Exact optimal solution for a class of dual control problems

This paper considers a discrete-time stochastic optimal control problem for which only measurement equation is partially observed with unknown constant parameters taking value in a finite set of stochastic systems. Because of the fact that the cost-to-go function at each stage contains variance and the non-separability of the variance is so complicated that the dynamic programming cannot be successfully applied, the optimal solution has not been found. In this paper, a new approach to the optimal solution is proposed by embedding the original non-separable problem into a separable auxiliary problem. The theoretical condition on which the optimal solution of the original problem can be attained from a set of solutions of the auxiliary problem is established. In addition, the optimality of the interchanging algorithm is proved and the analytical solution of the optimal control is also obtained. The performance of this controller is illustrated with a simple example.     

On a class of optimal control problems with an almost cost-free solution

For a class of linear singular optimal control problems with a nonunique singular arc, the solution of the corresponding nearly singular problem is analyzed and a limit solution based on formal singular perturbations is derived. A rigorous proof of the correctness of the result is given by constructing a convergent power series satisfying the Riccati equation of the nearly singular problem.     

Approximate solution of a class of optimal control problems in domains of arbitrary form

A combination of both fictitious domain and net methods is used for solution of optimal-control problems for elliptic systems. The proposed difference scheme has an order of accuracy ofO(h ^1/2) in the net norm L₂. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 138–146, January–February, 2000.     

An efficient NFEM for optimal control problems governed by a bilinear state equation

In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by a bilinear state equation. The state and adjoint state are approximated by the nonconforming $E{Q}_1^{rot}$ element, and the control is approximated by the orthogonal projection through the state and adjoint state. Some superclose and superconvergence properties are obtained by full use of the distinguish characters of this $E{Q}_1^{rot}$ element, such as the interpolation operator equals the Ritz projection, and the consistency error is one order higher than its interpolation error in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.     

Constructive solution of a bilinear optimal control problem for a Schrödinger equation

Often considered in numerical simulations related to the control of quantum systems, the so-called monotonic schemes have not been so far much studied from the functional analysis point of view. Yet, these procedures provide an efficient constructive method for solving a certain class of optimal control problems. This paper aims both at extending the results already available about these algorithms in the finite-dimensional case (i.e., the time-discretized case) and at completing those of the continuous case. This paper starts with some results about the regularity of a functional related to a wide class of models in quantum chemistry. These enable us to extend an inequality due to Łojasiewicz to the infinite-dimensional case. Finally, some inequalities proving the Cauchy character of the monotonic sequence are obtained, followed by an estimation of the rate of convergence.     

Multigrid optimization methods for linear and bilinear elliptic optimal control problems

Summary Multigrid optimization schemes that solve elliptic linear and bilinear optimal control problems are discussed. For the solution of these problems, the multigrid for optimization (MGOPT) method and the collective smoothing multigrid (CSMG) method are developed and compared. It is shown that though these two methods are formally similar, they provide different approaches to computational optimization with partial differential equations.      

System equivalence in a class of nonlinear optimal control problems

In a family of nonlinear optimal control problems, equivalence classes of problems that yield identical optimal closed-loop systems are delineated. The equivalence relations are established as one-to-one correspondences between the solutions of the Hamilton-Jacobi equations associated with the problems. Two such equivalence relations are presented. These allow an arbitrary problem in the family to be reduced to an equivalent basic standard form of problem. Also, a new formal procedure for solving this basic problem is given that allows the determination of coefficients in the series expansion of the optimal closed-loop system. The set of algebraic equations to be solved is considerably simpler than that derived by other methods.     

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.     

Regional optimal control of a bilinear wave equation

We study a regional optimal control problem of a bilinear wave equation evolving on a spatial domain Ω with a distributed controls. We search a distributed control which aims to minimise a given functional cost that contains the gap between a desired state and the reached one. This latter is defined only on a subregion ω of Ω. Therefore, we prove existence and we give characterisation of an optimal control. The obtained results lead to an algorithm that we illustrate by simulations.     

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.     

MGOPT with gradient projection method for solving bilinear elliptic optimal control problems

Multigrid optimization schemes that solve optimal control problems with bilinear elliptic partial differential equations are presented. For the solution of the control-unconstrained and control-constrained problems, finite difference discretization is utilized. To solve the control-unconstrained case, multigrid for optimization (MGOPT) method is considered and for the control-constrained case, MGOPT with gradient projection method is applied to solve the problem. Numerical experiments show the efficiency of these techniques.     

Lossless convexification of a class of optimal control problems with non-convex control constraints

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem, which is referred to as the lossless convexification of the optimal control problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. The solution approach is demonstrated on a number of planetary soft landing optimal control 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.     

Asymptotic series solution of singularly perturbed optimal control problems

An optimal control problem of some nonlinear differential equations containing a small parameter is considered. This problem leads to a two-point boundary value problem (TPBVP) whose differential order can be reduced by neglecting the small parameter. In this paper an explicit method of constructing an asymptotic power series solution of the TPBVP as the small parameter tends to zero is presented. The method allows a separation of slow and fast dynamics in the problem while reducing the differential order of the equations. The determination of asymptotic series terms is computationally simple since one needs to solve only initial or final value problems instead of TPBVP's. The method has computational similarities with quasilinearization and second variation techniques.     

Formulation and multigrid solution of Cauchy-Riemann optimal control problems

The formulation of optimal control problems governed by Cauchy-Riemann equations is presented. A distributed control mechanism through divergence and curl sources is considered with the boundary conditions of mixed type. A Lagrange multiplier framework is introduced to characterize the solution to Cauchy-Riemann optimal control problems as the solution of an optimality system of four first-order partial differential equations and two optimality conditions. To solve the optimality system, staggered grids and multigrid methods are investigated. It results that staggered grids provide a natural collocation of the optimization variables and second-order accurate solutions are obtained. The proposed multigrid scheme is based on a coarsening by a factor of three that results in a nested hierarchy of staggered grids. On these grids a distributed-Gauss-Seidel and gradient-based smoothing scheme is employed. Results of numerical experiments validate the proposed optimal control formulation and demonstrate the effectiveness of the staggered-grids multigrid solution procedure.     

Robust optimal control of bilinear DC-DC converters

This paper addresses the control problem of dc-dc converters. The control law synthesis considered here exploits the potential of LMI-based control approaches, which allow to cope with model uncertainty, disturbances and bilinearities to synthesize simple state-feedback controllers with a priori guarantee of stability in a large domain of initial and operating conditions. The aim of the paper is to contribute with a robust control framework to deal with the common requirements of regulated dc-dc converters. The correctness of the results has been verified both with numerical simulations and with experimental measurements from a laboratory prototype.     

The computational solution of optimal control problems with time lag

A gradient computational procedure is developed for discrete-time optimal control problems in which the current cost and rule governing state transitions depend on the entire past history of states and decisions. A condition necessary for optimality is also deduced in two ways and given physical interpretation.     

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.