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In this paper we show that, for optimal control problems involving equality and inequality constraints on the control function, the notions of normality and properness (or the Mangasarian–Fromovitz constraint qualification) of a trajectory relative to the set of constraints are equivalent. This is in contrast with some differences recently obtained between the theories of mathematical programming and optimal control, and it provides an important insight in the derivation of first and second order necessary optimality conditions for infinite dimensional problems.

The problem of synthesizing a control system using symbolic regression methods is considered. A new approach suggested in this work considers the control system synthesis problem as the set of optimal control problems and suggests a two-step algorithm to solve it. At the first step evolutionary algorithms are used to solve the optimal control problem numerically for each initial state from a given domain. As a result of the first step a set of optimal trajectories is obtained. At the second step symbolic regression methods are considered to solve the problem of approximating the found in the first step set of trajectories optimally. The result of the second step is a solution of the control system synthesis problem as a multidimensional control function of object’s state vector. Compared to the known approach to solve the control system synthesis problem using symbolic regression methods, the suggested approach provides betters results since the search of control function is carried out based on the set of optimal trajectories. Computational experiment presents the solution of the applied problem of synthesizing the optimal control system for the spacecraft landing on the Moon. It is experimentally demonstrated that the synthesized control system provides a close to optimal landing trajectory for any initial state from a given domain.

The paper introduces an approach to solve the optimal control problem applying symbolic regression methods. All known approaches to solve the optimal control problem are based on direct or indirect methods. Indirect methods may provide analytical solution but impose restrictions on the form and dimension of control object. In direct ones the optimal control problem is reduced to the nonlinear programming problem which provides general numerical solution. Also direct methods impose no restrictions on control object and have a wide range of application to complex applied problems. The approach proposed combines the advantages of both direct and indirect methods not involving the disadvantages mentioned previously. The suggested approach provides solution to the optimal control problem as a control function of time in explicit form. This approach benefits in case of applied optimal control problem requiring a solution in explicit form while indirect methods are inapplicable due to the restrictions imposed on the control system. The computational experiment of solving the optimal control problem for a mobile robot with phase constraints is presented. The results illustrate that the proposed approach finds an optimal control as a function of time which provides a mobile robot’s motion along the optimal trajectory avoiding phase constraints.

This paper is one of the few attempts to use the optimal control theory to find optimal quarantine strategies for eradication of the spread of the COVID-19 infection in the Mexican human population. This is achieved by introducing into the SEIR model a bounded control function of time that reflects these quarantine measures. The objective function to be minimized is the weighted sum of the total infection level in the population and the total cost of the quarantine. An optimal control problem reflecting the search for an effective quarantine strategy is stated and solved analytically and numerically. The properties of the corresponding optimal control are established analytically by applying the Pontryagin maximum principle. The optimal solution is obtained numerically by solving the two-point boundary value problem for the maximum principle using MATLAB software. A detailed discussion of the results and the corresponding practical conclusions are presented.

In this paper we solve the global optimality problem of determining probabilistic distance between two sets. We treat this problem as that of finding an optimal approximate solution of a fixed poin...

Traditional penalty-based methods might not achieve variable selection consistency when endogeneity exists in high-dimensional data. In this article we construct a regularization framework based on the two-stage control function model, so called the regularized control function (RCF) method, to estimate important covariate effects, select key instruments, and replace the CF-based hypothesis test with variable selection to identify truly endogenous predictors. Under appropriate conditions, we establish theoretical properties of the RCF estimators, including the consistency of coefficient estimation and model selection. Simulation results confirm that the RCF method is effective and superior to its main competitor, the penalized least squares (PLS) method. The proposed method also provides insightful and interpretable results on a real data analysis.

In this paper, we consider a nonlinear time-varying dynamical (NTVD) system with uncertain system parameters assigned to their nominal values in batch culture of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumoniae. Some important properties of the NTVD system are discussed. Our goal is to choose a time-varying function for the NTVD system. Thus, an optimal control problem (OCP) governed by the NTVD system and subject to continuous state inequality constraints arising from engineering specifications is proposed, where the time-varying function is the control function to be chosen such that system cost (the relative error between experimental data and the simulated output of the system) and system robustness (robustness of the system with respect to uncertain system parameters) is optimized. Based on the actual fermentation process, the time-varying function is specified by a four-piecewise linear function with unknown kinetic parameters and switching instants. The resulting OCP is approximated as a sequence of nonlinear mathematical programming subproblems by the time-scaling transformation, the constraint transcription and the locally smoothing approximation techniques. A parallel global optimization algorithm, based on a novel combination of limited information particle swarm optimization and local search strategy, is then developed to solve these subproblems. Numerical results show the effectiveness and applicability of our proposed algorithm.

In this paper, we consider an optimal control problem in which the control is almost smooth and the state and control are subject to terminal state constraints and continuous state and control inequality constraints. By introducing an extra set of differential equations for this almost smooth control, we transform this constrained optimal control problem into an equivalent problem involving both control function and system parameter vector as decision variables. Then, by the control parametrization technique and a time scaling transformation, the equivalent problem is approximated by a sequence of constrained optimal parameter selection problems, each of which is a finite dimensional optimization problem. For each of these constrained optimal parameter selection problems, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. This gives rise to a sequence of unconstrained optimal parameter selection problems; and each of which can be solved by existing optimization algorithms or software packages. Finally, a practical container crane operation problem is solved, showing the effectiveness and applicability of the proposed approach.

Economically optimal management of continuous cover forests can be obtained via a new approach to adaptive control function optimization We maximize our objective function the expected present value with consideration of stochastic prices timber quality variations and dynamically changing spatial competition The parameters of the control function are optimized via the first order optimum conditions based on a multivariate polynomial approximation of the objective function The second order maximum conditions are investigated and used to determine relevant parameter ranges The procedure is described and optimal results are derived for general function multi species CCF forests A numerically specified model with empirical data showed that if the stochastic price variations are not considered when the harvest decisions are taken the expected present value is reduced by

Reaction-diffusion models can be used to describe temporal and spatial dynamics of populations. When the interests is the relationship of a given species with the environment in which it is inserted, we can model this phenomenon through the equation of Fisher–Kolmogorov that, in a mathematical sense, consider the existence of a source term, that represents both the reproduction and the growth rate of a population up to the medium's carrying capacity. Through this equation it is possible to describe the behaviour of some animals, plants, bacteria and cells, for example, in order to predict the growth and invasion of these population, and control them if necessary. Thus, in this work we propose to introduce a control strategy in a biological system consisting of aquatic plants, in which the configuration is the same as the control function introduced in a logistic model. The solution to the reaction-diffusion model with control is obtained through the finite Fourier transform. This strategy, in practical terms, is based on the withdrawal of plants by human management. Computational simulations will be presented to illustrate the efficiency of the control strategy adopted.

This paper considers a class of nonlinear systems in which the control function is a time-varying state-delay. The optimal control problem is to optimize the time-varying delay and a set of time-invariant system parameters subject to lower and upper bounds. To solve this problem, we first parameterize the delay in terms of piecewise-quadratic basis functions, thus yielding a finite-dimensional approximate problem with continuous-time inequality constraints induced by the delay bounds. We then exploit the quadratic structure of the delay to convert these continuous-time constraints into a finite set of canonical point constraints. We also develop an efficient numerical method for computing the gradients of the system cost function. This method, which involves integrating an auxiliary impulsive system with time-varying advance backwards in time, can be combined with any existing gradient-based optimization algorithm to generate approximate solutions for the optimal control problem.

The paper considers the solution of the problem of optimal control system synthesis. It is proposed to solve this problem based on the approximation of the set of optimal trajectories using symbolic regression methods. At the first step the optimal control problem is solved for various initial states; at the second step symbolic regression method is used to approximate the obtained set of optimal trajectories. In the suggested approach the proximity of the solution to the optimal one is determined by the accuracy of the approximation. A computational experiment of solving the applied problem of optimal control system synthesis for a group of car-like mobile robots in space with dynamic and static phase constraints is presented. The experiment showed that the found synthesized control function allows to move robots by the trajectory close to the optimal one for any initial state from a given domain.

The optimal control problem with phase constraints is considered. A new indirect approach of synthesized optimal control is proposed as an alternative to direct methods. A comparative study of direct and indirect approaches is carried out on the problem of optimal control for a small group of mobile robots in the complex environment with phase constraints by evolutionary algorithms. With a direct approach to the numerical solution of the optimal control problem, the control function is searched in the form of piece-wise functional approximation. The indirect approach of synthesized optimal control comes from the engineering practice. Instead of reducing the optimal control problem to the problem of finite-dimensional optimization, we firstly make the object stable relative to some point in the state space by solving an additional task of synthesis of stabilizing control and then we find the coordinates of stabilization points as the desired parameters of optimal control.

Optimization is a widespread tooling with diverse applications in nearly each technical discipline. Its task is to find a control function that minimizes a performance index while meeting boundary as well as continuous constraints. Different approaches exist to solve corresponding problems. This paper briefly summarizes the most important ones and based on that presents an improvement that reduces the transcribed problem’s dimensionality. Apart from that, implementation details are given, enabling the real-time capable solution of the resulting optimization problem. The generic approach is finally applied to a real-world problem, i.e., the planning of a reference trajectory of a front-steered vehicle. Closing, simulation results as well as achieved runtimes are presented.

It is well known that optimal control problems with L1-control costs produce sparse solutions, i.e., the optimal control is zero on whole intervals. In this paper, we study a general class of convex linear-quadratic optimal control problems with a sparsity functional that promotes a so-called group sparsity structure of the optimal controls. In this case, the components of the control function take the value of zero on parts of the time interval, simultaneously. These problems are both theoretically interesting and practically relevant. After obtaining results about the structure of the optimal controls, we derive stability estimates for the solution of the problem w.r.t. perturbations and L2-regularization. These results are consequently applied to prove convergence of the Euler discretization. Finally, the usefulness of our approach is demonstrated by solving an illustrative example using a semismooth Newton method.

This paper considers a free terminal time optimal control problem governed by nonlinear time delayed system, where both the terminal time and the control are required to be determined such that a cost function is minimized subject to continuous inequality state constraints. To solve this free terminal time optimal control problem, the control parameterization technique is applied to approximate the control function as a piecewise constant control function, where both the heights and the switching times are regarded as decision variables. In this way, the free terminal time optimal control problem is approximated as a sequence of optimal parameter selection problems governed by nonlinear time delayed systems, each of which can be viewed as a nonlinear optimization problem. Then, a fully informed particle swarm optimization method is adopted to solve the approximate problem. Finally, two free terminal time optimal control problems, including an optimal fishery control problem, are solved by using the proposed method so as to demonstrate its applicability.

In this paper, we study optimal control problems for a new class of noninstantaneous impulsive differential equations arising from the dynamics of evolution processes in pharmacotherapy. We construct a suitable control function, which allows us to characterize the structure of controllability by using the terminal time subinterval instead of the global time interval. We apply fixed point approach to show the controllability results that are the foundation of optimal control theory. Next, we study existence of optimal control problems for a certain quadratic functional acting as the performance index. In addition, we design ILC updating laws for deterministic impulsive systems to generate a sequence of control functions to approximate the optimal control function. Further, we extend the deterministic results to random case by designing ILC updating laws with randomly varying trial length. Finally, several examples are given to demonstrate the validity of theoretical results and design methods. Here, we remark that ILC updating algorithm is adopted to find a desired optimal control for impulsive systems, which provides another effective way to solve optimization problem via computer techniques.

The paper is concerned with the optimal harvesting of a marine park, which is described by a parabolic heat equation with Neumann boundary conditions and a nonlinear source term. We consider a cost functional, which is linear with respect to the control; hence the optimal solution can belong to the class of measure-valued control strategies. For each control function, we prove existence and stability estimates for solutions of the parabolic equation. Moreover, we prove the existence of an optimal solution. Finally, some numerical simulations conclude the paper.

The objective of this paper is to study a pursuit differential game with finite or countably number of pursuers and one evader. The game is described by differential equations in l 2 -space, and integral constraints are imposed on the control function of the players. The duration of the game is fixed and the payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. However, we discuss the condition for finding the value of the game and construct the optimal strategies of the players which ensure the completion of the game. An important fact to note is that we relaxed the usual conditions on the energy resources of the players. Finally, some examples are provided to illustrate our result.

The paper deals with the optimal control problem, including the linear parabolic equation as a state problem. Pointwise constraints are imposed on the control function. The objective functional involves the observation function in the entire space-time domain. The optimal control problem is approximated by a finite dimensional problem with mesh approximation of the state equation by the explicit (forward Euler) mesh scheme with variable time steps. The existence of unique solutions for the continuous and mesh optimal control problems is proved. The Uzawa-type iterative method is used for solving the finite dimensional optimal control problem. The results of numerical experiments are presented.

The presence of weak instruments is translated into a nearly singular problem in a control function representation. Therefore, the math formula-norm type of regularization is proposed to implement the 2SLS estimation for addressing the weak instrument problem. The math formula-norm regularization with a regularized parameter O(n) allows us to obtain the Rothenberg (1984) type of higher-order approximation of the 2SLS estimator in the weak instrument asymptotic framework. The proposed regularized parameter yields the regularized concentration parameter O(n), which is used as a standardized factor in the higher-order approximation. We also show that the proposed math formula-norm regularization consequently reduces the finite sample bias. A number of existing estimators that address finite sample bias in the presence of weak instruments, especially Fuller's limited information maximum likelihood estimator, are compared with our proposed estimator in a simple Monte Carlo exercise.

We consider identification and estimation of nonseparable sample selection models with censored selection rules. We employ a control function approach and discuss different objects of interest based on (1) local effects conditional on the control function, and (2) global effects obtained from integration over ranges of values of the control function. We derive conditions for identification of these different objects and suggest strategies for estimation. Moreover, we provide the associated asymptotic theory. These strategies are illustrated in an empirical investigation of the determinants of female wages in the United Kingdom.

We present a mathematical model of cancer growth in the bladder that includes the immune cells, BCG, immune checkpoints and drug therapy (checkpoint inhibitor) in the form of a control function. The control function blocks the action of immune checkpoints on the immune system. Our aim here is to apply optimal control theory to find a control strategy that will minimize the number of cancer cells in the bladder and cost of control. Existence of the optimal control is stated and Pontryagin’s maximum principle is used to characterize the nature of the control function. The optimality system obtained gives a two-point boundary value problem; hence, we use the forward-backward sweep method to present the numerical solutions of the system. The optimality conditions and characterization of the control are discussed.

We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function representing the efficiency of reverse transcriptase inhibitors and consider the pharmacological delay associated to the control. Finally, we propose and analyze an optimal control problem with state and control delays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs., Comment: This is a preprint of a paper whose final and definite form is with 'Discrete and Continuous Dynamical Systems - Series B' (DCDS-B), ISSN 1531-3492 (print), ISSN 1553-524X (online), available at [http://aimsciences.org/journals/home.jsp?journalID=2]. Paper Submitted 12-July-2016; Revised 26-Nov-2016 and 01-March-2017; Accepted 21-Aug-2017

Optimisation of the control function for multiple automated interacting production stations is a complex problem, even for skilled and experienced operators or process planners. When using mathematical optimisation techniques, it often becomes necessary to use simulation models to represent the problem because of the high complexity (i.e. simulation-based optimisation). Standard optimisation techniques are likely to either exceed the practical time frame or under-perform compared to the manual tuning by the operators or process planners. This paper presents the Constructive cooperative coevolutionary (C3) algorithm, which objective is to enable effective simulation-based optimisation for the control of automated interacting production stations within a practical time frame. C3 is inspired by an existing cooperative coevolutionary algorithm. Thereby, it embeds an algorithm that optimises subproblems separately. C3 also incorporates a novel constructive heuristic to find good initial solutions and thereby expedite the optimisation. In this work, two industrial optimisation problems, involving interaction production stations, with different sizes are used to evaluate C3. The results illustrate that with C3, it is possible to optimise these problems within a practical time frame and obtain a better solution compared to manual tuning.

This paper reviews the approaches to explicit approximate Nonlinear Model Predictive Control (NMPC), based on orthogonal partition of the parameter space. They are classified with respect to the type of control problem, the type of model describing the nonlinear system, the presence of uncertainty, the type of control input, and the type of the approximating control function. A software tool for design of explicit NMPC by applying the described approaches is presented. It is used to design an explicit NMPC controller for a continuous stirred tank reactor.

In fed-batch culture, feeding substrates is to provide sufficient nutrition and reduce inhibitions simultaneously for cells growth. Hence, when and how much to feed substrates are important during the process. In this paper, a nonlinear impulsive controlls system, in which the volume of feeding is taken as the control function, is proposed to formulate the fed-batch fermentation process.In the system, both impulsive moments and jumps size of state are state-dependent. Some important properties of the system are investigated. To maximize the concentration of target product at the terminal time, an optimal control model involving the nonlinear state-dependent impulsive controlled system is presented.The optimal control problem is subject to the continuous state inequality constraint and the control constraint. The existence of optimal control is also obtained. In order to derive the optimality conditions, the optimal control model is transcribed into an equivalent one by treating the constraints. Finally, the optimality conditions of the optimal control model are obtained via calculus of variations.

The article proposes an analytical method for constructing control function that ensures transferring linear inhomogeneous stationary system from an initial state to a given final state. Conditions under which the specified transfer is guaranteed are presented. Mathematics Subject Classification: 34H15, 93C15, 93B05

This paper provides an overview of control function (CF) methods for solving the problem of endogenous explanatory variables (EEVs) in linear and nonlinear models. CF methods often can be justified in situations where “plug- in” approaches are known to produce inconsistent estimators of parameters and partial effects. Usually, CF approaches require fewer assumptions than maximum likelihood, and CF methods are computationally simpler. The recent focus on estimating average partial effects, along with theoretical results on nonparametric identification, suggests some simple, flexible parametric CF strategies. The CF approach for handling discrete EEVs in nonlinear models is more controversial but approximate solutions are available.

A class of systems with several non–ordinary Poisson input flows is studied. It is assumed that the flows are conflicting which means they cannot be served simultaneously. A service device carries out control function also. A probabilistic model for the class of the systems is constructed. Easily verifiable conditions of stationarity are determined analytically for two control algorithms: a cyclic algorithm for the homogeneous flows and a feedback algorithm with threshold priority and prolongations for the flows that differs in priority and intensity. A computer simulation model is described. Some examples of determining the quasi-optimal values of the control system parameters are given.

In this paper we introduce a numerical technique based on Fourier series for solving of nonlinear optimal control problems, where this approach is used for solving optimal control problem of an HIV infection treatment control model. In this paper, first by using healthy cells CD4 + T (T), infected cells CD4 + T (I), viral load (V) and also by using a drug inhibitor of reverse polygraph as a control function, a control model is presented for treatment of HIV infection. A cost function to minimize the cost of drug during the treatment is defined as well. To find the pair of trajectory and control of such nonlinear optimal control problem, we used Fourier series to approximate optimal pair of trajectory and control.

The control function approach (Heckman and Robb (1985)) in a system of linear simultaneous equations provides a convenient procedure to estimate one of the functions in the system using reduced form residuals from the other functions as additional regressors. The conditions on the structural system under which this procedure can be used in nonlinear and nonparametric simultaneous equations has thus far been unknown. In this paper, we define a new property of functions called control function separability and show it provides a complete characterization of the structural systems of simultaneous equations in which the control function procedure is valid.

We consider a three-dimensional nonlinear control model based on the Wodarz HIV model. The model phase variables are populations of the uninfected and infected target cells and the concentration of an antiretroviral drug. The drug intake rate is assumed to be a bounded control function. An optimal control problem of minimizing the cumulative infection level (the infected cells population) on a given time interval is stated and solved, and the types of the optimal control for different model parameters are found by analytical methods. We thereby reduce the two-point boundary value problem (TPBVP) for the Pontryagin maximum principle to a problem of the finite-dimensional optimization. Numerical results are presented to illustrate the optimal solution.

In this paper we introduce a numerical technique based on Legendre polynomials for solving of nonlinear optimal control problems, where this approach is used for solving optimal control problem of an HIV infection treatment control model. In this paper, first by using healthy cells CD4 + T (T), infected cells CD4 + T (I), viral load (V) and also by using a drug inhibitor of reverse polygraph as a control function, a control model is presented for treatment of HIV infection. A cost function to minimize the cost of drug during the treatment is defined as well. To find the pair of trajectory and control of such nonlinear optimal control problem, we used shifted Legendre polynomials to approximate optimal pair of trajectory and control.

The control parameterization method is a popular numerical technique for solving optimal control problems. The main idea of control parameterization is to discretize the control space by approximating the control function by a linear combination of basis functions. Under this approximation scheme, the optimal control problem is reduced to an approximate nonlinear optimization problem with a finite number of decision variables. This approximate problem can then be solved using nonlinear programming techniques. The aim of this paper is to introduce the fundamentals of the control parameterization method and survey its various applications to non-standard optimal control problems. Topics discussed include gradient computation, numerical convergence, variable switching times, and methods for handling state constraints. We conclude the paper with some suggestions for future research.

An approach to solving control problems for thermal processes with phase transitions is proposed. The problem of controlling the phase boundary evolution in the course of solidification in metal casting is studied by following this approach. A mathematical model of this process is underlain by a three-dimensional nonstationary two-phase initial–boundary value problem of the Stefan type. The control function is determined by solving the optimal control problem formulated in this work.

In this paper we consider a classical global optimization problem of obtaining minimum distance between pairs of closed sets by using an extension of the weak contraction mapping. The extension is accomplished with the help of three control functions. P-property of pairs of closed sets has been used. The result has several corollaries and an illustrative example. One corollary extends an existing work on best proximity point problem. The example shows that the extension is actual.

In this paper, we consider an optimal co with state and control constraints, where a prespecified sequence of active subsystems is given. Both the switching instants and the control function are to be chosen such that the cost functional is minimized. Firstly, we convert the problem into a conventional optimal control problem based on parameterization of the switching instants. Next, through discretizing the control space, applying a time-scaling transformation and using the smoothing technique, the optimal control problem is approximated by a non-linear optimization problem with linear constraints and bounds on the variables. Furthermore, an algorithm that finds a suboptimal solution to the original problem is proposed. One major advantage with the new algorithm implementation is the only need to solve a non-linear optimization problem with linear constraints and bounds on the variables. The gradients of these linear constraints and the objective function can be easily obtained. Therefore, the or...

The main control goal in fed-batch fermentation is to get a high concentration of production. In this paper, a new nonlinear dynamical system, in which the feed rate of glycerol is the control function and the switching instants between the batch and feed processes are the variables, is proposed to formulate the fed-batch fermentation process of glycerol bioconversion to 1,3-propanediol (1,3-PD). To maximize the concentration of 1,3-PD at the terminal time, an optimal control model involving the nonlinear system and subject to the continuous state constraints and the control constraint is presented. To seek the optimal solution of the constrained optimal control problem, the control parametrization enhancing transform together with the constraint transcription technique is first used to convert the constrained optimal control problem into a sequence of mathematical programming problems. An improved Particle Swarm Optimization is subsequently constructed to solve the resultant mathematical programming problem. Numerical results show that, by employing the obtained optimal strategy, 1,3-PD concentration at the terminal time can be increased considerably.

Bayesian model averaging provides the best possible estimate of a model, given the data. This article uses that approach twice: once to get a distribution of plausible models of the world, and again to find a distribution of plausible control functions. The resulting ensemble gives control instructions different from simply taking the single best-fitting model and using it to find a single lowest-error control function for that single model. The only drawback is, of course, the need for more computer time: this article demonstrates that the required computer time is feasible. The test problem here is from flood control and risk management.

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated.

We consider the control problem for the 2D model of the scattering TM-polarized electromagnetic waves. This problem arises when developing the design technologies of electromagnetic cloaking devices using optimization method. The role of control function is played by refraction index of the dielectric obstacle. We prove the solvability of the control problem, derive the optimality system, study the uniqueness and stability of solutions and propose an efficient numerical algorithm.

The aim of this paper is to formulate the weak maximum principle for hybrid optimal control problems. We give definitions of a hybrid system and a hybrid state trajectory. These definitions are kept at an equation-based level, to make the analysis helpful while constructing an appropriate numerical procedure. We then define an optimal control problem for a hybrid system. Next, we analyze the perturbations of a hybrid state trajectory in response to variations of a control function. We build first order approximations to the trajectory perturbations and utilize them to approximate the variations of cost and constraint functionals in the optimal control problem. Finally, we introduce the adjoint variables, which enables to formulate the weak maximum principle in an elegant way. We also give some remarks concerning a construction of a numerical procedure for hybrid optimal control problems.

By introduce the information entropy and the average-distance-amongst-points to analysis the population distribution in the process of evolution, and figured out the cause of the DE/best/* premature convergence is the control function of the current optimal individual to decrease the population diversity of the algorithm. Based on the number of base vectors, improved the DE algorithm by setting up the elite group, the elite differential evolution algorithm is proposed. Finally, several typical test functions are used to test the performance. The results show that the elite differential evolution algorithm has a good performance in the search success rate and the global search capability.

This paper attempts to propose a new method based on capabilities of artificial neural networks, in function approximation, to attain the solution of optimal control problems. To do so, we try to approximate the solution of Hamiltonian conditions based on the Pontryagin minimum principle (PMP). For this purpose, we introduce an error function that contains all PMP conditions. In the proposed error function, we used trial solutions for the trajectory function, control function and the Lagrange multipliers. These trial solutions are constructed by using neurons. Then, we minimize the error function that contains just the weights of the trial solutions. Substituting the optimal values of the weights in the trial solutions, we obtain the optimal trajectory function, optimal control function and the optimal Lagrange multipliers.

A mathematical model for leukemia therapy based on the Gompertzian law of cell growth is studied. It is assumed that the chemotherapeutic agents kill leukemic as well as normal cells. Effectiveness of the medicine is described in terms of a therapy function. Two types of therapy functions are considered: monotonic and non-monotonic. In the former case the level of the effect of the chemotherapy directly depends on the quantity of the chemotherapeutic agent. In the latter case the therapy function achieves its peak at a threshold value and then the effect of the therapy decreases. At any given moment the amount of the applied chemotherapeutic is regulated by a control function with a bounded maximum. Additionally, the total quantity of chemotherapeutic agent which can be used during the treatment process is bounded too. The problem is to find an optimal strategy of treatment to minimize the number of leukemic cells while at the same time retaining as many normal cells as possible. With the help of Pontryagin’s Maximum Principle it was proved that the optimal control function has at most one switch point in both monotonic and non-monotonic cases for most relevant parameter values. A control strategy called alternative is suggested. This strategy involves increasing the amount of the chemotherapeutical medicine up to a certain value within the shortest possible period of time, and holding this level until the end of the treatment. The comparison of the results from the numerical calculation using the Pontryagin’s Maximum Principle with the alternative control strategy shows that the difference between the values of cost functions is negligibly small.

We bridge mathematical number theory with that of optimal control and show that a generalised Fibonacci sequence enters the control function of finite horizon dynamic optimisation problems with one state and one control variable. In particular, we show that the recursive expression describing the first-order approximation of the control function can be written in terms of a generalised Fibonacci sequence when restricting the final state to equal the steady state of the system. Further, by deriving the solution to this sequence, we are able to write the first-order approximation of optimal control explicitly. Our procedure is illustrated in an example often referred to as the Brock-Mirman economic growth model.