Asymptotics of the optimal time in a time-optimal control problem with a small parameter

A time-optimal control problem for a singularly perturbed linear autonomous system is considered. The main difference between this case and the case of systems with fast and slow variables studied earlier is that the eigenvalues of the matrix at the fast variables do not satisfy the standard requirement of negativity of the real part. We obtain and justify a complete power asymptotic expansion in the sense of Erdélyi of the optimal time and optimal control in a small parameter at the derivatives in the equations of the system.     

Krylov-Bogolyubov averaging of asymptotically autonomous differential equations

We apply the Krylov and Bogolyubov asymptotic integration procedure to asymptotically autonomous systems. First, we consider linear systems with quasi-periodic coefficient matrix multiplied by a scalar factor vanishing at infinity. Next, we study the asymptotically autonomous Van-der-Pol oscillator.

     

Quadratic systems with a linear reflecting function

We construct a set of nonautonomous differential systems with quadratic right-hand side and with a linear reflecting function. We also construct a set of systems whose reflecting function coincides with that of a linear autonomous system.     

Methods for computing the local null controllability time of linear control systems

We present methods for computing the local null controllability time in the class of linear control systems (not necessarily autonomous) whose families of null controllability sets are monotone with respect to inclusion.     

Multiple periodic orbits of asymptotically linear autonomous convex Hamiltonian systems

We investigate multiple periodic solutions of convex asymptotically linear autonomous Hamiltonian systems. Our theorem generalizes a result by Mawhin and Willem.     

Geometric integration of non-autonomous linear Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.     

On the error in computing Lyapunov exponents by QR Methods

We consider the error introduced using QR methods to approximate Lyapunov exponents. We give a backward error statement for linear non-autonomous systems, and further discuss nonlinear autonomous problems. In particular, for linear systems we show that one approximates a ``nearby' discontinuous problem where how nearby is measured in terms of local errors and a measure of non-normality. For nonlinear problems we use a type of shadowing result. This work was supported in part under NSF Grants DMS/FRG-0139895 and DMS/FRG-0139824     

On bilinear systems with coinciding attainable sets

We consider the problem of determining to what degree the parameters of two autonomous bilinear systems must agree, given that their attainable set functions coincide. Our results generalize some known results for the linear case.     

Approximate solution to the resource consumption minimization problem. I. Construction of a quasioptimal control

For linear systems with constrained control and fixed transition time, we propose two methods for solving the resource consumption minimization problem approximately. We prove that the switching moments of resource-quasioptimal controls are independent of the initial conditions and constant for autonomous systems. Some region of the initial conditions is found for which the constraints on the control are never violated.     

On new resonances and normal forms of autonomous systems with one zero eigenvalue

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has one zero eigenvalue, while the other eigenvalues lie outside the imaginary axis. We study the reducibility of such systems to polynomial normal form.     

Finitely smooth local equivalence of autonomous systems with one zero root

In this paper, in a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has one zero eigenvalue, while the other eigenvalues lie outside the imaginary axis. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.     

Large time behaviour for functional differential equations with dominant eigenvalues of arbitrary order

The spectral theory for linear autonomous neutral functional differential equations (FDE) yields explicit formulas for the large time behaviour of solutions. Our results are based on resolvent computations and Dunford calculus, applied to establish explicit formulas for the large time behaviour of solutions of FDE. We investigate in detail a class of two-dimensional systems of FDE.     

Constructing autonomous boundary-value linear systems from acausal input-output functions

Afrequency-domain realization theory is developed for the class of autonomous, but not necessarily stationary, boundary-value linear systems. It is shown that this realization problem, which consists of constructing autonomous boundary-value linear systems from prescribed input-output functions (weighting patterns), reduces to the factorization of several rational matrices in two variables having separable denominators. This factorization problem is examined and a method is given for constructing minimal factorizations for such rational matrices.     

Data-driven optimal tracking control of switched linear systems

This paper addresses the optimal tracking control for switched linear systems with unknown dynamics. We convert the problem into an optimal control problem of the augmented switched systems. In view of the augmented systems, we propose a data-driven switched linear quadratic regular algorithm for obtaining the optimal switching signal under unknown system dynamics. It is proved that the optimal switching signal will not cause Zeno behavior and can make the system stable. Besides, with the proposed algorithm, we just need to identify an autonomous system instead of the original systems, which has fewer parameters to be determined. A numerical example is given to illustrate the validity of the main results.     

On the Asymptotic Integration of a System of Linear Differential Equations with a Small Parameter in the Coefficients of a Part of Derivatives

We propose an asymptotic method for the integration of one type of systems of linear differential equations with a small parameter in the coefficients of a part of derivatives.     

Reduction of underdetermined systems of ordinary differential equations: II

We study the equivalence of underdetermined systems of ordinary differential equations and present a local classification of systems that are linear in derivatives, are sufficiently regular, and consist of two equations.     

Normal form of real differential equations

In a neighborhood of a fixed point we consider an autonomous analytic system of ordinary differential equations. We establish the existence of a normalizing transformation for which the normal form retains the properties of the original system such as reality and invariance with respect to a linear change of variables. For real systems we consider the problem of existence of an analytic transformation into normal form and the problem of existence of a finitely smooth transformation into a linear system.     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

MULTIPLE PERIODIC SOLUTIONS OF ASYMPTOTICALLY LINEAR NONAUTONOMOUS HAMILTONIAN SYSTEMS

This paper studies the existence of periodic solutions for nonautonomous asymptoticallylinear Hamiltonian systems.By using the Z_p index theory some multiplicity results fornonautonomous systems are given,which generalize some results for autonomous systemsdue to Amman and Zenhder.     

Finite Spectrum Assignment for Completely Regular Differential-Algebraic Systems with Aftereffect

For linear autonomous completely regular differential-algebraic systems with commensurable delays in the state and control, we study the problem of constructing a state feedback that ensures a finite spectrum for the closed-loop system. We propose criteria for spectral reducibility and weak spectral reducibility whose proofs contain the synthesis schemes of appropriate controllers. Several illustrative examples are given.