Elena V. Goncharova

Optimization of Measure-Driven Hybrid Systems

We study an optimal control problem for a state-dependent impulse system described by a measure differential equation. Axa0specific time reparameterization technique is developed to reduce the impulsive control problem to the one with bounded controls. Necessary conditions of optimality are obtained by interpreting the Maximum Principle in the reduced problem. An impulsive control improvement scheme is outlined. The results of numeric simulation are presented.

Asymptotic behavior of attainable sets of a linear impulse control system

Abstract We study the long term asymptotic behavior of attainable sets and their shapes of linear time-invariant impulse control systems. We give an exhaustive description of attractors arising and the related dynamics. The results are compared with [4], [3].

Measure-controlled dynamic systems: Polyhedral approximation of their reachable set boundary

An algorithm for polyhedral approximation of the reachable set of impulsive dynamic control systems is designed. The boundary points of the reachable set are determined by recursively generating and solving a family of auxiliary optimal impulsive control problems with state-linear objective functional. The impulsive control problem is solved with an algorithm that implicitly reduces the problem an ordinary optimal control problem. The reduced problem thus obtained is solved with an algorithm based on local approximations of the reachable set.

Small-Time Reachable Sets of Linear Systems with Integral Control Constraints: Birth of the Shape of a Reachable Set

The paper is concerned with small-time reachable sets of a linear dynamical system under integral constraints on control. The main result is the existence of a limit shape of the reachable sets as time tends to zero. A precise estimate for the rate of convergence is given.

Optimal impulsive control problem with state and mixed constraints: The case of vector-valued measure

The paper is concerned with a nonlinear optimal impulsive control problem with trajectories of bounded variation. Vector Lebesgue-Stieltjes measures play the part of controls. Studied were state and mixed constraints in the conventional and fast times, as well as joint conditions for trajectory and impulsive control. A method of reduction to the classical problem of optimal control was developed relying on the discontinuous time reparameterization. The original and reduced problems were established to be equivalent.

Limit Behavior of Reachable Sets of Linear Time-Invariant Systems with Integral Bounds on Control

In this paper, a linear dynamic system is considered under Lp-constraint on control. We establish the existence of the limit shape of reachable sets as time goes to infinity. Asymptotic formulas are obtained for reachable sets and their shapes. The results throw a bridge between the cases of geometric bounds on control and constraints on the total impulse of control, and create a unified picture of the structure of the limit shapes of reachable sets.

Asymptotics for Shapes of Singularly Perturbed Reachable Sets

We study shapes of reachable sets of singularly perturbed linear control systems. The fast component of a phase vector is assumed to be governed by a hyperbolic linear system. We show that the shapes of reachable sets have a limit as the parameter of singular perturbation tends to zero. We also find a sharp estimate for the rate of convergence. A precise asymptotics for the support function of the normalized reachable sets is presented.

Gradient refinement methods for optimal impulse control problems

An optimal control problem is considered for a system described by a differential equation with measures; a certain constraint is imposed on the total variation of controlmeasure. Involving the method of discontinuous time reparameterization, an interpretation is performed for the procedures of weak control variation in an auxiliary reduced problem, and new refinement methods are developed for impulsive processes. An example is provided.

Asymptotics for singularly perturbed reachable sets

We study, in the spirit of [1], reachable sets for singularly perturbed linear control systems The fast component of the phase vector is assumed to be governed by a strictly stable linear system It is shown in loc.cit that the reachable sets converge as the small parameter e tends to 0, and the rate of convergence is O(eα), where 0<α<1 is arbitrary In fact, the said rate of convergence is elog1/e Under an extra smoothness assumption we find the coefficient of elog1/e in the asymptotics of the support function of the reachable set.

Improvement of Discrete Control Processes in Problems with Mixed Constraints

The discrete problem of optimal control with mixed constraints was considered from the standpoint of numerical solution. Computational methods are usually based on solving a series of improvement problems. The nonlinear discrete problem and that of nonlinear functional improvement for the variational system were compared. A procedure for constructing an admissible control process improving the original problem from the solution of the reduced problem was described.