An approximate method for solving a class of nonlinear optimal control problems

This paper presents a novel computational approach to generate the suboptimal solutions for a class of nonlinear optimal control problems (OCP's) with a quadratic performance index. Our method is based on the one‐dimensional differential transform method (DTM) and new polynomials that are called DT's polynomials. This method simplifies the difficulties and massive computational work for calculating the differential transform of nonlinear function. The convergence of proposed method are discussed in detail. This method consists of a new modified version of the DTM together with a shooting method such as procedure, for solving the extreme conditions obtained from the Pontryagin's maximum principle. The results reveal that the proposed methods are very effective and simple. Comparisons are made between new DTM generated results, results from literature, and MATLAB bvp4c generated results, and good agreement is observed. Copyright © 2013 John Wiley & Sons, Ltd.     

A gradient algorithm for solution of the optimal control problem for hybrid switching systems

In this article, an algorithm is presented for solving the optimal control problem for the general form of a hybrid switching system. The cost function comprises terminal, running and switching costs. The controlled system is an autonomous hybrid switching system with jumps either at some switching times or some time varying switching manifolds. The proposed algorithm is an extension of the first-order gradient method for the conventional optimal control problem. The algorithm requires a low computational effort. The system's dynamical equations together with a set of algebraic equations are solved at each iteration in order to find the descent direction. The convergence of algorithm is proved and examples are provided to demonstrate the efficiency of the algorithm for different types of hybrid switching system optimal control problems.     

Optimal control analysis of a tuberculosis model with exogenous re-infection and incomplete treatment

A new mathematical model of tuberculosis (TB) featuring exogenous re-infection and incomplete treatment is presented and analyzed. The model divides total population into susceptible, latently infected, actively infected (uninformed and enlightened), and treatment classes. The model with or without incomplete treatment exhibits backward bifurcation phenomenon, which is caused by the presence of exogenous re-infection. However, further investigation reveals that the absence of incomplete treatment has the potential to reduce the backward bifurcation range significantly. The global dynamics of the TB model without exogenous re-infection is completely determined by the basic reproduction number under certain modifications on parameters. Furthermore, the model is extended to include three time-dependent control functions, such as public awareness campaign, treatment effort, and preventive control against incomplete treatment. The existence of the optimal control for the nonautonomous model is proven and the three controls are characterized using Pontryagin's maximum principle. Numerical simulations are performed to show the significance of singular implementation of each of the controls and combination of the three controls in minimizing the TB burden in the population.     

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large‐scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal control problem for an elliptic equation which has exactly two solutions

An optimal control problem governed by a class of semilinear non‐well‐posed elliptic equations is discussed in this paper. It is pointed out that the state equation we considered has exactly two solutions and the control domain and cost functional maybe non‐convex. Using the relaxed controls, Pontryagin's maximum principle for the optimal pairs is obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

Quadratic and linear controls developing an optimal treatment for the use of BCG immunotherapy in superficial bladder cancer

Our prime objective of the study is to exhibit the advantage to introduce a quadratic control in place of linear control in a cost function to be minimized, and that is associated to an optimal control problem that we formulate for a pre‐validated model of bacillus Calmette‐Guérin (BCG) immunotherapy in superficial bladder cancer. The compartmental model of interest is in the form of a nonlinear system of four ordinary differential equations that describe interactions between the used BCG strain, tumor cells, and immune responses. Previous studies reported that the optimal dose of BCG for treating bladder cancer is yet unknown. Hence, we aim to establish the optimization approach that can be applied for determining the values of the optimal BCG concentration along the therapy period to stimulate immune‐system cells and reduce cancer cells growth during BCG intravesical therapy. Pontryagin's maximum principle and the generalized Legendre–Clebsch condition are employed to provide the explicit formulations of the sought optimal controls. The optimality system is resolved numerically based on a fourth‐order iterative Runge–Kutta progressive‐regressive scheme, which is used to solve a two‐point boundary value problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Singular linear quadratic optimal control for singular stochastic discrete‐time systems

The finite time horizon singular linear quadratic (LQ) optimal control problem is investigated for singular stochastic discrete‐time systems. The problem is transformed into positive LQ one for standard stochastic systems via two equivalent transformations. It is proved that the singular LQ optimal control problem is solvable under two reasonable rank conditions. Via dynamic programming principle, the desired optimal controller is presented in terms of matrix iterative form. One simulation is provided to show the effectiveness of the proposed approaches. Copyright © 2012 John Wiley & Sons, Ltd.     

A novel continuous genetic algorithm for the solution of optimal control problems

In this paper, the Continuous Genetic Algorithm (CGA), previously developed by the principal author, is applied for the solution of optimal control problems. The optimal control problem is formulated as an optimization problem by the direct minimization of the performance index subject to constraints, and is then solved using CGA. In general, CGA uses smooth operators and avoids sharp jumps in the parameter values. This novel approach possesses two main advantages when compared to other existing direct and indirect methods that either suffer from low accuracy or lack of robustness. First, our method can be applied to optimal control problems without any limitation on the nature of the problem, the number of control signals, and the number of mesh points. Second, high accuracy can be achieved where the performance index is globally minimized while satisfying the constraints. The applicability and efficiency of the proposed novel algorithm for the solution of different optimal control problems is investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

Synthesis of optimal control in a mathematical model of tumour–immune dynamics

An optimal control problem for a mathematical model of tumour–immune dynamics under the influence of chemotherapy is considered. The toxicity effect of the chemotherapeutic agent on both tumour and immunocompetent cells is taken into account. A standard linear pharmacokinetic equation for the chemotherapeutic agent is added to the system. The aim is to find an optimal strategy of treatment to minimize the tumour volume while keeping the immune response not lower than a fixed permissible level as far as possible. Sufficient conditions for the existence of not more than one switching and not more than two switchings without singular regimes are obtained. The surfaces in the extended phase space, on which the last switching appears, are constructed analytically.Copyright © 2013 John Wiley & Sons, Ltd.     

A second‐order optimization algorithm using quadric control updates for multistage optimal control problems

This paper describes a trajectory optimization algorithm that generates a quadric control update, which satisfies the constraints and necessary conditions to the second order. The algorithm is designed to solve multistage optimization problems. The algorithm is tested against a commercially available Sequential Quadratic Programming algorithm on problems with linear dynamics and linear and nonlinear constraints. This algorithm is a departure from previous methods because it explicitly satisfies the constraints to the second order. Copyright © 2009 John Wiley & Sons, Ltd.     

Minimal time crisis versus minimum time to reach a viability kernel: A case study in the prey‐predator model

In this work, we provide a reformulation of the minimal time crisis problem associated with a constraint set as a free terminal time control problem. The proof requires the existence of a nonempty viability kernel that is reachable from the state space. In addition, we suppose a uniform lower bound between two consecutive crossing times of the constraint set. Thanks to this result, we compute an optimal synthesis for the minimal time crisis problem governed by the prey‐predator dynamics, with a controlled mortality on the predators. Finally, we compare the time spent in the crisis set by optimal trajectories of the minimal time crisis problem with the minimum time problem to reach the viability kernel. This is made possible by means of an exact characterization of the viability kernel.     

Optimal feedback gains determination method for nominal reentry guidance

This paper presents a new method for solving feedback gains in guidance based on optimal control theory. The determination of feedback gains is a critical problem for nominal guidance, because landing precision benefits from appropriate feedback gains. The magnitude of the bank angle is determined by minimizing the total error of final altitude as well as downrange requirement to the landing site. The problem is formulated as an optimal control problem by means of the Pontryagin's maximum principle, and backward integration is used to solve the problem based on homogeneous property. The proposed approach not only can avoid linearizing the nonlinear dynamic equations, but also are applicable to different guidance equations. Validation shows that the proposed approach can provide consistent results with that of Apollo. Despite employment of nominal guidance in the entire reentry of lunar mission, the proposed approach demonstrates good performance in all simulation cases while subject to significant disturbances. Copyright © 2015 John Wiley & Sons, Ltd.     

Solution algorithms for non-linear singularly perturbed optimal control problems

The applicability and usefulness of several classical and other methods for solving the two-point boundaryvalue problem which arises in non-linear singularly perturbed optimal control are assessed. Specific algorithms of the Picard, Newton and averaging types are formally developed for this class of problem. The computational requirements associated with each algorithm are analysed and compared with the computational requirement of the method of matched asymptotic expansions. Approximate solutions to a linear and a non-linear problem are obtained by each method and compared.     

A Riccati-equation-based algorithm for continuous-time optimal control problems

In this paper we consider continuous-time unconstrained optimal control problems. We propose a computational method which is essentially based on the closed-loop solutions of the linear quadratic optimal control problems. In the proposed algorithm, Riccati differential equations play an important role. We prove that accumulation points generated by the present algorithm, if they exist, satisfy the weak necessary conditions for optimality, under some assumptions including Kalman's sufficient conditions for the bounded Riccati solutions. In addition, we also propose the simple but effective technique to guarantee the boundedness of the solutions of Riccati equations. Lastly, we illustrate the usefulness of the present algorithm through simulation experiences. Copyright © 1998 John Wiley & Sons, Ltd.     

A differential dynamic programming algorithm for differential games

We develop and prove the convergence of a first‐order differential dynamic programming algorithm for the solution of a zero‐sum two‐person differential game with perfect information. The algorithm extends a first‐order strong variation algorithm for optimal control given by Mayne and Polak. Assuming separability of the Hamiltonian, we decompose the differential game problem into two control subproblems, C₁ and C₂. The objective is to determine a point (u^*, v^*) in U×V, where U and V are the control spaces for C₁ and C₂, respectively, that satisfies an integral form of Pontryagin's maximum principle for differential games. Copyright © 2001 John Wiley & Sons, Ltd.     

Environmental effects of tourism industry investments: an inter‐temporal trade‐off

Efficient investment programmes in touristic infrastructure have to take into consideration that any kind of tourism reduces the environmental quality. Since pollution shows negative repercussions as concerns the attractiveness of a touristic region, tourism planners have to determine a trade‐off between adequate services for tourists and a clean environment. To deal with this problem in a dynamic context, a three‐state optimal control model is formulated. It turns out that, even if pollution reduction is not a goal in itself, the profit‐maximizing tourism industry should care for ecological conservation. The paper further shows that persistent periodic investment policies are optimal for realistic parameter sets, and provides an economic intuition for such behaviour. From an economic point of view, this result implies that expansionary periods with high investment are followed by periods of stagnation with low investment. Copyright © 2002 John Wiley & Sons, Ltd.     

Finite-time optimal pinning control and synchronization for partially interdependent networks

For partially interdependent networks composed of two subnetworks, the finite-time optimal pinning control problem is investigated. Among them, only a part of the nodes between the two subnetworks are interdependent on each other. In the network, the coupling relationship between any two nodes of the network is a continuous nonlinear function. Based on the pinning control, the optimal control theory, Kalman's controllability rank conditions, and introducing the Lagrange function and applying controllers to the partial nodes of the network, we propose some characterization indicators which ensure that the two subnetworks of partially interdependent networks can synchronize to the equilibrium points of their isolate systems, separately. Finally, some numerical simulations are performed on the partially interdependent networks including two small world subnetworks. The results show that the optimal pinning control method proposed in this article can greatly reduce the control costs and achieve the ideal synchronous status quickly.     

Combined homotopy and neighboring extremal optimal control

This paper presents a new approach to trajectory optimization for nonlinear systems. The method exploits homotopy between a linear system and a nonlinear system and neighboring extremal optimal control, in combination with few iterations of a convergent optimizer at each step, to iteratively update the trajectory as the homotopy parameter changes. To illustrate the proposed method, a numerical example of a three‐dimensional orbit transfer problem for a spacecraft is presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Bilinear quadratic optimal control: a recursive approach

This paper deals with the time-varying bilinear quadratic optimal control problem. Using Adomian's decomposition method, we shall first derive a functional expansion for the input–output map of the system, then transform the cost functional so that it yields the optimal control in a recursive manner. The optimal tracking problem is considered to illustrate the theory. An alternative method is derived which is proved to be more ‘robust’. © 1997 John Wiley & Sons, Ltd.