynamic systems described by index‐1 differential‐algebraic equations (DAEs). The types of problems considered include optimal control problems and parameter identification problems. The numerical techniques are employed to transform the dynamic system optimization problem into a parameter optimization problem by: (i) parameterizing the control input as piecewise constant on a fixed mesh, and (ii) approximating the DAEs using a linearly implicit Runge‐Kutta method. The resultant nonlinear programming (NLP) problem is solved via a sequential quadratic programming technique. The program dsoa is evaluated using 83 nontrivial optimal control problems that have appeared in the literature. Here we compare the performance of the algorithm using two different NLP problem solvers, and two techniques for computing the derivatives of the functions that define the problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Expanded proximate time‐optimal servo control of permanent magnet synchronous motor

This paper extends the existing proximate time‐optimal servomechanism control methodology to the more typical second‐order servo systems with a damping element. A parameterized design of expanded proximate time‐optimal servomechanism control law with a speed‐dependent linear region is presented for rapid and smooth set‐point tracking using a bounded input signal. The control scheme uses the time‐optimal bang‐bang control law to accomplish maximum acceleration or braking whenever appropriate and then smoothly switches into a linear control law to achieve a bumpless settling. The closed‐loop stability is analyzed, and then the control scheme is applied to the position–velocity control loop in a permanent magnet synchronous motor servo system for set‐point position regulation. Numerical simulation has been conducted, followed by experimental verification based on a TMS320F2812 digital signal controller board. The results confirm that the servo system can track a wide range of target references with superior transient performance and steady‐state accuracy. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐order maximum principles for the stability analysis of positive bilinear control systems

We consider a continuous‐time positive bilinear control system, which is a bilinear control system with Metzler matrices. The positive orthant is an invariant set of such a system, and the corresponding transition matrix is entrywise nonnegative for all time. Motivated by the stability analysis of positive linear switched systems under arbitrary switching laws, we define a control as optimal if it maximizes the spectral radius of the transition matrix at a given final time. We derive high‐order necessary conditions for optimality for both singular and bang–bang controls. Our approach is based on combining results on the second‐order derivative of a simple eigenvalue with the generalized Legendre‐Clebsch condition and the Agrachev–Gamkrelidze second‐order optimality condition. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear multi‐model time‐optimization

A linear optimization problem with unknown parameters from a given finite set is tackled. The problem is to find the robust time‐optimal control transferring a given initial point to a convex terminal compact set M for all unknown parameters in a shortest time. The robust maximum principle for this minimax problem is formulated. It gives a necessary and sufficient condition of robust optimality. Under natural conditions, the existence and uniqueness of robust optimal controls are proven when the resource set is a convex polytope. Several illustrating examples, including a bang–bang robust optimal control, are considered in detail. Copyright © 2002 John Wiley & Sons, Ltd.     

Bi‐criteria optimal control of redundant robot manipulators using LVI‐based primal‐dual neural network

In this paper, a bi‐criteria weighting scheme is proposed for the optimal motion control of redundant robot manipulators. To diminish the discontinuity phenomenon of pure infinity‐norm velocity minimization (INVM) scheme, the proposed bi‐criteria redundancy‐resolution scheme combines the minimum kinetic energy scheme and the INVM scheme via a weighting factor. Joint physical limits such as joint limits and joint‐velocity limits could also be incorporated simultaneously into the scheme formulation. The optimal kinematic control scheme can be reformulated finally as a quadratic programming (QP) problem. As the real‐time QP solver, a primal‐dual neural network (PDNN) based on linear variational inequalities (LVI) is developed as well with a simple piecewise‐linear structure and global exponential convergence to optimal solutions. Since the LVI‐based PDNN is matrix‐inversion free, it has higher computational efficiency in comparison with dual neural networks. Computer simulations performed based on the PUMA560 manipulator illustrate the validity and advantages of such a bi‐criteria neural optimal motion‐control scheme for redundant robots. Copyright © 2009 John Wiley & Sons, Ltd.     

Approximation of integro-differential equations associated with piecewise deterministic process

Aim of this paper is to present an approximation scheme for optimal control problems of piecewise deterministic processes and corresponding integro-differential Hamilton–Jacobi–Bellman equations. The method is based on a discrete dynamic programming approach. We discretize the continuous process and the cost functional obtaining a discrete time optimal control problem. The corresponding dynamic programming equation gives an approximation of the integro-differential equation. The main feature of the method is the uniform convergence to the value function of the continuous control problem, which can be characterized as the unique weak solution (in viscosity sense) of the dynamic programming equation. Moreover, under appropriate assumptions, an error estimate on the truncation error is derived. It is worth noting that the method provides approximate feedback controls at any point of the grid without extra computations. An application of the approximation scheme to the numerical solution of an optimal control problem for a storage process is also detailed. © 1997 John Wiley & Sons, Ltd.