Elena Litsyn

Control of Lyapunov Exponents via Hybrid Feedback Algorithms

Abstract For a control linear system in the plane, which has two complex eigenvalues in the absence of the control, asymptotic convergence / divergence of solutions at an arbitrary rate in the case of incomplete observations is proved. It is also shown that given an arbitrary open interval lying to the left of the real part of the systems eigenvalues there always exists a hybrid feedback control which puts the upper and lower Lyapunov exponents of the control system into this interval.

Efficient criteria for the stabilization of planar linear systems by hybrid feedback controls

We suggest some criteria for the stabilization of planar linear systems via linear hybrid feedback controls. The results are formulated in terms of the input matrices. For instance, this enables us to work out an algorithm which is directly suitable for a computer realization. At the same time, this algorithm helps to check easily if a given linear 2×2 system can be stabilized (a) by a linear ordinary feedback control or (b) by a linear hybrid feedback control.

On a damping problem for functional differential equations

We consider a damping problem for the equation i.e. the problem of moving the system to an equilibrium beginning at a time moment T. We find a control u satisfying by reducing the damping problem to a boundary value problem for a functional differential equation.

Classification of linear planar systems with hybrid feedback control

Continuing the authors’ studies of hybrid dynamical systems, i.e. differential equations governed by finite automata, an efficient and complete classification of control linear systems in the plane is offered. The set of all such systems is divided into equivalence classes which are explicitly characterized by some quantitative invariants. The canonical representatives in each class are determined. It is shown how to use this classification to find out whether a given system is stabilizable or not.

On Positivity of Solutions of Delayed Differential Equation with State Dependent Impulses

We consider the delay differential equation x(t) + p(t)x(t - ?(t)) = f(t), t?[0,?) x(?) = ?(?), ?<0 with state dependent impulses. We give sufficient conditions for positivity of solutions of the Cauchy and periodic problems as well as conditions for positivity of solutions of the initial problem with a condition on the right end of the interval [0, ?]. We also formulate sufficient conditions for nonoscillation of solutions of the homogeneous equation (f = 0, ? = 0) on the halfline.

Controlling linear functional differential equation

Necessary conditions for the optimal control of a linear system of neutral type functional differential equations are obtained. We show that a success in reducing the initial problem to the problem of solvability of a certain boundary value problem is, in principle, a question of ones ability to construct adjoint operators to the operators appearing in the initial control problem.