Indirect Numerical Method Research Articles

Loss control regions in optimal control problems

This paper addresses optimal control problems with loss control regions. In that context the state space is partitioned into disjoint subsets, referred to as regions, which are classified into two types: control regions and loss control regions. When the state belongs to a control region, the control is permanent (i.e. the control value is authorized to be modified at any time). On the contrary, when the state belongs to a loss control region, the control must remain constant as long as the state belongs to this region. The objective of this paper is twofold. First, we reformulate the above setting into a hybrid optimal control problem (with spatially heterogeneous dynamics) involving moreover a regionally switching parameter, and we prove a corresponding hybrid maximum principle: hence first-order necessary optimality conditions in a Pontryagin form are obtained. Second, this paper proposes a two-steps numerical scheme to solve optimal control problems with loss control regions. It is based on a direct numerical method (applied to a regularized problem) which initializes an indirect numerical method (applied to the original problem and based on the aforementioned necessary optimality conditions). This numerical approach is applied to several illustrative examples.

ESTIMATING THE QUALITY FUNCTIONAL IN OPTIMAL CONTROL VARIATIONAL CALCULUS TASKS UNDER THE CHANGES OF THE LEFT-HAND ARGUMENT OF BOUNDARY CONDITION

Estimating the quality functional in optimal control variational calculus tasks under the changes of boundary condition is considered. In control systems, there is a certain permissible change of the controlled parameter based on which the control system is designed. Unlike ordinary control systems, in optimal control systems, there is another indicator - the quality functional (optimality criteria), which also depends on the controlled parameters. Therefore, the permissible range of changes of controlled parameters in optimal control systems should be determined based on the quality functional value’s tolerances. Therefore, estimating the quality functional under the changes of boundary conditions has theoretical and practical importance. It is particularly important for problems solved by numerical methods because it is quite simple for problems solved analytically. In this article, the estimation methods of the quality functional under the changes of left-hand argument of boundary condition in the form of Taylor series with an accuracy of the second member in optimal control variational calculus tasks solved by the indirect nu-merical method are developed. According to the developed method, both the optimization function and the quality functional are functions of the left-hand argument of boundary condition. In other words, the boundary condition parameter is an argument of the optimi-zation function. The mathematical apparatus for determining the specified series coeffi-cients has been developed. These coefficients depend on the partial derivatives of the op-timization function by boundary parameter. The boundary conditions for partial deriva-tive functions of the optimal function by boundary parameters, and the determination methods for these functions using the known optimal function are studied. The truth of the Taylor series coefficients, and the algorithm for their determination are shown experimen-tally.

The application of a universal separating vector lemma to optimal sampled-data control problems with nonsmooth Mayer cost function

<p style='text-indent:20px;'>In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set. This so-called <i>universal separating vector lemma</i> is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by implementing an indirect numerical method.</p>

A computational method based on the generalized Lucas polynomials for fractional optimal control problems

Nonorthogonal polynomials have many useful properties like being used as a basis for spectral methods, being generated in an easy way, having exponential rates of convergence, having fewer terms and reducing computational errors in comparison with some others, and producing most important basic polynomials. In this regard, this paper deals with a new indirect numerical method to solve fractional optimal control problems based on the generalized Lucas polynomials. Through the way, the left and right Caputo fractional derivatives operational matrices for these polynomials are derived. Based on the Pontryagin maximum principle, the necessary optimality conditions for this problem reduce into a two-point boundary value problem. The main and efficient characteristic behind the proposed method is to convert the problem under consideration into a system of algebraic equations which reduces many computational costs and CPU time. To demonstrate the efficiency, applicability, and simplicity of the proposed method, several examples are solved, and the obtained results are compared with those obtained with other methods.

Fuel-Optimal Ascent Trajectory Problem for Launch Vehicle via Theory of Functional Connections

In this study, we develop a method based on the Theory of Functional Connections (TFC) to solve the fuel-optimal problem in the ascending phase of the launch vehicle. The problem is first transformed into a nonlinear two-point boundary value problem (TPBVP) using the indirect method. Then, using the function interpolation technique called the TFC, the problem’s constraints are analytically embedded into a functional, and the TPBVP is transformed into an unconstrained optimization problem that includes orthogonal polynomials with unknown coefficients. This process effectively reduces the search space of the solution because the original constrained problem transformed into an unconstrained problem, and thus, the unknown coefficients of the unconstrained expression can be solved using simple numerical methods. Finally, the proposed algorithm is validated by comparing to a general nonlinear optimal control software GPOPS-II and the traditional indirect numerical method. The results demonstrated that the proposed algorithm is robust to poor initial values, and solutions can be solved in less than 300 ms within the MATLAB implementation. Consequently, the proposed method has the potential to generate optimal trajectories on-board in real time.

New class of hybrid explicit methods for numerical solution of optimal control problems

Forward-backward sweep method (FBSM) is an indirect numerical method used for solving optimal control problems, in which the differential equation arising from this method is solved by the Pontryagin’s maximum principle. In this paper, a set of hybrid methods based on explicit 6th-order RungeKutta method is presented for the FBSM solution of optimal control problems. Order of truncation error, stability region, and numerical results of the new hybrid methods were compared with those of the 6th-order Runge Kutta method. Numerical results show that new hybrid methods are more accurate than the 6th-order Runge–Kutta method and that their stability regions are also wider than that of the 6th-order Runge–Kutta method.

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. We obtain, in particular, a nonpositive averaged Hamiltonian gradient condition associated with an adjoint vector being a function of bounded variation. As a well known challenge, theoretical and numerical difficulties may arise due to the possible pathological behavior of the adjoint vector (jumps and singular part lying on parts of the optimal trajectory in contact with the boundary of the restricted state space). However, in our case with sampled-data controls, we prove that, under certain general hypotheses, the optimal trajectory activates the running inequality state constraints at most at the sampling times. Due to this so-called bouncing trajectory phenomenon, the adjoint vector experiences jumps at most at the sampling times (and thus in a finite number and at precise instants) and its singular part vanishes. Taking advantage of these informations, we are able to implement an indirect numerical method which we use to solve three simple examples.

Tempered stable process, first passage time, and path-dependent option pricing

In this paper, we will discuss an approximation of the characteristic function of the first passage time for a Levy process using the martingale approach. The characteristic function of the first passage time of the tempered stable process is provided explicitly or by an indirect numerical method. This will be applied to the perpetual American option pricing and the barrier option pricing. Numerical illustrations are provided for the calibrated parameters using the market call and put prices.

Calculation of critical parameters for spontaneous combustion for some complex geometries using an indirect numerical method

In the theory of spontaneous combustion, identifying the critical value of the Frank- Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature. doi:10.1017/S1446181117000578

CALCULATION OF CRITICAL PARAMETERS FOR SPONTANEOUS COMBUSTION FOR SOME COMPLEX GEOMETRIES USING AN INDIRECT NUMERICAL METHOD

In the theory of spontaneous combustion, identifying the critical value of the Frank-Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature.

Optimization of Multiphase Aircraft Trajectories Using Hybrid Optimal Control

An approach to optimize multiphase trajectories of commercial transport aircraft is presented. The approach is based on the theory of hybrid optimal control, and it is applied to the case of minimum-fuel trajectories. The multiphase trajectories are composed of three types of phases, climb, cruise, and descent, in a given sequence. In each phase, the optimal control is scalar and of the bang–singular–bang type, and the optimal path is formed by a singular arc and two minimum/maximum-control arcs joining the singular arc with the initial and final switching points. An indirect numerical method is developed, which takes into account the structure of the solution directly in the algorithm and exploits the singular character of the problem. In the analysis, the effects of horizontal winds are taken into account; general along-track wind and crosswind profiles are considered, dependent both on altitude and along-track position. The optimal trajectories are computed for a model of a Boeing 767-300ER performing a climb–cruise–descent trajectory with a one-step cruise climb, and for different wind profiles. The influence of the wind profiles on the optimal results is analyzed and shown to be important.

Natural Convection Heat Transfer from Two Perpendicular Heated Surfaces Embedded In Porous Cavity

The effect of modified Rayleigh number, positions and lengths of two perpendicular heated surfaces on natural convection heat transfer was studied numerically, where the two surfaces are embedded in square cavity filled with saturated porous medium. All walls of the cavity are kept at constant temperature. Indirect numerical method was used to solve the governing equations, which are: the nondimensional Darcy flow equation as well as the non-dimensional energy equation, which were solved numerically by finite difference method using Gauss-Seidel iteration coupled with (Successive Under Relaxation) technique. This study covered a wide range of modified Rayleigh number range (100-1000), nine positions and the ratios of length of vertical surface to the horizontal (0.5,1, 2). It was found that the positions of two heated surfaces have small effect on the heat transfer rate, but any increase in the length of two surfaces leads to an increase in average Nusselt number, but an increase in the length of vertical surface leads to more increase in the average Nusselt number

Sparse random matrices: the eigenvalue spectrum revisited

We revisit the derivation of the density of states of sparse random matrices. We derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit. Using the iterative scheme introduced by Biroli and Monasson [J. Phys. A 32, L255 (1999)] we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration. The combination of the two methods yields a very simple simple geometric interpretation of the tails of the spectrum. We test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum.

An indirect numerical method for the solution of a class of optimal control problems with singular arcs

An indirect numerical method is presented that solves a class of optimal control problems that have a singular arc occurring after an initial nonsingular arc. This method iterates on the subset of initial costate variables that enforce the junction conditions for switching to a singular arc, and the time of switching off of the singular arc to a final nonsingular arc, to reduce a terminal error function of the final conditions to zero. This results in the solution to the two-point boundary-value problem obtained using the minimum principle and some necessary conditions for singular arcs. The main advantage of this method is that the exact solution to the two-point boundary-value problem is obtained. The main disadvantage is that the sequence of controls for the problem must be known to apply this method. Two illustrative examples are presented.

An indirect numerical method for the solution of a class of optimal control problems with singular arcs

An indirect numerical method for the solution of a class of optimal control problems with singular arcs