Algorithm For Optimal Control Problems Research Articles

First-Order Necessary Conditions in Optimal Control

In an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided conditions under which accumulation points satisfy a condition requiring a certain optimality function, used in the algorithms to generate search directions, to be nonnegative for all controls. The aim of this paper is to clarify the nature of this optimality condition, which we call the first-order minimax condition, and of a related integrated form of the condition, which, also, is implicit in past algorithm convergence analysis. We consider these conditions, separately, when a pathwise state constraint is, and is not, included in the problem formulation. When there are no pathwise state constraints, we show that the integrated first-order minimax condition is equivalent to the minimum principle and that the minimum principle (and equivalent integrated first-order minimax condition) is strictly stronger than the first-order minimax condition. For problems with state constraints, we establish that the integrated first-order minimax condition and the minimum principle are, once again, equivalent. But, in the state constrained context, it is no longer the case that the minimum principle is stronger than the first-order minimax condition, or vice versa. An example confirms the perhaps surprising fact that the first-order minimax condition is a distinct optimality condition that can provide information, for problems with state constraints, in some circumstances when the minimum principle fails to do so.

Extension technology and extrema selections in a stochastic multistart algorithm for optimal control problems

The paper proposes a method for finding the minimum value of a functional in nonlinear nonconvex optimal control problems. The method takes advantage of the hidden convexity property of the controlled differential equations systems. Application of the multistart idea with extrema selection procedures makes it possible to create software that does not strongly depend on the problem size and supplies additional information about the object under investigation. Three test problems are considered to show specific properties of using the stochastic multistart algorithm and extension numerical technology.

A domain decomposition algorithm for optimal control problems governed by elliptic PDEs with random inputs

In this study, we apply a domain decomposition algorithm based on optimization technique to optimal control problems governed by an elliptic partial differential equation with random inputs. The domain decomposition is implemented by introducing an auxiliary optimal control problem, which results in a multi-objective optimization problem. We prove the existence of a solution to the resulting optimization problem as well as the convergence to the optimal solution of original control problem. Solutions of the domain decomposition problem are determined from an optimality system and error estimates for finite element approximations are analyzed. Finally, some numerical experiments are provided to confirm theoretical results.

A fast gradient projection algorithm for time fractional optimal control problem

In this paper a fast gradient projection algorithm for optimal control problems governed by time fractional diffusion equation is developed. The state equation is discretized by piecewise linear FE for space variable and L1 scheme for time variable, while the control variable is approximated by variational discretization. Based on the block triangular Toeplitz structure of the coefficient matrix of the discretized state equation and adjoint state equation, a fast gradient projection algorithm is designed for the control problem. Numerical examples are carried out to illustrate the effectiveness of the algorithms.

An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

In this work, we present an adaptive Newton-type method to solve nonlinear constrained optimization problems, in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive strategy is based on a goal-oriented a posteriori error estimation for the discretization and for the iteration error. The iteration error stems from an inexact solution of the nonlinear system of first-order optimality conditions by the Newton-type method. This strategy allows one to balance the two errors and to derive effective stopping criteria for the Newton iterations. The algorithm proceeds with the search of the optimal point on coarse grids, which are refined only if the discretization error becomes dominant. Using computable error indicators, the mesh is refined locally leading to a highly efficient solution process. The performance of the algorithm is shown with several examples and in particular with an application in the neurosciences: the optimal electrode design for the study of neuronal networks.

A gradient algorithm for optimal control problems with model-reality differences

In this paper, we propose a computational approach to solve a model-based optimal control problem. Our aim is to obtain the optimal solution of the nonlinear optimal control problem. Since the structures of both problems are different, only solving the model-based optimal control problem will not give the optimal solution of the nonlinear optimal control problem. In our approach, the adjusted parameters are added into the model used so as the differences between the real plant and the model can be measured. On this basis, an expanded optimal control problem is introduced, where system optimization and parameter estimation are integrated interactively. The Hamiltonian function, which adjoins the cost function, the state equation and the additional constraints, is defined. By applying the calculus of variation, a set of the necessary optimality conditions, which defines modified model-based optimal control problem, parameter estimation problem and computation of modifiers, is then derived. To obtain the optimal solution, the modified model-based optimal control problem is converted in a nonlinear programming problem through the canonical formulation, where the gradient formulation can be made. During the iterative procedure, the control sequences are generated as the admissible control law of the model used, together with the corresponding state sequences. Consequently, the optimal solution is updated repeatedly by the adjusted parameters. At the end of iteration, the converged solution approaches to the correct optimal solution of the original optimal control problem in spite of model-reality differences. For illustration, two examples are studied and the results show the efficiency of the approach proposed.

A Shooting Algorithm for Optimal Control Problems with Singular Arcs

In this article, we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss–Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system), we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated with the perturbed problem. We present numerical tests that validate our method.

Solving Optimal Control Problem Using Max-Min Ant System

An improved ant colony algorithm for optimal control problems with box constrain on control functions is presented. The hypercube of the feasible controls as well as the time interval are initially discretized to approximate control problem into a discrete parameter selection problem. Then, the ant colony algorithm is applied to search for optimum parameters of approximated problem while a proper local search is also introduced to iteratively enhance the quality of solution. The results of numerical simulation on MATLAB environment illustrate the effectiveness of this method.

HN-adaptive spectral element discretization of optimal control problems for environmental applications

In this work, we propose an hN adaptive algorithm for optimal control problems whose space discretization is based on the spectral element method. We focus on a special class of problems described by quadratic cost functional and linear advection-diffusion state equation. This model has been employed to forecast and suitably reduce, by an optimization process, the pollutant concentration in air on a specific area of interest. Moreover, a local adaptive algorithm is employed to achieve a desired accuracy on the cost functional by reducing the computational cost of the problem.

A memory-reduced implementation of the Newton-CG method in optimal control of nonlinear time-dependent PDEs

Derivative-based solution algorithms for optimal control problems of time-dependent nonlinear PDE systems require multiple solutions of backward-in-time adjoint systems. Since these adjoint systems, in general, depend on the primal state, and thus on forward information, the storage requirement for such solution algorithms is very large. This paper proposes stable and memory-efficient checkpointing techniques for evaluating gradients and Hessian times increment for such solution algorithms, and presents numerical tests with the instationary Navier–Stokes system which demonstrate that huge memory savings are achieved by the proposed approach while the increase in runtime is moderate. More precisely, a memory reduction of two orders of magnitude causes only a slow down factor of two in run-time.

Homotopy Algorithm for Optimal Control Problems with a Second-order State Constraint

This paper deals with optimal control problems with a regular second-order state constraint and a scalar control, satisfying the strengthened Legendre-Clebsch condition. We study the stability of structure of stationary points. It is shown that under a uniform strict complementarity assumption, boundary arcs are stable under sufficiently smooth perturbations of the data. On the contrary, nonreducible touch points are not stable under perturbations. We show that under some reasonable conditions, either a boundary arc or a second touch point may appear. Those results allow us to design an homotopy algorithm which automatically detects the structure of the trajectory and initializes the shooting parameters associated with boundary arcs and touch points.

Well-Posedness of the Shooting Algorithm for State Constrained Optimal Control Problems with a Single Constraint and Control

This paper deals with the shooting algorithm for optimal control problems with a scalar control and a regular scalar state constraint. Additional conditions are displayed, under which the so-called alternative formulation is equivalent to Pontryagin's minimum principle. The shooting algorithm appears to be well-posed (invertible Jacobian) iff (i) the no-gap second-order sufficient optimality condition holds, and (ii) when the constraint is of order $q \geq 3$, there is no boundary arc. Stability and sensitivity results without strict complementarity at touch points are derived using Robinson's strong regularity theory, under a minimal second-order sufficient condition. The directional derivatives of the control and state are obtained as solutions of a linear quadratic problem.

Inexact Central Path Following Algorithms for Optimal Control Problems

A new approach to the numerical solution of optimal control problems including control and state constraints is presented. Like hybrid methods, the approach aims at combining the advantages of direct and indirect methods. Unlike hybrid methods, however, our method is directly based on interior point concepts in function space—realized via an adaptive multilevel scheme applied to the complementarity formulation and numerical continuation along the central path. An adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact Newton methods. Finally, the performance of our new type of algorithm is documented by a simple example within the range of our present theory, and by the successful treatment of the well-known intricate windshear problem outside this range.

Computation of optimal Mars trajectories via combined chemical/electrical propulsion, Part 3: Compromise solutions

The success of the solar-electric ion engine powering the DS1 spacecraft has paved the way toward the use of low-thrust electrical engines in future planetary/interplanetary missions. Vis-à-vis a chemical engine, an electrical engine has a higher specific impulse, implying a possible decrease in propellant mass; however, the low-thrust aspect discourages the use of an electrical engine in the near-planet phases of a trip, since this might result in an increase in flight time. Therefore, a fundamental design problem is to find the best combination of chemical propulsion and electrical propulsion for a given mission, for example, a mission from Earth to Mars. With this in mind, this paper is the third of a series dealing with the optimization of Earth–Mars missions via the use of hybrid engines, namely the combination of high-thrust chemical engines for planetary flight and low-thrust electrical engines for interplanetary flight. We look at the deep-space interplanetary portion of the trajectory under rather idealized conditions. The two major performance indexes, the propellant mass and the flight time, are in conflict with one another for the following reason: any attempt at reducing the former causes an increase in the latter and vice versa. Therefore, it is natural to consider a compromise performance index involving the scaled values of the propellant mass and flight time weighted respectively by the compromise factor C and its complement 1 - C . We use the compromise factor as the parameter of the one-parameter family of compromise trajectories. Analyses carried out with the sequential gradient-restoration algorithm for optimal control problems lead to results which can be highlighted as follows. Thrust profile. Generally speaking, the thrust profile of the compromise trajectory includes three subarcs: the first subarc is characterized by maximum thrust in conjunction with positive (upward) thrust direction; the second subarc is characterized by zero thrust (coasting flight); the third subarc is characterized by maximum thrust in conjunction with negative (downward) thrust direction. Effect of the compromise factor. As the compromise factor increases, the propellant mass decreases and the flight time increases; correspondingly, the following changes in the thrust profile take place: (a) the time lengths of the first and third subarcs (powered phases) decrease slightly, meaning that thrust application occurs for shorter duration; also, the average value of the thrust direction in the first and third subarcs decreases, implying higher efficiency of thrust application wrt the spacecraft energy level; as a result, the total propellant mass decreases; (b) the time length of the second subarc (coasting) increases considerably, resulting in total time increase. Minimum time trajectory. If C = 0 , the resulting minimum time trajectory has the following characteristics: (a) the time length of the coasting subarc reduces to zero and the three-subarc trajectory degenerates into a two-subarc trajectory; (b) maximum thrust is applied at all times and the thrust direction switches from upward to downward at midcourse. Minimum propellant mass trajectory. If C = 1 , the resulting minimum propellant mass trajectory has the following characteristics: (a) the thrust magnitude has a bang-zero-bang profile; (b) for the powered subarcs, the thrust direction is tangent to the flight path at all times.

On feasibility of launch vehicle designs

We study various launch vehicle designs for a spacecraft to be launched vertically and ascending to an Earth orbit located at Space Station altitude ( h = 463 km). We optimize the trajectory from the maximum payload viewpoint with respect to the angle of attack time history and thrust setting time history. We consider single-stage, double-stage, and triple-stage configurations. To perform the study, we employ a newly developed member of the family of sequential gradient-restoration algorithms for optimal control problems, the multiple-subarc sequential gradient-restoration algorithm. Assuming that the engine specific impulse has the best value available today ( I sp = 450 s), we study the effect of the structural factor ε (ratio of the structural mass to sum of structural mass and propellant mass) on the payload and then develop simple relations connecting the structural factor to the payload. With these relations, we are able to predict the limiting value of the structural factor for which the payload vanishes. To two significant digits, the limiting value is approximately ε ≅ 0.11 for a single-stage configuration, ε ≅ 0.27 for a double-stage configuration, and ε ≅ 0.33 for a triple-stage configuration. Based on these results, we conclude that, given the current values of the engine specific impulse and spacecraft structural factor, a single-stage configuration is not feasible at this particular time, while double-stage and triple-stage configurations are feasible. The above limiting values refer to multistage configurations designed with uniform structural factor. Higher payloads and higher limiting values for the last stage structural factor are possible for configurations designed with nonuniform structural factor, that is, with a last-stage structural factor much higher than the structural factors of the previous stages. For example, a double-stage configuration with ε 1 = 0.10, ε 2 = 0.25 can deliver to orbit a payload mass equal to 4.7% at the takeoff mass; a triple-stage configuration with ε 1 = ε 2 = 0.10, ε 3 = 0.25 can deliver to orbit a payload mass equal to 6.8% of the takeoff mass.

Optimization of parameters in the second-order improving algorithms for optimal control problems

The paper describes modifications of the second-order improving algorithms for optimal control problems. The algorithms presented are based on the localization technique and involve some parameters, which regulate the depth of the functional descent on each iteration. The modifications proposed allow one to organize a search for the best values of parameters on each of the steps.

Using logarithmic penalties in the shooting algorithm for optimal control problems

AbstractThe paper deals with optimal control problems of ordinary differential equations with bound control constraints. We analyse the logarithmic penalty method for converting the problem into an unconstrained one, the latter being solved by a shooting algorithm. Convergence of the value function and optimal controls is obtained for linear quadratic problems, and more generally when the control variable enters linearly in the state equation and in a quadratic way in the cost function. We display some numerical results on two examples: an aircraft maneuver, and the stabilization of an oscillating system. Copyright © 2003 John Wiley & Sons, Ltd.

ANNSA: a hybrid artificial neural network/simulated annealing algorithm for optimal control problems

This paper introduces a numerical technique for solving nonlinear optimal control problems. The universal function approximation capability of a three-layer feedforward neural network has been combined with a simulated annealing algorithm to develop a simple yet efficient hybrid optimisation algorithm to determine optimal control profiles. The applicability of the technique is illustrated by solving various optimal control problems including multivariable nonlinear problems and free final time problems. Results obtained for the different case studies considered agree well with those reported in the literature.

拘束条件を有する最適制御問題の一数値解法

拘束条件を有する最適制御問題の一数値解法

Implementation of distributed iterative algorithm for optimal control problems on several parallel architectures

We consider an optimal control problem without constraint. The solution of this problem leads to a large scale block linear system. We present a two-stage method which is well suited to an asynchronous implementation with flexible communication whereby every processor can have access at any time and without any fixed rule to the current value of the components of the iteration vector updated by any other processor. The implementation is carried out on three types of architecture: a super computer with distributed memory CRAY T3E, a shared memory symmetric multiprocessor (SMP) and a high speed network of SMP.