Elena V. Khoroshilova

Extragradient-type method for optimal control problem with linear constraints and convex objective function

The paper presents a method for solving optimal control problem with free right end and linear differential equations constraints. The proposed iterative process of extragradient-type is formulated in the functional subspace of piecewise continuous controls of L2. The convergence of the method is proved.

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.

Saddle point approach to solving problem of optimal control with fixed ends

In a Hilbert space, the problem of terminal control with linear dynamics and fixed ends of the trajectory is considered. The integral objective functional has a quadratic form. In contrast to the traditional approach, the problem of terminal control is interpreted not as an optimization problem, but as a saddle-point problem. The solution to this problem is a saddle point of the Lagrange function with components in the form of controls, phase and conjugate trajectories. A saddle-point method is proposed, the convergence of the method in all components of the solution is proved.

On Methods of Terminal Control with Boundary-Value Problems: Lagrange Approach

A dynamic model of terminal control with boundary value problems in the form of convex programming is considered. The solutions to these finite-dimensional problems define implicitly initial and terminal conditions at the ends of time interval at which the controlled dynamics develops. The model describes a real situation when an object needs to be transferred from one state to another. Based on the Lagrange formalism, the model is considered as a saddle-point controlled dynamical problem formulated in a Hilbert space. Iterative saddle-point method has been proposed for solving it. We prove the convergence of the method to saddle-point solution in all its components: weak convergence—in controls, strong convergence—in phase and conjugate trajectories, and terminal variables.

Regularized extragradient method for finding a saddle point in an optimal control problem

We propose a regularized variant of the extragradient method of saddle point search for a convex-concave functional defined on solutions of control systems of linear ordinary differential equations. We assume that the input data of the problem are given inaccurately. Since the problem under consideration is, generally speaking, unstable under a disturbance in the input data, we propose a regularized variant of the extragradient method, investigate its convergence, and construct a regularizing operator. The regularization parameters of the method agree asymptotically with the disturbance level of the input data.

Sensitivity function: Properties and applications

The sensitivity function induced by a convex programming problem is examined. Its monotonicity, subdifferentiability, and closure properties are analyzed. A relation to the Pareto optimal solution set of the multicriteria convex optimization problem is established. The role of the sensitivity function in systems describing optimization problems is clarified. It is shown that the solution of these systems can often be reduced to the minimization of the sensitivity function on a convex set. Numerical methods for solving such problems are proposed, and their convergence is proved.

An extragradient method for finding the saddle point in an optimal control problem

We propose an extragradient method for finding the saddle point of a convex-concave functional defined on solutions of controlled systems of linear ordinary differential equations. We prove the convergence of the method.

Minimizing a sensitivity function as boundary-value problem in terminal control

A problem of terminal control with linear dynamics on a finite time interval is considered. The right-hand end of trajectory is defined implicitly as a solution of boundary-value problem. The problem is reduced to finding a saddle point of the Lagrange function, and linear dynamics is regarded as an equality-type constraint. Dual extraproximal iterative method for solving the problem was proposed, and convergence of the method to problem solution in all components was proved.

Controlled dynamic model with boundary-value problem of minimizing a sensitivity function

We formulate a controlled dynamic model with a boundary-value problem of minimizing a sensitivity function under constraints. Solution of boundary-value problem implicitly defines a terminal condition for the dynamic model. In the model, a unique trajectory corresponds to each control taken from a bounded set. The problem is to select the control such that the corresponding trajectory takes an object from an arbitrary initial state to the terminal state. In this paper, the dynamic model is treated as a problem of stabilization, and the terminal state of the object is interpreted as a state of equilibrium. If under the influence of external disturbances the object loses equilibrium then this object is returned back by selecting the appropriate control. A saddle-point method for solving problem is proposed. We prove its convergence to solution of the problem in all the variables.