Discrete Optimal Control Problem Research Articles

An Optimal Penalty Constant For Discrete Optimal Control Regulator Problems

Indiscriminate and imprecise use of penalty constants can lead to substantial computational problems and consequently erroneous conclusions and deductions. This paper view this, with utmost seriousness and thus establish an optimal penalty constant φ that optimizes the minimization of the cost functional while solving discrete optimal control problems using extended conjugate gradient method (ECGM). I employed some measures that examine optimal control of dynamic processes that can be described by Differential Algebraic Equations (DAEs) that entails integer restriction on some or all of the control functions. I established the construction of an optimal penalty constant which can be employed in the extended conjugate gradient method algorithm for discrete optimal control regulator problems. Comparative results emanating from the use of different penalty constants/algorithms on some problems are given. The obtained numerical results reveal that the proposed optimal penalty constant expression is efficient.

Optimal Sizing and Control of Battery Energy Storage Systems for Hybrid Turboelectric Aircraft

<div class="section abstract"><div class="htmlview paragraph">Hybrid-electric gas turbine generators are considered a promising technology for more efficient and sustainable air transportation. The Ohio State University is leading the NASA University Leadership Initiative (ULI) <i>Electric Propulsion: Challenges and Opportunities</i>, focused on the design and demonstration of advanced components and systems to enable high-efficiency hybrid turboelectric powertrains in regional aircraft to be deployed in 2030. Within this large effort, the team is optimizing the design of the battery energy storage system (ESS) and, concurrently, developing a supervisory energy management strategy for the hybrid system to reduce fuel burn while mitigating the impact on the ESS life. In this paper, an energy-based model was developed to predict the performance of a battery-hybrid turboelectric distributed-propulsion (BHTeDP) regional jet. A study was conducted to elucidate the effects of ESS sizing and cell selection on the optimal power split between the turbogenerators (TGs) and ESS. To this extent, the supervisory energy management strategy is formulated into a discrete time optimal control problem and solved via dynamic programming. The performance of BHTeDP was compared to a turboelectric distributed-propulsion (TeDP) next-gen aircraft that assumes improvements in weight, drag, and engine efficiency consistent with regional jet entering operation in 2035.</div></div>

Optimal Stefan problem

We consider the inverse multiphase Stefan problem with homogeneous Dirichlet boundary condition on a bounded Lipschitz domain, where the density of the heat source is unknown in addition to the temperature and the phase transition boundaries. The variational formulation is pursued in the optimal control framework, where the density of the heat source is a control parameter, and the criteria for optimality is the minimization of the $L_2-$norm declination of the trace of the solution to the Stefan problem from a temperature measurement on the whole domain at the final time. The state vector solves the multiphase Stefan problem in a weak formulation, which is equivalent to Dirichlet problem for the quasilinear parabolic PDE with discontinuous coefficient. The optimal control problem is fully discretized using the method of finite differences. We prove the existence of the optimal control and the convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control. In particular, the convergence of the method of finite differences for the weak solution of the multidimensional multiphase Stefan problem is proved. The proofs are based on achieving a uniform $L_{\infty}$ bound and $W_2^{1,1}$ energy estimate for the discrete multiphase Stefan problem.

A simple proof of the discrete time geometric Pontryagin maximum principle on smooth manifolds

We establish a geometric Pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, c) constraints on the frequency spectrum of the optimal control trajectories. Our proof follows, in spirit, the path to establish geometric versions of the Pontryagin maximum principle on smooth manifolds indicated in Chang (2011) in the context of continuous-time optimal control.

Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform $L_{\infty}$ bound, and $W_2^{1,1}$-energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient.

Optimal control of coefficients in parabolic free boundary problems modeling laser ablation

Inverse Stefan type free boundary problem for the second order parabolic equation arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. New PDE constrained optimal control framework in Hilbert–Besov spaces introduced in Abdulla (2013) is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proved. Numerical algorithm based on the Fréchet differentiability and the gradient method in Hilbert–Besov spaces is implemented. The results of the numerical experiments for the identification of the free boundary and diffusion coefficient are presented.

Hp spectral element approximation for integral state constrained optimal control problems governed by harmonic equations

In this paper, an integral state constrained optimal control problem governed by first bi-harmonic equation and its hp spectral element approximation are considered. We first present the optimality conditions of the continuous and discrete optimal control problems, respectively. Then, a priori error estimates of hp spectral element approximation for the optimal control problem are derived rigorously. Moreover, a posteriori error estimates are established for the coupled state and control approximations in detail. Such estimators can be used to construct reliable adaptive methods for optimal control problems.

ONE MODEL OF MIGRATION FLOWS CONTROL

The paper presents a discrete optimal control problem with constraints on which a method for calculation of migration flows, where qualified and unskilled workers are distinguished, is discussed. At the same time, the optimality criterion in the problem is associated with the achievement of the maximum output with the minimization of the total number of migrants. Numerical calculations are provided that illustrate the sustainable growth scenario over a 10 year period. The work objectives included the development of an approach for calculating the necessary size of working-age population migration and its components to achieve optimal output growth. A macromodel is proposed, which is a discrete optimal control problem. An algorithm for the control synthesis is pro-posed. Numerical modeling is carried out. The obtained results can be used in migration flows planning and management processes.

Series Solution of Stochastic Dynamic Programming Equations

Abstract In this paper we consider discrete time stochastic optimal control problems over infinite and finite time horizons. We show that for a large class of such problems the Taylor polynomials of the solutions to the associated Dynamic Programming Equations can be computed degree by degree.

Discretization of Optimal Control Problems Governed by p-Laplacian Elliptic Equations

In this paper, a state-constrained optimal control problem governed by p-Laplacian elliptic equations is studied. The feasible control set or the cost functional may be nonconvex, and the purpose is to obtain the convergence of a solution of the discretized control problem to an optimal control of the relaxed continuous problem.

Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints

We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. While the first two types of constraints are already included in the existing versions of the Pontryagin maximum principle, it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system. We provide two different proofs of our Pontryagin maximum principle in this article, and include several special cases fine-tuned to control-affine nonlinear and linear system models. In particular, for minimization of quadratic cost functions and linear time invariant control systems, we provide tight conditions under which the optimal controls under frequency constraints are either normal or abnormal.

Dual Forecasting Algorithm for Technological Structural Matrices in Dynamic Input-Output Models

Based on the global Krotov successive improvement method, we propose a dual computational algorithm for a discrete optimal control problem corresponding to a convex large-scale quadratic programming problem with a separable functional that arises in the prediction of the direct costs (structural) matrix in dynamic input-output models. With decomposition, we are able to use a special form of the constraint matrix to reduce the problem dimension.

First- and second-order necessary conditions with respect to components for discrete optimal control problems

This paper is devoted to the study of discrete optimal control problems. We aim to obtain more constructive optimality conditions under weakened convexity assumptions. Based on a new approach introduced in this work, an optimality condition with respect to every component is obtained in the form of a global maximum principle. In addition, an optimality condition with respect to one of the components of a control in the form of the global maximum principle and with respect to another component of a control in the form of the linearized maximum principle are obtained. Furthermore, various second-order optimality conditions in terms of singular and quasi-singular controls with respect to the components are obtained on the fly.

Investigation of one linear non-smooth two-parameter discrete optimal control problem

The object of research is the linear optimal control problem described by discrete two-parameter systems under the assumption that the controlled process is stepwise.The work is aimed at deriving the necessary first-order optimality conditions in the case of a non-smooth quality function. And also to establish the necessary conditions of second-order optimality in stepwise control problems for discrete two-parameter systems. The paper investigates one linear two-parameter discrete optimal control problem with a non-smooth quality criterion. A special increment of the quality functional is calculated. Cases under the condition of a convex set are considered. The concept of special control in the problem under study is given. A number of necessary optimality conditions for the first and second orders are established. And also the necessary second-order optimality conditions are obtained in terms of directional derivatives. In the case of a linear quality criterion, the necessary and sufficient optimality condition is proved using the increment formula by analogous arguments. Under the assumption that the set is convex, a special increment of the quality criterion for admissible control is defined.The methods of calculus of variations and optimal control, the theory of difference equations are used. The result is obtained for the optimality of a special, first-order control, in the case of convexity of the set. The case when the minimized functional is linear is considered. In this case, a necessary and sufficient condition is obtained for the optimality of the admissible control.Thanks to the research results, it is possible to obtain the necessary first-order optimality conditions in terms of directional derivatives in the stepwise problem of optimal control of discrete two-parameter systems. As well as the necessary conditions of optimality of the second order in the case of convexity of the control domain and the necessary optimality conditions of special controls.The theoretical results obtained in the work are of interest in the theory of optimal control of step systems and can be used in the further development of the theory of necessary optimality conditions for step control problems.

Discrete time optimal control with frequency constraints for non-smooth systems

We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) smooth frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of type (b), while less standard in the literature, effectively serve the purpose of describing important properties of inertial actuators and systems. The conjunction of constraints of type (a) and (b) is a relatively new phenomenon in optimal control but is important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability of our theory.

Numerical solution of a parabolic optimal control problem arising in economics and management

We investigate a parabolic optimal control problem that serves as a mathematical model for a class of problems of economics and management. The problem is to minimize the objective functional that is not convex and not coercive, and the state equation is a two-dimensional linear parabolic diffusion-advection equation controlled by the coefficients of the advective part. The main property of a solution of state equation is its non-negativity for a non-negative initial data. We prove that implicit (backward Euler) and fractional steps (operator splitting) approximations of the state equation have strictly positive solutions for positive time and use this fact to prove the existence of a solution for a discrete optimal control problem without imposing any additional constraints on the control function and mesh parameters. We derive first order optimality conditions and apply gradient-type iterative methods for the constructed discrete optimal control problem. Numerical tests confirm theoretically valid results and demonstrate the effectiveness of the proposed solution methods.

Numerical Solution for Classical Optimal Control Problem Governing by Hyperbolic Partial Differential Equation via Galerkin Finite Element-Implicit method with Gradient Projection Method

This paper deals with the numerical solution of the discrete classical optimal control problem (DCOCP) governing by linear hyperbolic boundary value problem (LHBVP). The method which is used here consists of: the GFEIM " the Galerkin finite element method in space variable with the implicit finite difference method in time variable" to find the solution of the discrete state equation (DSE) and the solution of its corresponding discrete adjoint equation, where a discrete classical control (DCC) is given. The gradient projection method with either the Armijo method (GPARM) or with the optimal method (GPOSM) is used to solve the minimization problem which is obtained from the necessary condition for optimality of the DCOCP to find the DCC.An algorithm is given and a computer program is coded using the above methods to find the numerical solution of the DCOCP with step length of space variable , and step length of time variable . Illustration examples are given to explain the efficiency of these methods. The results show the methods which are used here are better than those obtained when we used the Gradient method (GM) or Frank Wolfe method (FWM) with Armijo step search method to solve the minimization problem.

Efficient Iterative Method for Solving Optimal Control Problem Governed by Diffusion Equation with Time Fractional Derivative

We solve finite-difference approximations of a linear-quadratic optimal control problem governed by Dirichlet boundary value problem with fractional time derivative. The state equation of the problem is approximated using locally one-dimensional difference schemes. The stability estimates of discrete state equations necessary for studying the convergence of iterative solution methods for the constructed discrete optimal control problems are proved. The rate of convergence of the proposed iterative method is obtained and the optimal iterative parameter is found. The results of numerical tests for a model problem are presented.

Necessary First-Order and Second-Order Optimality Conditions in Discrete-Time Stochastic Systems

In this paper, first-order and second-order necessary conditions for optimality for discrete-time stochastic optimal control problems governed by discrete-time Ito equations are established. A new discrete-time backward stochastic equation and discrete-time backward stochastic matrix equation are introduced. Based on the discrete-time backward stochastic Ito equation, a discrete-time stochastic maximum principle for the stochastic discrete optimal control problems is obtained. Moreover, using the discrete-time backward stochastic matrix equation the second-order necessary conditions for optimality are obtained.

Fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation

In this paper, we mainly investigate the fractional spectral collocation discretization of optimal control problem governed by a space-fractional diffusion equation. Existence and uniqueness of the solution to optimal control problem is proved. The continuous first order optimality condition is derived. The eigenfunctions of two classes of fractional Strum–Liouville problems are used as basis functions to approximate state variable and adjoint state variable, respectively. The fractional spectral collocation scheme for the control problem is constructed based on ‘first optimize, then discretize’ approach. Note that the solutions of fractional differential equations are usually singular near the boundary, a generalized fractional spectral collocation scheme for the control problem is proposed based on ‘first optimize, then discretize’ approach. A projected gradient algorithm is designed based on the discrete optimality condition. Numerical experiments are carried out to verify the effectiveness of the proposed numerical schemes and algorithm.