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.     

A numerical solution scheme for softening problems involving total strain control

Nonlinear structural problems in which material softening is present constitute a severe challenge to solution algorithms. Most of the existing computational codes are based on using an incremental procedure with iterations and suitable constraints. Among the constraints are load control, direct or indirect displacement control, and arc-length control involving a combination of load and displacement parameters. For many problems in which softening and localization occur, these algorithms fail at some point in the post-peak regime. In an attempt to remedy this problem, an alternative constraint condition is proposed whereby a combination of the total strain components is prescribed at the most critical point in the body. The critical point is defined to be that point where a suitable measure of strain is a maximum. Numerical solutions for both plane strain and plane stress problems are given to illustrate the ability of the procedure to capture post-peak responses of structures governed by materially nonlinear behavior.     

On numerical algorithm and interactive visualization for optimal control problems

We present methods for the visualization of the numerical solution of optimal control problems. The solution is based on dynamic programming techniques where the corresponding optimal value function is approximated on an adaptively refined grid. This approximation is then used in order to compute approximately optimal solution trajectories. We discuss requirements for the efficient visualization of both the optimal value functions and the optimal trajectories and develop graphic routines that in particular support adaptive, hierarchical grid structures, interactivity and animation. Several implementational aspects using the Graphics Programming Environment ‘GRAPE’ are discussed. Received: 4 December 1997 / Accepted: 5 August 1998     

General formulation for the numerical solution of optimal control problems

A new improved computational method for a class of optimal control problems is presented. The state and the costate (adjoint) variables are approximated using a set of basis functions. A method, similar to a variational virtual work approach with weighing coefficients, is used to transform the canonical equations into a set of algebraic equations. The method allows approximating functions that need not satisfy the initial conditions a priori. A Lagrange multiplier technique is used to enforce the terminal conditions. This enlarges the space from which the approximating functions can be chosen. Orthogonal polynomials are used to obtain a set of simultaneous equations with fewer non-zero entries. Such a sparse system results in substantial computational economy. Two examples, a time-invariant system and a time-varying system with quadratic performance index, are solved using three different sets of orthogonal polynomials and the power series to demonstrate the feasibility and efficiency of this method.     

Direct scheme in optimal control problems with fast and slow motions

The nonlinear optimal control problems described by singularly perturbed models are studied in order to obtain the ways of decomposition and to construct the suboptimal control sequences. A new scheme, the so-called direct scheme, applying the Vasil'eva boundary layer functions method in control theory is introduced. It consists in directly expanding the problem's conditions into postulated asymptotic series and then in successively solving lower-dimensional approximate decomposition problems of optimal control. By means of the direct scheme a new property of an asymptotic expansion — the relaxation — is obtained, i.e. the decrease of the performance index with each new control approximation. Illustrating examples are given.     

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.     

A modified explicit numerical scheme for the two-dimensional sine-Gordon equation

Abstract

A fourth-order rational approximant to the matrix-exponential term in a three-time-level recurrence relation is used to transform the two-dimensional sine-Gordon equation into a second-order initial-value problem. The resulting nonlinear system is solved using an appropriate predictor–corrector (P-C) scheme in which the predictor is an explicit one of second order. The procedure of the corrector is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the nonlinear method and the predictor–corrector are analysed for local truncation error and stability. The MPC scheme has been tested on line and circular ring solitons known from the literature, and numerical experiments have proved that there is an improvement in accuracy over the standard predictor–corrector implementation.     

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 unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.     

State-constrained optimal control problems governed by coupled nonlinear wave equations with memory

This work considers the optimal control problems governed by coupled nonlinear wave equations with memory, where the state constraint is a type of integral. First, we investigate the existence of the optimal solution for the optimal control problem, and then we deduce the maximum principle for the optimal solution with state constraint. Moreover, we give a concrete example which is covered by the main result.     

A sufficient condition for optimal control problems with time delays

This paper considers a class of systems modeled by ordinary differential equations which contain a delay in both the state and control variables. A sufficient condition for the optimal control of such systems is presented. The condition is a modification of a sufficient condition for optimal control problems without time delays. The application of the result is illustrated by examples.     

A residual-based compact scheme of optimal order for hyperbolic problems

A new fourth-order dissipative scheme on a compact 3 × 3 stencil is presented for solving 2D hyperbolic problems. It belongs to the family of previously developed residual-based compact schemes and can be considered as optimal since it offers the maximum achievable order of accuracy on the 3 × 3-point stencil. The computation of 2D scalar problems demonstrates the excellent accuracy and efficiency properties offered by this new RBC scheme with respect to existing second- and third-order versions.     

A numerical scheme for mixed boundary value problems in elasticity

A numerical scheme, based upon the Kobayashi-Tranter method with certain modifications, is given for axisymmetric punch and crack problems in elasticity. The problems are reduced to solving a system of linear algebraic equations instead of a Fredholm integral equation of the second kind. A standard program thus allows the treatment of a range of different cases.The indentation of a rigid punch on an elastic layer overlying an elastic foundation is formulated in this fashion and numerical results for various cases are presented.     

A modified genetic algorithm for optimal control problems

This paper studies the application of a genetic algorithm to discrete-time optimal control problems. Numerical results obtained here are compared with ones yielded by GAMS, a system for construction and solution of large and complex mathematical programming models. While GAMS appears to work well only for linear quadratic optimal control problems or problems with short horizon, the genetic algorithm applies to more general problems equally well.     

A canonical structure for constrained optimal control problems

We consider a general class of optimal control problems with regional pole and controller structure constraints. Our goal is to show that for a fairly general class of regional pole and controller structure constraints, such constrained optimal control problems can be transformed to a new one with a canonical structure. A three-step transformation procedure is used to achieve our goal, which essentially amounts to repeated augmentations of plant dynamics and repeated reductions of the controller. The transformed problem is one of the standard optimal static output feedback with a decentralized and repeated structure.     

System sensitivity for optimal problems with singular control

Extensions of the differential performance index sensitivity to small parameter variations are presented in this paper. The system considered is assumed to be linear with free end time and initial and final target manifolds.

Results, previously reported for systems with bang.bang type implementation or control law resulting in from a minimum-fuel type performance index implementation. were all subject to the normality condition constraint. In this paper the normality requirements were removed from the system considered, thus extending the results to the singular control case. A universal expression thus presented which relates the variations of the performance index to that of the initial and final manifolds and a term that represents a physical constant, related to the Hamiltonian and which is independent of the way by which the optimal control was implemented.     

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

This paper presents a new approach for the efficient solution of singular optimal control problems (SOCPs). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of solution. At first, the SOCP is converted into a binary optimal control problem. Then, by utilising the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the optimal control and to compute the approximating solution. The main advantages of the present method are that: (1) without a priori information, the structure of optimal control is detected; (2) it produces good results even using a small number of collocation points; (3) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.     

A feasible directions algorithm for optimal control problems with control and terminal inequality constraints

Control constraints, because they are nondifferentiable in the space of control functions, are difficult to cope with in optimization algorithms not of the penalty function type. In feasible directions type algorithms a search direction is obtained by solving a subproblem which has a fixed constraint setSchosen a priori to make this subproblem simple to solve. In this paper control constraints are dealt with by using them to define, at the current controlu, a constraint setS(u)for the search direction subproblem. This enables control constraints to be easily handled.