Sequential synthesis of time-optimal control for linear systems with disturbances

A method of sequential synthesis of time-optimal control for a linear system with unknown disturbances is considered. A system of linear algebraic equations is obtained which relates the increments of phase coordinates to the increments of initial conditions of a normalized adjoint system and to the increment of control completion time. Evaluations consist in solving repeatedly a system of linear algebraic equations and integrating a matrix differential equation on the displacement intervals of control switching times and on the displacement interval of final control time. A procedure of correcting the switching times and the completion time in moving along the phase trajectory of a controllable object is examined. Simple and constructive conditions are specified for a discontinuous mode to occur, for a representation point to move along the switching manifolds, and for the optimal control structure to transform in moving along the phase trajectory of a system with uncontrollable disturbance. A computational algorithm is presented. It is proved that a sequence of controls converges locally at a quadratic rate and globally to a time-optimal control.     

Stabilization of linear dynamical systems by optimal controls of linear-quadratic problems

A method of obtaining a bounded feedback which stabilizes a linear dynamical system is described. The attainment of a positional solution of a special auxilliary linear-quadratic problem of optimal control using an optimal regulator is the basis of the method. The results are illustrated for the well-known problem of the stabilization of a pendulum in the upper unstable positions by means of inertial controls.     

On the maximum principle for time-optimal controls for a class of distributed-boundary control problems

In this note, we present the necessary conditions of optimality for time-optimal controls for a class of distributed-boundary control problems in general Banach spaces using the semigroup theory. Theorem 3.1 is based on a recent general maximum principle due to Barbu (Ref. 1), which was proved for strictly convex reflexive Banach spaces. Theorem 3.2 generalizes this result (for time-optimal control problems) by lifting the assumption.This work was supported by the National Science and Engineering Council of Canada under Grant No. 7109.     

Linearized maximum principle for neutral-type,variable-structure optimal problems with delays in controls

The authors state and study an optimal problem for variable-structure systems described by neutral-type quasilinear differential equations with discontinuous initial condition and incommensurable delays. The variation of the system structure means that in the process of motion, at a certain instant of time not known in advance, the object considered can pass from one law of motion to another, and, moreover, the initial condition for each of the subsequent states of the system depends on the state of the next to the last. The discontinuity of the initial condition means that at the initial instant of time, the value of the initial function and that of the trajectory do not coincide in general. The necessary optimality conditions are proved in the form of the linearized integral maximum principle for controls and initial functions and in the form of inequalities and equalities for the initial and final instants of structure change. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 42, Optimal Control, 2006.     

Asymptotics of the optimal time in a singular perturbed linear time-optimal problem

We consider an abstract attainability problem under constraints of asymptotic character and describe a general approach to constructing “nonsequential” attraction sets in the space of generalized elements formalized as finitely additive measures. We also study the existence and structure of an asymptotic formula universal in the range of “asymptotic constraints” and topologies of the space of generalized elements in the case when the space of ordinary solutions is not necessarily compactifiable.     

Variation formulas of solutions and optimal control problems for differential equations with retarded argument

The authors prove a theorem on the continuous dependence of solutions of nonlinear systems of differential equations with variable delay on the perturbations of initial data (initial instant, initial function, and initial value of the trajectory) and the right-hand side in the case where these perturbations are small in the Euclidean and integral topology, respectively. The variation formulas of solutions of a differential equation with discontinuous and continuous initial condition are deduced; as compared with those known earlier, these formulas take into account the variation of the initial instant and the discontinuity and continuity of the initial data. A necessary condition for criticality of mappings defined on a finitely locally convex set is obtained. The quasiconvexity of filters in studying optimal problems with delays in controls is proved. Necessary optimality conditions and existence theorems are proved for optimal problems with variable delays in phase coordinates and controls having a nonfixed initial instant, a discontinuous and a continuous initial condition, and functional and boundary conditions of general form. Necessary optimality conditions are obtained for optimal problems with variable structure and delays. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 25, Optimal Control, 2005.     

Analytical and numerical construction of the optimal outcome function for a class of time-optimal problems

Analytical and numerical algorithms are proposed for constructing the optimal outcome function and its Lebesgue set for the time-optimal control problem with a circular velocity indicatrix. Our approach to the solution of the time-optimal control problem essentially utilizes the specific dynamic properties of the controlled system. The circular vectogram of possible velocities enables us to interpret the cross sections of the controllability set as wavefronts whose source is uniformly distributed on the boundary of the target set. Procedures have been developed for analytical and numerical construction of the evolution of wavefronts based on prior (given the geometry of the target set boundary) identification of their nonsmoothness sets. An essential feature of the construction is the point-to-set distance function. We investigate the differential properties of this function and identify the manifolds on which it loses its smoothness. The proposed wavefront construction algorithms are of independent interest in so far as they enable us to investigate the geometry of the sets and compute their nonconvexity measure. The results are useful not only when studying the evolution of reachability sets of controlled systems, but also for computing the eikonal in geometrical optics and investigating the solutions of the wave equation. __________ Translated from Prikladnaya Matematika i Informatika, No. 27, pp. 65–79, 2007.     

Maximum principle for optimal control problems involving impulse controls with nonsmooth data

This paper is concerned with the stochastic maximum principle for impulse optimal control problems of forward–backward systems, where the coefficients of the forward part are Lipschitz continuous. The domain of the regular controls is not necessarily convex. We establish a Pontryagins maximum principle for this control problem by applying Ekelands variational principle to a sequence of approximated control problems with smooth coefficients of the initial problems.     

A method of optimal real-time computation of a linear system with retarded control

A new method of solving time-optimal control problems in real time is developed. The method is based on the following: 1) approximating the attainability sets with a family of hyperplanes; 2) subdividing the whole computational process into the computations performed beforehand and those that are carried out while the control takes place; 3) integrating the differential equations only over the displacement intervals of the final time point and the switching time points. The computational cost of the method is evaluated. The peculiarities of calculating the optimal control of a linear system with retarded control in real time are considered. The results of simulation and numerical calculations are presented.     

Construction of asymptotically optimal controls for control and game problems

Recently the connection between control and game problems and Backward Stochastic Differential Equations has been established. This allows us to use an approximation scheme for such equations in order to construct an ɛ-optimal control. Received: 13 November 1995 / Revised version: 11 February 1998     

The Optimal Homotopy Analysis Method for solving linear optimal control problems

In this paper, an Optimal Homotopy Analysis Method (Optimal HAM) is applied to solve the linear optimal control problems (OCPs), which have a quadratic performance index. This approach contains at most two convergence-control parameters which depend on the control system and is computationally rather efficient. A squared residual error for the system is defined, which can be used to find the unknown optimal convergence-control parameters by using Mathematica package BVPh (version 2.0). The results of comparisons among the proposed method, the homotopy perturbation method (HPM), the Adomian decomposition method (ADM), the differential transform method (DTM) and the homotopy analysis method (HAM) provide verification for the validity of the proposed approach. Moreover, numerical results are presented by several examples involving scalar and 2nd-order systems to clarify the efficiency and high accuracy of the proposed approach.     

Some remarks on existence of optimal controls for Mayer problems with singular components

Summary This paper proves an existence theorem for optimal controls for systems governed by ordinary differential equations and a large class of functional differential equations of neutral type. Extensions beyond earlier work are made as a result of employing a new closure theorem originally used by Cesari and Suryanarayana in their study of Pareto optima and which is, in turn, based on the Fatou lemma for vector-valued functions as proved by Schmeidler. The use of these techniques simplifies the standard arguments for existence in the presence of singular components and allows the use of very weak semi-normality conditions. It also permits the consideration of a significantly larger class of hereditary systems than has been treated in the existing literature.     

Regularity properties of optimal controls for problems with time-varying state and control constraints

In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.     

Convergence properties of optimal controls of pseudo-parabolic problems

Approximation properties of pseudo-parabolic optimal controls to the parabolic optimal control are considered for two concrete problems.     

Convergence of optimal controls in some optimization problems

We consider the behavior of bundles of optimal controls when the initial state of the system goes to some given vector. We investigate three types of optimization problems: problems with fixed process length and a free right endpoint; problems with fixed process length and a fixed right endpoint; time-optimal problems. The article is a review of recent results obtained by the author. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 301–314, 2004.     

Function-space quasi-Newton algorithms for optimal control problems with bounded controls and singular arcs

Two existing function-space quasi-Newton algorithms, the Davidon algorithm and the projected gradient algorithm, are modified so that they may handle directly control-variable inequality constraints. A third quasi-Newton-type algorithm, developed by Broyden, is extended to optimal control problems. The Broyden algorithm is further modified so that it may handle directly control-variable inequality constraints. From a computational viewpoint, dyadic operator implementation of quasi-Newton methods is shown to be superior to the integral kernel representation. The quasi-Newton methods, along with the steepest descent method and two conjugate gradient algorithms, are simulated on three relatively simple (yet representative) bounded control problems, two of which possess singular subarcs. Overall, the Broyden algorithm was found to be superior. The most notable result of the simulations was the clear superiority of the Broyden and Davidon algorithms in producing a sharp singular control subarc.This research was supported by the National Science Foundation under Grant Nos. GK-30115 and ENG 74-21618 and by the National Aeronautics and Space Administration under Contract No. NAS 9-12872.     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

Classification of stable time-optimal controls on 2-manifolds

In this paper, we provide a topological classification via graphs of time-optimal flows for generic control systems of the form , x ∈ M, |u| ≤ 1, on two-dimensional orientable compact manifolds, also proving the structural stability of generic optimal flows. More precisely, adding some additional structure to topological graphs, more precisely, rotation systems, and owing to a theorem of Heffter, dating back to the 19th century, we prove that there is a one-to-one correspondence between graphs with rotation systems and couples formed by a system and the 2-D manifold of minimal genus in which the system can be embedded. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.