Continuous-time Optimal Control Problem Research Articles

On the Gradient Projection Method for Optimal Control Problems with Nonnegative $\mathcal{L}^2 $ Inputs

Local convergence and active constraint identification theorems are proved for gradient-projection iterates in the cone of nonnegative ${\cal L}^2$ functions on $[0,1]$. The theorems are based on recently established infinite-dimensional extensions of the Kuhn-Tucker sufficient conditions and are directly applicable to a large class of continuous-time optimal control problems with smooth nonconvex nonquadratic objective fractions and Hamiltonians that are quadratic in the control input $u$.

Computation of the digital lqg regulator and tracker for time-varying systems

Digital optimal control problems, i.e. problems where a continuous-time system is controlled by a digital computer, are very often approximated by either discrete-time or continuous-time optimal control problems. A digital controller based on one of these approximations requires a small sampling time and constitutes only an approximate solution. The digital LQG regulator and tracker constitute solutions to real digital control problems which involve sampled-data, piecewise constant controls and integral criteria. Until now only the numerical computation of the digital LQG regulator in the case of time-invariant system and criterion matrices has been considered in the literature. The control of non-linear stochastic systems about state trajectories is very often performed by an LQG regulator based on the linearized dynamics about the trajectory, which constitute a time-varying system. We present a numerical procedure to compute the digital LQG regulator and tracker in the case where the system and criterion matrices are time-varying. Finally we present a numerical example.

Dynamic user optimal traffic assignment on congested multidestination networks

An equivalent continuous time optimal control problem is formulated to predict the temporal evolution of traffic flow pattern on a congested multiple origin-destination network, corresponding to a dynamic generalization of Wardropian user equilibrium. Optimality conditions are derived using the Pontryagin minimum principle and given economic interpretations, which are generalizations of similar results previously reported for single-destination networks. Analyses of sufficient conditions for optimality and of singular controls are also given. Under the steady-state assumptions, the model is shown to be a proper dynamic extension of Beckmann's mathematical programming problem for a static user equilibrium traffic assignment.

Time-axis decomposition of large-scale optimal control problems

Continuous-time optimal control problems can rarely be solved directly but have to be approximated with discrete analogues. Shorter time steps lead to more accurate approximations, but result in formulations that are often too big for computer memory. This paper presents a technique for decomposing the problem along the time axis and iterating toward a solution in a leader-follower framework.

Dynamic Network Traffic Assignment Considered as a Continuous Time Optimal Control Problem

Two continuous time formulations of the dynamic traffic assignment problem are considered, one that corresponds to system optimization and the other to a version of user optimization on a single mode network using optimal control theory. Pontryagin's necessary conditions are analyzed and given economic interpretations that correspond to intuitive notions regarding dynamic system optimized and dynamic user optimized traffic flow patterns. Notably, we offer the first dynamic generalization of Beckmann's equivalent optimization problem for static user optimized traffic assignment in the form of an optimal control problem. The analysis further establishes that a constraint qualification and convexity requirements for the Hamiltonian, which together ensure that the necessary conditions are also sufficient, are satisfied under commonly encountered regularity conditions.

ON MINIMUM TIME OPTIMAL CONTROL OF BATCH CRYSTALLIZATION OF SUGAR

The problem of controlling the syrup feed rate and evaporation rate in a batch sugar crystallization process is solved as a continuous time optimal control problem. Constraints on nucleation boundary, flow viscosity, and pan-capacity are also incorporated. As opposed to transformation techniques by addition of slack variables used in the literature, these continuous state constraints are considered directly via an appropriate constraint transcription. This short communication reports on some new computed results based on a model by Ress [1]. Even though the present approach is more direct and simpler, the results obtained are consistent with that of a discrete model

Diagonally Modified Conditional Gradient Methods for Input Constrained Optimal Control Problems

Many iterative methods for constrained minimization problems conform to the general scheme, $u_{i + 1} \in A_i (\Omega ,F,u_i )$, $u_1 \in \Omega $ where $\Omega $ is a set of feasible vectors in a Banach space X, F is a payoff function whose minimum is sought in $\Omega $ and $A_i $ is a map with range in the set of subsets of $\Omega $. If $\{ (\Omega _i ,F_i )\} $ is a sequence of related problems that approximate $\{ (\Omega ,F)\} $ in some sense, with $\Omega _i \subset \Omega _{i + 1} \subset \Omega $, then the corresponding diagonal modification of the original algorithm generates iterates via the recursion, $u_{i + 1} \in A_i (\Omega _i ,F_i ,u_i )$, $u_1 \in \Omega _1 $. If $(\Omega _i ,F_i )$ is properly selected, the diagonal modification can compute approximate solutions for $(\Omega ,F)$ efficiently in circumstances where the original algorithm is difficult or impossible to implement for $(\Omega ,F)$. In particular, this happens for certain gradient-related descent methods and increasingly refined finite-dimensional approximations to infinite-dimensional optimal control problems. Convergence rate estimates are obtained for a diagonally modified conditional gradient method, and this algorithm is applied to bounded input continuous-time optimal control problems.

New algorithms for unconstrained nonlinear optimal control problems

A family of new first-order algorithms for solving continuous time optimal control problems is presented. The algorithms make use of the Riccati matrix differential equation and are capable of solving the linear quadratic problem in one step. The paper includes an analysis of the convergence of the proposed algorithms in the space of relaxed controls, as well as the proof of the reduction of the cost functional at each iteration and numerical examples.

A review of direct sufficient conditions in optimal control theory

The development of conditions sufficient to ensure that a proposed solution to a continuous-time optimal control problem is indeed a solution has proceeded along two general lines. One of these, with origins in field theory, is identified with the works of Weierstrass, Hamilton, Jacobi and Bellman; the other line, which we call the direct theory, is related to the saddle-point theory of mathematical programming, to Pontryagin's theory of optimal processes, and to some extent to the variational calculus. The direct sufficient conditions are the subject of this paper. It is found that four important direct sufficient conditions are of varying scope and that one of these conditions subsumes the others. The differences among the conditions are illustrated by both theorem and example, and speculations are offered on the possibilities for the development of additional direct sufficient conditions. The most powerful results found are mustered in the paper's final theorem.

On difference approximations of optimal control systems

The approximation of continuous-time optimal control problems by sequences of finite-dimensional (discrete-time) optimization problems, arising from difference replacement of derivatives, is investigated. Necessary and sufficient condition for the convergence of discrete (finite difference)approximations with respect to a functional is obtained under minimal assumptions and estimates of convergence rate are found. The results obtained permit the justification of numerical methods for solving optimal control problems on a computer and the investigation of a number of interrelated qualitative aspects of the optimization of continuous and discrete control systems.

On Krasovskii's method for solving the continuous time-optimal control problem

It. is shown that the iterative procedure proposed by Krasovskii for solving the continuous time-optimal control problem possesses an excellent rate of convergence. Furthermore, a new iterative procedure for solving the subproblems that arise from Krasovskii's technique is presented, together with its convergence proof. An illustrative example is given

Continuous-time optimal control theory for cost functionals including discrete state penalty terms

In this short paper, continuous-time optimal control problems are treated, where the cost functional includes additional discrete state penalty terms K_{j}(x(t_{j})) at specified times t_{j}(j = 1,..., N - 1) in the interior of the time interval [t_{0}, t_{N}] of the problem. Necessary conditions for a control to be optimal are given in the form of a minimum principle. The costate is necessarily discontinuous at the times t_{j}(j = 1,..., N - 1) . Consequently, the optimal control is discontinuous at t j , in general. A numerical example is given.

Further Aspects of Turnpike Theory in Continuous Time With Applications

In this paper, the Turnpike theorem for continuous time optimal control problems is extended. This extension is due to a relaxation of the rather strong assumptions used in previous analysis. The Turnpike results are then applied to an economic model of optimal growth in the presence of several replenishable resources and to the Kolgomorov’s prey-predator ecosystem.