Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l₂ space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.     

Asymptotic solution of a singularly perturbed optimal control problem related to the reconstruction of a defective curve

The introduction of “cheap” controls for minimizing the simplest energy functional in an optimal control problem related to the reconstruction of a defective curve necessitates solving a singularly perturbed variational problem with fixed time and fixed ends. The construction of a uniform zero asymptotic approximation to the optimal control in the latter problem permits one to conclude that the optimal trajectories in the original optimal control problem combine a uniform motion in the interior of the time interval with rapid transition layers at the boundaries of the control interval.     

Solution of an optimal control problem with phase constraints by the double variation method

An optimal control problem with an integral quality index specified in a finite time interval is formulated for a model of economic growth that leads to emission of greenhouse gases. The controlled system is linear with respect to control. The problem contains phase constraints that abandon emission of greenhouse gases above some predefined time-dependent limit. As is known, optimal control problems with phase constraints fall beyond the sphere of efficient application of the Pontryagin maximum principle because, for such problems, this principle is formulated in a complicated form difficult for analytic treatment in particular situations. In this study, the analytic structure of the optimal control and phase trajectories is constructed using the double variation method.     

Accumulation of switchings in distributed parameter problems

In recent times, optimal control theory for distributed parameter systems has been actively studied; among them, an important place is occupied by the class of systems describing oscillation processes. This work studies linear control distributed parameter systems of hyperbolic type. The minimization problem of a quadratic functional on the trajectories of the system is considered. By using the Fourier method, the problem is reduced to studying optimal solutions for a countable control system of ordinary differential equations. For Galerkin’s approximations of this system, it is proved that the optimal control is a chattering control, i.e., it has infinitely many switchings on a finite interval of time. The construction of the optimal synthesis uses the results of the theory of singular regimes and regimes with with more and more frequent switchings. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

Optimal synthesis in a control problem with Lipschitz input data

We propose a numerical algorithm for constructing an optimal synthesis in the control problem for a nonlinear system on a fixed time interval. We estimate the difference between the values of the cost functional on optimal trajectories and on the trajectories constructed according to this algorithm. The operation of the algorithm is illustrated by solving model examples on the plane.     

Optimal control of the viscous weakly dispersive Degasperis-Procesi equation

In this paper, we study the optimal control problem for the viscous weakly dispersive Degasperis-Procesi equation. We deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Then, according to optimal control theories and distributed parameter system control theories, the optimal control of the viscous weakly dispersive Degasperis-Procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive Degasperis-Procesi equation is proved.     

Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

     

Optimal observation of controlled elastic vibrations of a beam in the presence of measurement errors

The problem of constructing of an optimal operation for restoring the state of controlled elastic vibrations of a beam in the presence of measurement errors is investigated. By the method of separation of variables, the problem is reduced to an observation problem with an actual output signal for an infinite system of ordinary differential equations. For each harmonic, a universal optimal operation that restores the deflection of the beam from equilibrium and the velocities of all points of the beam is constructed by amplifying the ideal part of the signal produced by the system.     

Solutions for a class of optimal control problems with time delay,part 1

The optimal control of a system whose state is governed by a nonlinear autonomous Volterra integrodifferential equation with unbounded time interval is considered. Specifically, it is assumed that the delay occurs only in the state variable. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson (Ref. 1) to a situation where the trajectories are not necessary bounded. Also, we study the structure of approximate solutions for the problem on a finite interval.The author thanks A. Leizarowitz for fruitful discussions.     

Determination/estimation of an optimal replacement interval under minimal repair

In this paper the problem of estimation of an optimal replacement interval for a system which is minimally repaired at failures is studied. The problem is investigated both under a parametric and a nonparametric form of the failure intensity of the system. It is assumed that observational data from n systems are available. Some asymptotic results are shown. A graphical procedure for determining/estimating an optimal replacement interval is presented. The procedure is particularly valuable for sensitivity analyses, for example with respect to the costs involved.     

Methods of constructing optimal stabilizers

The problem of transforming a linear dynamical system in the neighbourhood of a state of equilibrium [1,2] is solved using the special problem of the damping of the system by controls of minimum intensity after a finite time interval. The possibility of using other problems of optimal control is discussed. The main attention is devoted to constructing algorithms of the operation of a device (a stabilizer) which is able, in real time, to generate a stabilizing control circulating in the closed optimal system when unknown perturbations operate constantly [3, 4]. The proposed method is based on the constructive theory of optimal control [5, 6]. Another form of this theory for solving the problem of stabilization is presented in [7](see also [8]).     

Optimization of periodic systems

The problem of designing a regulator, optimal by a quadratic performance criterion, on an infinite time interval is examined for a linear periodic system. It is assumed that the control plant's motion is described by a system of linear periodic finite-difference equations. Controllable plants whose motion is described by differential and by finite-difference equations on different parts of the period are analyzed as well. The optimal regulator design problem is reduced to the determination of a periodic solution of an appropriate Riccati equation. An algorithm for constructing such a solution is derived. It is noted that this result can be used in periodic optimization problems /1/ and in the design of a stabilization system for a pacing apparatus.     

An adaptive parameter approach for an approximate solution of optimal control problems

In this paper, we consider a class of nonlinear optimal control problems (Bolza-problems) with constraints of the control vector, initial and boundary conditions of the state vectors. The time interval is fixed. Our approach to parametrize both the state functions and the control functions is described by general piecewise polynomials with unknown coefficients (parameters), where a fixed partition of the time interval is used. Here each of these functions in a suitable way individually will be approximated by such polynomials. The optimal control problem thus is reduced to a mathematical programming problem for these parameters. The existence of an optimal solution is assumed. Convergence properties of this method are not considered in this paper.     

Classical characteristics of the bellman equation in constructions of grid optimal synthesis

We consider optimal control problems with fixed final time and terminal-integral cost functional, and address the question of constructing a grid optimal synthesis (a universal feedback) on the basis of classical characteristics of the Bellman equation. To construct an optimal synthesis, we propose a numerical algorithm that relies on the necessary optimality conditions (the Pontryagin maximum principle) and sufficient conditions in the Hamiltonian form. We obtain estimates for the efficiency of the numerical method. The method is illustrated by an example of the numerical solution of a nonlinear optimal control problem.     

A complete asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with geometric constraints

We consider an optimal control problem for solutions of a boundary value problem on an interval for a second-order ordinary differential equation with a small parameter at the second derivative. The control is scalar and is subject to geometric constraints. Expansions of a solution to this problem up to any power of the small parameter are constructed and justified.     

Existence Theorem in the Optimal Control Problem on an Infinite Time Interval

We consider the optimal control problem on an infinite time interval. The system is linear in the control, the functional is convex in the control, and the control set is convex and compact. We propose a new condition on the behavior of the functional at infinity, which is weaker than the previously known conditions, and prove the existence theorem for the solution under this condition. We consider several special cases and propose a general abstract scheme.     

Optimal control with connected initial and terminal conditions

An optimal control problem with linear dynamics is considered on a fixed time interval. The ends of the interval correspond to terminal spaces, and a finite-dimensional optimization problem is formulated on the Cartesian product of these spaces. Two components of the solution of this problem define the initial and terminal conditions for the controlled dynamics. The dynamics in the optimal control problem is treated as an equality constraint. The controls are assumed to be bounded in the norm of L₂. A saddle-point method is proposed to solve the problem. The method is based on finding saddle points of the Lagrangian. The weak convergence of the method in controls and its strong convergence in state trajectories, dual trajectories, and terminal variables are proved.     

Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.     

A class of optimal state-delay control problems

We consider a general nonlinear time-delay system with state-delays as control variables. The problem of determining optimal values for the state-delays to minimize overall system cost is a non-standard optimal control problem–called an optimal state-delay control problem–that cannot be solved using existing optimal control techniques. We show that this optimal control problem can be formulated as a nonlinear programming problem in which the cost function is an implicit function of the decision variables. We then develop an efficient numerical method for determining the cost function’s gradient. This method, which involves integrating an auxiliary impulsive system backwards in time, can be combined with any standard gradient-based optimization method to solve the optimal state-delay control problem effectively. We conclude the paper by discussing applications of our approach to parameter identification and delayed feedback control.