Optimal boundary control problem for ill‐posed elliptic equation in domains with rugous boundary. Existence result and optimality conditions

The main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular.     

Optimality conditions for boundary control of non‐well‐posed elliptic equations

In this paper, we discuss a boundary optimal control problem governed by a class of non‐well‐posed semi‐linear elliptic equations. By considering a well‐posed penalization problem and taking limit in the optimality system for penalization problem, we obtain the necessary optimality conditions for optimal pairs of initial control problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal shape design and unilateral boundary value problems: Part I

In the first part we give a general existence theorem and a regularization method for an optimal control problem where the control is a domain in R″ and where the system is governed by a state relation which includes differential equations as well as inequalities. In the second part applications for optimal shape design problems governed by the Dirichlet-Signorini boundary value problem are presented. Several numerical examples are included.     

Optimal control of a delayed breast cancer stem cells nonlinear model

In this article, we consider a nonlinear model, which is governed by an ordinary differential equations system with time delays in state and control. The model is used in order to describe the growth of breast cancer cells under therapy. We seek optimal therapies to minimize the number of cancer cells as well as the total quantity of drug used in the treatment. In this way, we formulate an optimal control problem. We prove the existence of an optimal therapy and use Pontryagin's maximum principle in order to find optimality conditions, which characterize such optimal therapy. At last, both numerical results and conclusion are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Shape optimization analysis: First- and second-order necessary conditions

In this paper, first- and second-order necessary conditions for optimality are studied for a domain optimization problem. The optimization problem considered is the minimization of an objective function defined on the domain boundary through the solution of a boundary value problem. In order to derive the first and second variations of the objective function due to boundary variation, the first and second variations of the solution of the boundary value problem are calculated using a perturbation technique. An iterative shape optimization algorithm for potential flow problems in R² with Dirichlet boundary conditions is presented. In the algorithm a boundary element method (BEM) is employed to solve the Laplace equation numerically. The validity and accuracy of the algorithm have been verified on a problem where the final solution is known. Finally, the problem of designing a 90° bend for two-dimensional potential flow is solved.     

Infinite horizon optimal control for mean-field stochastic delay systems driven by Teugels martingales under partial information

In this article, we discuss an infinite horizon optimal control of the stochastic system with partial information, where the state is governed by a mean-field stochastic differential delay equation driven by Teugels martingales associated with Lévy processes and an independent Brownian motion. First, we show the existence and uniqueness theorem for an infinite horizon mean-field anticipated backward stochastic differential equation driven by Teugels martingales. Then applying different approaches for the underlying system, we establish two classes of stochastic maximum principles, which include two necessary conditions and two sufficient conditions for optimality, under a convex control domain. Moreover, compared with the finite horizon optimal control, we add the transversality conditions to the two kinds of stochastic maximum principles. Finally, using the stochastic maximum principle II, we settle an infinite horizon optimal consumption problem driven by Teugels martingales associated with Gamma processes.     

Second-order necessary optimality conditions for a discrete optimal control problem with nonlinear state equations

In this article, we study second-order necessary optimality conditions for a discrete optimal control problem with a nonconvex cost function, nonlinear state equations and mixed constraints. In order to achieve these conditions, we first establish an abstract result on the second-order necessary optimality conditions for a mathematical programming problem and then we derive the second-order necessary optimality conditions for a discrete optimal control problem. The main result of this article is illustrated by two examples.     

Solving mixed‐integer optimal control problems by branch&bound: a case study from automobile test‐driving with gear shift

The article discusses the application of the branch&bound method to a mixed integer non‐linear optimization problem (MINLP) arising from a discretization of an optimal control problem with partly discrete control set. The optimal control problem has its origin in automobile test‐driving, where the car model involves a discrete‐valued control function for the gear shift. Since the number of variables in (MINLP) grows with the number of grid points used for discretization of the optimal control problem, the example from automobile test‐driving may serve as a benchmark problem of scalable complexity. Reference solutions are computed numerically for two different problem sizes. A simple heuristic approach suitable for optimal control problems is suggested that reduces the computational amount considerably, though it cannot guarantee optimality anymore. Copyright © 2005 John Wiley & Sons, Ltd.     

Fractional Chebyshev functional link neural network-optimization method for solving delay fractional optimal control problems with Atangana-Baleanu derivative

In this article, we propose a higher order neural network, namely the functional link neural network (FLNN), for the model of linear and nonlinear delay fractional optimal control problems (DFOCPs) with mixed control-state constraints. We consider DFOCPs using a new fractional derivative with nonlocal and nonsingular kernel that was recently proposed by Atangana and Baleanu. The derivative possesses more important characteristics that are very useful in modelling. In the proposed method, a fractional Chebyshev FLNN is developed. At the first step, the delay problem is transformed to a nondelay problem, using a Padé approximation. The necessary optimality condition is stated in a form of fractional two-point boundary value problem. By applying the fractional integration by parts and by constructing an error function, we then define an unconstrained minimization problem. In the optimization problem, trial solutions for state, co-state and control functions are utilized where these trial solutions are constructed by using single-layer fractional Chebyshev neural network model. We then minimize the error function using an unconstrained optimization scheme based on the gradient descent algorithm for updating the network parameters (weights and bias) associated with all neurons. To show the effectiveness of the proposed neural network, some numerical results are provided.     

A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

In this article, we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions. Let Ω and ω be two bounded domains of R^dsuch that ω⊂Ω. For a linear elliptic problem in Ω\ω with Robin boundary condition on the boundary γ of ω, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given; they suggest optimal order of convergence.     

Time‐optimal control of flexible systems subject to friction

This paper presents a technique for the determination of time‐optimal control profiles for rest‐to‐rest maneuvers of a mass–spring system, subject to Coulomb friction. A parameterization of the control input that accounts for the friction force, resulting in a linear analysis of the system is proposed. The optimality condition is examined for the control profile resulting from the parameter optimization problem. The development is illustrated on a single input system where the control input and the friction force act on the same body. The variation of the optimal control structure as a function of final displacement is also exemplified on the friction benchmark problem. Copyright © 2007 John Wiley & Sons, Ltd.     

On a control problem governed by a linear partial differential equation with a smooth functional

In this paper, the optimal control problem for the Helmholtz equation with non‐local boundary conditions is considered. The necessary and sufficient conditions of optimality in a maximum principle form have been obtained. We note that this problem is basically different from classical‐type problems because it is impossible to use Green's formula and we cannot rewrite it in the variational form widely used in the literature. So it is impossible to use all the theory that has been developed for optimal control problems with classical boundary conditions. Copyright © 2009 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.     

Optimal control of nonzero sum game mean‐field delayed Markov regime‐switching forward‐backward system with Lévy processes

This article investigates the optimal control problem of nonzero sum game mean‐field delayed Markov regime‐switching forward‐backward stochastic system with Lévy processes associated with Teugels martingales over the infinite time horizon. Based on the transversality conditions, assumption of convex control domain, infinite‐horizon version of stochastic maximum principle (Nash equilibrium), and necessary condition for optimality are established. Finally, the Nash equilibrium for the optimization problem in the financial market is considered to illustrate the observed theoretical results.     

Numerical solution of a drug displacement problem with bounded state variables

The interaction of the two drugs warfarin and phenylbutazone has previously been considered as a time-optimal control problem with state inequality constraints. We include bounds for the control and show that necessary optimality conditions and junction conditions for bounded state variables lead to boundary value problems with switching and junction conditions. A special version of the multiple-shooting algorithm is employed for solving the different types of boundary value problems. The switching structure of the optimal control is determined for a range of parameters in the state constraint. Owing to the special structure of the control, a state space solution is obtained in a first step which reduces the numerical complexity of the problem. It is shown how the numerical results can be used to compute the generalized gradient of the optimal value function explicitly.     

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.     

Optimal control of HIV treatment and immunotherapy combination with state and control delays

In this paper, we propose and analyze an optimal control problem where human immunodeficiency virus treatment and immunotherapy are described by two control functions that are subject to time delays representing pharmacological and absorption delays, respectively. The goal is to propose effective optimal control solutions for the combination of human immunodeficiency virus treatment and immunotherapy, ensuring a functional behavior of the immune system. The incubation period is mathematically represented by a time delay in the virus load, and the local asymptotic and Hopf bifurcation analysis of the CTL-equilibrium point of the uncontrolled delayed system is studied. We obtain optimal controls of bang-singular type both for the nondelayed and delayed optimal control problem with and without state constraints. We study boundary arcs of state constraints and junction properties of the control and adjoint variables at entry and exit points of boundary arcs. Moreover, we derive an explicit formula of the multiplier associated with the state constraint.     

Optimal design problems for curvilinear shallow elements of structures

New problem statements for the optimal design of thin-wall structural elements are considered by means of optimal control theory. The distribution of initial curvature of shallow curvilinear structural elements in a non-strain state is taken as the control function.^1–3 Integral stiffness is considered as the optimized performance index. The necessary optimality condition and the partial differential equations for the adjoint variables are derived. An analysis of these relations is carried out, and the initial optimal control problem is reduced to a boundary problem of the bending of an uncurved element. Problems of optimal design of plates under transverse loads, as well as under tensile or compression forces acting in the middle surface, are studied. Analogous problems of optimal design for shallow curvilinear plates on an elastic foundation are also investigated. Some two-dimensional analytical solutions for optimal plates under loads of different types are obtained.     

Incorporating a class of constraints into the dynamics of optimal control problems

A method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations. The considered class of constraints comprises up to m input constraints and m state constraints with well‐defined relative degree, where m denotes the number of inputs of the given nonlinear system. Starting from an equivalent normal form representation, the constraints are incorporated into a new system dynamics by means of saturation functions and differentiation along the normal form cascade. This procedure leads to a new unconstrained OCP, where an additional penalty term is introduced to avoid the unboundedness of the saturation function arguments if the original constraints are touched. The penalty parameter has to be successively reduced to converge to the original optimal solution. The approach is independent of the method used to solve the new unconstrained OCP. In particular, the constraints cannot be violated during the numerical solution and a successive reduction of the constraints is possible, e.g. to start from an unconstrained solution. Two examples in the single and multiple input case illustrate the potential of the approach. For these examples, a collocation method is used to solve the boundary value problems stemming from the optimality conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft

In this article, the minimum time and fuel consumption of an aircraft in its climbing phase are studied. The controls are the thrust and the lift coefficient and state constraints are taken into account: air slope and speed limitations. The application of the maximum principle leads to parameterize the optimal control and the multipliers associated with the state constraints with the state and the costate and leads to describe a multipoint boundary value problem, which is solved by multiple shooting. This indirect method is the numerical implementation of the maximum principle with state constraints and it is initialized by the direct method, both to determine the optimal structure and to obtain a satisfying initial guess. The solutions of the boundary value problems we define give extremals, which satisfy necessary conditions of optimality with at most 2 boundary arcs. Note that the aircraft dynamics has a singular perturbation but no reduction is performed.