Legendre wavelet method for solving variable-order nonlinear fractional optimal control problems with variable-order fractional Bolza cost

This paper deals with a numerical method for solving variable-order fractional optimal control problem with a fractional Bolza cost composed as the aggregate of a standard Mayer cost and a fractional Lagrange cost given by a variable-order Riemann–Liouville fractional integral. Using the integration by part formula and the calculus of variations, the necessary optimality conditions are derived in terms of two-point variable-order boundary value problem. Operational matrices of variable-order right and left Riemann–Liouville integration are derived, and by using them, the two-point boundary value problem is reduced into the system of algebraic equations. Additionally, the convergence analysis of the proposed method has been considered. Moreover, illustrative examples are given to demonstrate the applicability of the proposed method.     

A numerical technique for solving fractional optimal control problems

This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann-Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

An approximate method for numerically solving fractional order optimal control problems of general form

In this article, we discuss fractional order optimal control problems (FOCPs) and their solutions by means of rational approximation. The methodology developed here allows us to solve a very large class of FOCPs (linear/nonlinear, time-invariant/time-variant, SISO/MIMO, state/input constrained, free terminal conditions etc.) by converting them into a general, rational form of optimal control problem (OCP). The fractional differentiation operator used in the FOCP is approximated using Oustaloup’s approximation into a state-space realization form. The original problem is then reformulated to fit the definition used in general-purpose optimal control problem (OCP) solvers such as RIOTS_95, a solver created as a Matlab toolbox. Illustrative examples from the literature are reproduced to demonstrate the effectiveness of the proposed methodology and a free final time OCP is also solved.     

A direct second-variational method for unconstrained optimal control problems

A direct second-variational algorithm is proposed for the iterative solution of optimal control problems in continuous, as well as discrete, systems. While the classical method using the Riccati transformation is shown to follow from the direct method, the latter has certain promising features in terms of structural simplicity, ease of programming, and substantial reduction in computation time. The process involves a predictor-corrector algorithm for a forward sweep in the state equations, followed by a backwards sweep in the adjoint equations. Applied to discrete systems, this method functions under conditions less stringent than in the classical scheme. Linearized quasioptimal feedback control is examined from the point of view of the direct method and a scheme for successive improvement in quasi-optimal policy is proposed for systems that suffer small perturbations from a given optimal trajectory.     

A computational method for solving two‐dimensional nonlinear variable‐order fractional optimal control problems

In this paper, a new class of two‐dimensional nonlinear variable‐order fractional optimal control problems (V‐OFOCPs) is introduced where the variable‐order fractional derivative is defined in the Caputo type. The general procedure for solving theses systems is expanding the state variable and the control variable based on the Legendre cardinal functions in the matrix form. Hence, we derive their operational matrix of derivative (OMD) and operational matrix of variable‐order fractional derivative (OMV‐OFD). More significantly, some properties of these basis functions are proved to be exploited in our approach. Using these achieved results, we simply expand the matrix form of the nonlinear performance index in terms of the Legendre cardinal functions and subsequently convert it to an algebraic equation. We emphasize that it is a valuable advantage of applying cardinal functions in approximation theory. Then, we implement the OMD and the OMV‐OFD of the Legendre cardinal functions to transform the variable‐order fractional dynamical system to a system of algebraic equations. Next, the method of constrained extremum is applied to adjoin the constraint equations including the given dynamical system and the initial‐boundary conditions to the performance index by a set of undetermined Lagrange multipliers. Finally, the necessary conditions of the optimality are derived as a system of nonlinear algebraic equations including the unknown coefficients of the state variable, the control variable and the Lagrange multipliers. The applicability and efficiency of the proposed approach are investigated through the various types of test problems.     

Optimality conditions and a solution scheme for fractional optimal control problems

We formulate necessary conditions for optimality in Optimal control problems with dynamics described by differential equations of fractional order (derivatives of arbitrary real order). Then by using an expansion formula for fractional derivative, optimality conditions and a new solution scheme is proposed. We assumed that the highest derivative in the differential equation of the process is of integer order. Two examples are treated in detail.     

A hybrid approach to optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functional I subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the state x(t), the control u(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.The approach taken is a sequence of two-phase processes or cycles, each composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functions x(t), u(t), π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable.The technique employed is of the hybrid type, in an attempt to combine some of the best features of both the indirect and direct approaches. While a predetermined number and sequence of subarcs are assumed (a feature of direct methods), enforcement of the state inequality constraint is obtained through a Valentine-type representation (a feature of indirect methods).By properly choosing the analytical form of certain non-differential constraints to be satisfied by the augmented control along each subarc composing the extremal arc (these nondifferential constraints are arrived at through the Valentine-type transformation), one ensures satisfaction of the state inequality constraint everywhere. Specifically, strict inequality is enforced for the subarcs internal to the state boundary and strict equality is enforced for the subarcs lying on the state boundary.To facilitate the numerical solution on digital computers, the actual time θ is replaced by the normalized time t, defined in such a way that each subarc composing the extremal arc has a normalized time length Δt = 1. In this way, variable-time corner conditions and variable-time terminal conditions are transformed into fixed-time corner conditions and fixed-time terminal conditions. The actual times θ₁, θ₂, τ at which (i) the state boundary is entered, (ii) the state boundary is exited, and (iii) the terminal manifold is reached are regarded to be components of the parameter π being optimized.The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.     

A direct method for a class of optimal control problems

A direct method is developed for the solution of a class of minimum energy control problems. The method is applicable to linear and nonlinear, stationary and time-varying systems described by input-output functional relations. It is based on the expansion of the kernels of the system and of the input, the control, in terms of a set of functions that are characteristic of the kernels. The optimality is measured by the integral of a positive definite quadratic form of the input over the control time interval. The characteristic expansions reduce the optimal control problem to that of solving a finite set of algebraic equations.     

Piecewise fractional Legendre functions for nonlinear fractional optimal control problems with ABC fractional derivative and non-smooth solutions

In this study, to solve fractional problems with non-smooth solutions (which include some terms in the form of piecewise or fractional powers), a new category of basis functions called the orthonormal piecewise fractional Legendre functions is introduced. The upper bound of the error of the series expansion of these functions is obtained. Two explicit formulas for computing the Riemann–Liouville and Atangana–Baleanu fractional integrals of these functions are derived. A direct method based on these functions and their fractional integral is proposed to solve a family of optimal control problems involving the ABC fractional differentiation whose solutions are non-smooth in the above expressed forms. By the proposed technique, solving the original fractional problem turns into solving an equivalent system of algebraic equations. The established method accuracy is studied by solving some examples.     

On the method of dynamic programming for linear-quadratic problems of optimal control in hybrid systems

For a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. Consideration was given to the hybrid control systems with autonomous switching, as well as the corresponding problems of the hybrid linear-quadratic optimal control based on the recently suggested principle of maximum. Interrelations between the hybrid principle of maximum and the method of dynamic programming for the systems of this class were discussed. The classical formalism was extended, the corresponding Riccati equations were obtained, and discontinuity of the “hybrid” Riccati matrix was proved. The computational aspects of the established theoretical results were considered.     

The conjugate gradient method for optimal control problems

This paper extends the conjugate gradient minimization method of Fletcher and Reeves to optimal control problems. The technique is directly applicable only to unconstrained problems; if terminal conditions and inequality constraints are present, the problem must be converted to an unconstrained form; e.g., by penalty functions. Only the gradient trajectory, its norm, and one additional trajectory, the actual direction of search, need be stored. These search directions are generated from past and present values of the objective and its gradient. Successive points are determined by linear minimization down these directions, which are always directions of descent. Thus, the method tends to converge, even from poor approximations to the minimum. Since, near its minimum, a general nonlinear problem can be approximated by one with a linear system and quadratic objective, the rate of convergence is studied by considering this case. Here, the directions of search are conjugate and hence the objective is minimized over an expanding sequence of sets. Also, the distance from the current point to the miminum is reduced at each step. Three examples are presented to compare the method with the method of steepest descent. Convergence of the proposed method is much more rapid in all cases. A comparison with a second variational technique is also given in Example 3.     

Mixed coordination method for long-horizon optimal control problems

We present a new approach to solving long-horizon, discrete-time optimal control problems using the mixed coordination method. The idea is to decompose a long-horizon problem into subproblems along the time axis. The requirement that the initial state of a subproblem equal the terminal state of the preceding subproblem is relaxed by using Lagrange multipliers. The Lagrange multipliers and initial state of each subproblem are then selected as high-level variables. The equivalence of the two-level formulation and the original problem is proved for both convex and non-convex cases. The low-level subproblems are solved in parallel using extended differential dynamic programming (DDP). An efficient way to find the gradient and hessian of a low-level objective function with respect to high-level variables is developed. The high-level problem is solved using the modified Newton method. An effective procedure is developed to select initial values of multipliers based on the initial trajectory. The method can convexify the high-level problem while maintaining the separability of an originally non-convex problem. The method performs better and faster than one-level DDP for both convex and non-convex test problems.     

Pseudo-operational matrix method for the solution of variable-order fractional partial integro-differential equations

The main purpose of this paper is to utilize the collocation method based on fractional Genocchi functions to approximate the solution of variable-order fractional partial integro-differential equations. In the beginning, the pseudo-operational matrix of integration and derivative has been presented. Then, using these matrices, the proposed equation has been reduced to an algebraic system. Error estimate for the presented technique is discussed and has been implemented the error algorithm on an example. At last, several examples have been illustrated to justify the accuracy and efficiency of the method.

     

Degenerate problems of optimal control for discrete-continuous (hybrid) systems

The notion of the degenerate problem of optimal control for the discrete-continuous systems was formulated. The main approaches to the problems of this class that were developed for the uniform continuous and discrete systems such as the transformations to the derivative systems and the method of multiple maxima, a special technique to define the Krotov functions under the like sufficient conditions, were extended to the discrete-continuous systems. The fields of possible efficient applications were indicated, and an example was presented.     

Generation of finite element mesh with variable size over an unbounded 2D domain

With the advance of the finite element, general fluid dynamic and traffic flow problems with arbitrary boundary definition over an unbounded domain are tackled. This paper describes an algorithm for the generation of finite element mesh of variable element size over an unbounded 2D domain by using the advancing front circle packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within an open domain. The sequence of construction of the packing circles is determined by the shortest distance from the fictitious centre in such a way that the generation front is more or less a circular loop with occasional minor concave parts due to element size variation. As soon as a circle is added to the generation front, finite elements are directly generated by properly connecting frontal segments with the centre of the new circle. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of circle packing, this reduces a lot of geometrical checks for intersection with frontal segments, and a linear time complexity for mesh generation can be achieved. In case the boundary of the domain is needed, simply generate an unbounded mesh to cover the entire object. As the element adjacency relationship of the mesh has already been established in the circle packing process, insertion of boundary segments by neighbour tracing is fast and robust. Details of such a boundary recovery procedure are described, and practical meshing problems are given to demonstrate how physical objects are meshed by the unbounded meshing scheme followed by the insertion of domain boundaries.     

A nonsmooth Newton’s method for control-state constrained optimal control problems

We investigate optimal control problems subject to mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer–Burmeister function the minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate Banach spaces. This nonlinear and nonsmooth equation is solved by a nonsmooth Newton’s method. We will show the local quadratic convergence under certain regularity conditions and suggest a globalization strategy based on the minimization of the squared residual norm. A numerical example for the Rayleigh problem concludes the article.     

A least-squares/penalty method for distributed optimal control problems for Stokes equations

The purpose of this paper is to construct an unconstrained optimal control problem by using a least-squares approach for the constrained distributed optimal control problem associated with incompressible Stokes equations. The constrained equations are reformulated to the equivalent first-order system by introducing vorticity, and then the least-squares functional corresponding to the system is enforced via a penalty term to the objective functional. The existence of a solution of the unconstrained optimal control problem is proved, and the convergence of this solution to that of unpenalized one is demonstrated as the penalty parameter tends to zero. Finite element approximations with error estimates are studied, and the relevant computational experiments are presented.