Discrete Optimal Control Problem Research Articles

Augmented Lagrange-SQP Methods with Lipschitz-Continuous Lagrange Multiplier Updates

Augmented Lagrangian-SQP methods using Lipschitz-continuous Lagrange multiplier updates are analyzed. Kantorovich-style convergence results are proved and applied to the discretization of optimal control problems. The existence of stationary points for the discretized problems is also discussed.

Second Order Sufficiency Criteria for a Discrete Optimal Control Problem

In this work, we derive second order necessary and sufficient optimality conditions for a discrete optimal control problem with one variable endpoint and the other fixed, and with equality control constraints. In particular, the positivity of the second variation, which is a discrete quadratic functional with appropriate boundary conditions, is characterized in terms of the nonexistence of intervals conjugate to 0, the existence of a certain conjoined basis of the associated linear Hamiltonian difference system, or the existence of a symmetric solution to the implicit and explicit Riccati matrix equations. Some results require a certain minimal normality assumption, and are derived using the sensitivity analysis technique.

Discrete Optimal Control: Second Order Optimality Conditions

In this paper, we present a survey and refinement of our recent results in the discrete optimal control theory. For a general nonlinear discrete optimal control problem (P) , second order necessary and sufficient optimality conditions are derived via the nonnegativity ( I S 0) and positivity ( I >0) of the discrete quadratic functional I corresponding to its second variation. Thus, we fill the gap in the discrete-time theory by connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations. Necessary conditions for I S 0 are formulated in terms of the positivity of certain partial discrete quadratic functionals, the nonexistence of conjugate intervals, the existence of conjoined bases of the associated linear Hamiltonian system, and the existence of solutions to Riccati matrix equations. Natural strengthening of each of these conditions yields a characterization of the positivity of I and hence, sufficiency criteria for the original problem (P) . Finally, open problems and perspectives are also discussed.

On Characterizations of Proper Efficiency for Nonconvex Multiobjective Optimization

In this paper, nonconvex multiobjective optimization problems are studied. New characterizations of a properly efficient solution in the sense of Geoffrion's are established in terms of the stability of one scalar optimization problem and the existence of an exact penalty function of a scalar constrained program, respectively. One of the characterizations is applied to derive necessary conditions for a properly efficient control-parameter pair of a nonconvex multiobjective discrete optimal control problem with linear constraints.

Discrete optimal control problems and dynamic games with p players

Abstract The discrete optimal control problem with minimal integral-time cost criterion by a trajectory is studied. Some game models with p players for this problem are formulated and an algorithm for finding the optimal solution in the sense of Nash is proposed

A Control Algorithm to Mixed-Quantized Discrete Linear Optimal Control Systems.

Problem to date: Similar in nature to the knapsack problem and the indivisible investment problem, there exists a static optimal problem the variables of which have both discrete and continuous values as the optimum variables: that is, a mixed-integer programming problem as one optimal problem of the mixed programming problems. A relaxed method or a group method etc. has been used for these problems hitherto. Just like the dynamical indivisible investment problem and the resource allotment problem/the disposition of personnel problem through several periods, there are few optimal control methods for the mixed-quantized dynamical optimal control problem which has both discrete and continuous values. The proposed method in this paper: For the mixed-quantized optimal control problem in which the state equation is linear, the control problem is given by the formulation of Halkin’s discrete time optimal control problem. The mixed-quantized discrete maximum principle is given as the algorithm for this control problem. Where, for the maximization of Hamiltonian in each discrete time, the relaxed method—which improved branch and bound method—is used. Effects obtained in this paper: As the applications in this control problem, the above dynamical investment and the allotment problem through the several periods etc. are considered. In this paper, the solution to the discrete time mixed-quantized optimal control problem is given, and the efficiency of this method (mixed-quantized discrete maximum principle)—which is applied infrequently to this field—is shown, along with a numerical example.

A self-stabilizing Pantoja-like indirect algorithm for optimal control

In 1983 Pantoja described a computationally efficient stagewise construction of the Newton direction for the discrete time optimal control problem. Automatic Differentiation can be used to implement Pantoja's algorithm and calculate the Newton direction, without truncation error, and without extensive manual re-writing of targetfunction code to form derivatives. Pantoja's algorithm is direct, in that the independent variables are the control vectors at each timestep. In this paper we formulate an indirect analogue of Pantoja's algorithm, in which the only independent variables are the components of a costate vector corresponding to the initial timestep. This reformulated algorithm gives exactly the Newton step for the initial costate with respect to a terminal transversality condition: at each timestep we solve implicit equations for the current controlsand successor costates. A remarkable feature of the indirect algorithm is that it is straiehtforward to comensate for the effect of non-zero residuals in the implicit costate equations. .The indirect reformulation of Pantoja's algorithm set out in this paper is a suitable basis for verified optimization using interval methods.

Stability analysis of iterative optimal control algorithms modelled as linear unit memory repetitive processes

The theory of unit memory repetitive processes is used to investigate local convergence and stability properties of algorithms for the solution of discrete optimal control problems. In particular, the properties are addressed of a method for finding the correct solution of an optimal control problem where the model used for optimisation is different from reality. Limit profile and stability concepts of unit memory linear repetitive process theory are employed to demonstrate optimality and to obtain necessary and sufficient conditions for convergence. Two main stability theorems are obtained from different approaches and their equivalence is proved. The theoretical results are verified through simulation and numerical analysis, and it is demonstrated that repetitive process theory provides a useful tool for the analysis of iterative algorithms for the solution of dynamic optimal control problems.

Discrete Optimal Control: The Accessory Problem and Necessary Optimality Conditions

In this work we derive the accessory problem for a general nonlinear discrete optimal control problem. We obtain necessary conditions for optimality—nonnegativity of the second variation. As the second variation is a discrete quadratic functional, we proceed by deriving necessary conditions for the nonnegativity of such functionals. A certain controllability (or normality) assumption is required.

Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept

A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations. The Lagrangian formalism yields the first-order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive Galerkin finite element method. The mesh design in the resulting reduced models is controlled by residual-based a posteriori error estimates. These are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error. In this case, the computed state and costate variables can be used as sensitivity factors multiplying the local cell-residuals in the error estimators. This results in a generic and simple algorithm for mesh adaptation within the optimization process. This method is developed and tested for simple boundary control problems in semiconductor models.

On the Newton Interior-Point Method for Nonlinear Programming Problems

Interior-point methods have been developed largely for nonlinear programming problems. In this paper, we generalize the global Newton interior-point method introduced in Ref. 1 and we establish a global convergence theory for it, under the same assumptions as those stated in Ref. 1. The generalized algorithm gives the possibility of choosing different descent directions for a merit function so that difficulties due to small steplength for the perturbed Newton direction can be avoided. The particular choice of the perturbation enables us to interpret the generalized method as an inexact Newton method. Also, we suggest a more general criterion for backtracking, which is useful when the perturbed Newton system is not solved exactly. We include numerical experimentation on discrete optimal control problems.

LQG optimal control of constrained discrete time systems

In this paper, we study the discrete time LQG optimal control problem, with convex quadratic con straints. We show that the Separation Theorem does not hold. However, a generalization of this result which we call the Quasi-Separation Theorem applies. The optimal control is obtained by solving a convex, finite-dimensional optimization problem that is equivalent to solving a sequence of unconstrained LQG control problems.

Necessary optimality conditions for bicriterion discrete optimal control problems

AbstractIn management science and system engineering, problems with two incommensurate objectives are often detected. Bicriterion optimization finds an optimal solution for the problems. In this paper it is shown that bicriterion discrete optimal control problems can be solved by using a parametric optimization technique with relaxed convexity assumptions. Some necessary optimality conditions for discrete optimal control problems subject to a linear state difference equation are derived. It is shown that in this case no adjoint equation is required.

Cheap Newton steps for optimal control problems: automatic differentiation and Pantoja's algorithm

In this paper we discuss Pantoja's construction of the Newton direction for discrete time optimal control problems. We show that automatic differentiation (AD) techniques can be used to calculate the Newton direction accurately, without requiring extensive re-writing of user code, and at a surprisingly low computational cost: for an N-step problem with p control variables and q state variables at each step, the worst case cost is 6(p + q + 1) times the computational cost of a single target function evaluation, independent of N, together with at most p 3/3 + p 2(q + 1) + 2p(q + 1)2 + (q + l)3, i.e. less than (p + q + l)3, floating point multiply-and-add operations per time step. These costs may be considerably reduced if there is significant structural sparsity in the problem dynamics. The systematic use of checkpointing roughly doubles the operation counts, but reduces the total space cost to the order of 4pN floating point stores. A naive approach to finding the Newton step would require the solution of an Np × Np system of equations together with a number of function evaluations proportional to Np, so this approach to Pantoja's construction is extremely attractive, especially if q is very small relative to N. Straightforward modifications of the AD algorithms proposed here can be used to implement other discrete time optimal control solution techniques, such as differential dynamic programming (DDP), which use state-control feedback. The same techniques also can be used to determine with certainty, at the cost of a single Newton direction calculation, whether or not the Hessian of the target function is sufficiently positive definite at a point of interest. This allows computationally cheap post-hoc verification that a second-order minimum has been reached to a given accuracy, regardless of what method has been used to obtain it.

A Control Technique for a Quantized Discrete Optimal Control System with Terminal Free Conditions. 2nd Report. In the case of Systems Described by Non-linear State Equations.

Control systems whose controls can only take given discrete-level values are called quantized control systems. Optimization of such systems has conventionally been performed by mathematical programming methods. In this report, the authors formulate a quantized discrete non-linear optimal control problem with terminal free conditions using Halkin's formulation and derive the quantized discrete maximum principle for discrete-level valued solutions, which is a Pontryagin-type necessary control condition. An algorithm based on this maximum principle, that is, a two points boundary condition-typed algorithm is presented, along with two numerical examples.

Suboptimal local feedback control for a class of constrained discrete time nonlinear control problems

In this paper, we consider a class of constrained discrete time optimal control problems involving general nonlinear dynamics with fixed terminal time. A method to solve the feedback control problem for a class of unconstrained continuous time nonlinear systems has been proposed previously. In that work, the solution is based on synthesizing an approximate suboptimal feedback controller locally in the neighbourhood of a certain nominal optimal trajectory. This paper expands on the same theme by considering problems involving discrete time systems. Taking advantage of the nature of discrete time systems, a further reduction on the computational effort of synthesising the feedback controller is made possible. Also, this paper extends the applicability of the method to constrained systems. For illustration, a numerical example is solved using the proposed method.

Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems

In this paper, a family of trust-region interior-point sequential quadratic programming (SQP) algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise, e.g., from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed, for a different class of problems, by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418--445] and they exploit trust-region techniques for equality-constrained optimization. Thus, they allow the computation of the steps using a variety of methods, including many iterative techniques. Global convergence of these algorithms to a first-order Karush--Kuhn--Tucker (KKT) limit point is proved under very mild conditions on the trial steps. Under reasonable, but more stringent, conditions on the quadratic model and on the trial steps, the sequence of iterates generated by the algorithms is shown to have a limit point satisfying the second-order necessary KKT conditions. The local rate of convergence to a nondegenerate strict local minimizer is q-quadratic. The results given here include, as special cases, current results for only equality constraints and for only simple bounds. Numerical results for the solution of an optimal control problem governed by a nonlinear heat equation are reported.

On interior-point Newton algorithms for discretized optimal control problems with state constraints*

In this paper we consider a class of nonlinear programming problems that arise from the discretization of optimal control problems with bounds on both the state and the control variables. For this class of problems, we analyze constraint qualifications and optimality conditions in detail. We derive an affine-scaling and two primal-dual interior-point Newton algorithms by applying, in an interior-point way, Newton's method to equivalent formi of the first-order optimality conditions. Under appropriate assumptions, the interior-point Newton algorithms are shown to be locally well-defined with q-quadratic rate of locaj convergence. By using the structure of the problem, the linear algebra of these algorithms can be reduced to the null space of the Jacobian of the equality constraints. The similarities between the three algorithms are pointed out, and their corresponding versions for th^ general nonlinear programming problem are discussed

Application of a novel optimal control algorithm to a benchmark fed - batch fermentation process

A novel algorithm for solving nonlinear discrete time optimal control problems with model-reality differences is presented. The technique uses Dynamic Integrated System Optimisation and Parameter Estimation (DISOPE) which has been designed to achieve the correct optimal solution in spite of deficiencies in the mathematical model employed in the optimisation procedure. A method based on Broyden's ideas is used for approximating some derivative trajectories required. Ways for handling con straints on both manipulated and state variables are described. Further, a method for coping with batch-to- batch dynamic variations in the process, which are common in practice, is introduced. It is shown that the iterative procedure associated with the algorithm naturally suits applications to batch processes. The algorithm is success fully applied to a benchmark problem consisting of the input profile optimisation of a fed-batch fermentation process.

Numerical solutions of an optimal control problem governed by a ginzburg-landau model in superconductivity

A numerical method to solve a constrained optimal control problem governed by a generalized Ginzburg-Landau model which describes the phase transitions taking place in the superconducting thin films with variable thickness. The method is based on a finite element method to approximate the state equations and an exterior penalty function algorithm to solve the discrete optimal control problem. The convergence of the discrete optimal solutions to the continuous optimal solutions and the convergence of the exterior penalty function algorithm are proved. The objective of the work is to study efficient numerical methods for exploring the possibilities of controlling the motion of vortices in the superconducting thin films through the external magnetic field.