Order Periodic Systems Research Articles

A Variational Approach to the Problem of Continuous Dependence of Solutions to Second Order Periodic System

In the paper a second order differential system with a periodic boundary data is considered. Using some variational methods sufficient conditions for the continuous dependence of trajectories on controls are proved. The obtained results are applied then to the optimal control problem governed by the above system and the cost functional of a Bolza-type. In the end of the paper, an example of periodic optimal control problem demonstrating the applicability of the results is presented.

Solutions and multiple solutions for second order periodic systems with a nonsmooth potential

Solutions and multiple solutions for second order periodic systems with a nonsmooth potential

Conformal spectral theory for the monodromy matrix

For any N x N monodromy matrix we define the Lyapunov function which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy. We show that the endpoints of each gap are periodic (anti-periodic) eigenvalues or resonances (real branch points). Moreover, the following results are obtained: 1) We define the quasimomentum as an analytic function on the Riemann surface of the Lyapunov function; various properties and estimates of the quasimomentum are obtained. 2) We construct the conformal mapping with imaginary part given by the Lyapunov exponent, and we obtain various properties of this conformal mapping, which are similar to the case of the Hill operator. 3) We determine various new trace formulae for potentials and the Lyapunov exponent. 4) We obtain a priori estimates of gap lengths in terms of the Dirichlet integral. We apply these results to the Schrodinger operators and to first order periodic systems on the real line with a matrix-valued complex self-adjoint periodic potential.

Multiple solutions for nonautonomous second order periodic systems

We study nonautonomonus second order periodic systems with a nonsmooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large classes of both subquadratic and superquadratic potentials at infinity.

Nontrivial solutions for superquadratic nonautonomous periodic systems

We consider a nonautonomous second order periodic system with an indefinite linear part. We assume that the potential function is superquadratic, but it may not satisfy the Ambrosetti-Rabinowitz condition. Using an existence result for $C^1$-functionals having a local linking at the origin, we show that the system has at least one nontrivial solution.

On superquadratic periodic systems with indefinite linear part

We consider a second order periodic system with an indefinite linear part and a potential function which is superquadratic but does not satisfy the AR-condition. Using Morse critical groups, we show that the system has at least one nontrivial solution.

Existence and multiplicity of solutions for second order periodic systems with the p-Laplacian and a nonsmooth potential

In this paper we study nonlinear periodic systems driven by the ordinary p-Laplacian with a nonsmooth potential. We prove an existence theorem using a nonsmooth variant of the reduction method. We also prove two multiplicity results. The first is for scalar problems and uses the nonsmooth second deformation lemma. The second is for systems and it is based on the nonsmooth local linking theorem.

Spectral estimates for matrix-valued periodic Dirac operators

We consider the first order periodic systems perturbed by a $2N\ts 2N$ matrix-valued periodic potential on the real line. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has branch points, which we call resonances. We prove the existence of real or complex resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy (in terms of the Fourier coefficients of the potential). We show that there exist two types of gaps: i) stable gaps, i.e., the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, i.e., the endpoints are resonances (real branch points). Moreover, we determine various new trace formulae for potentials and the Lyapunov exponent.

Two nontrivial solutions for periodic systems with indefinite linear part

We consider second order periodic systems with a nonsmooth potential and an indefinite linear part. We impose conditions under which the nonsmooth Euler functional is unbounded. Then using a nonsmooth variant of the reduction method and the nonsmooth local linking theorem, we establish the existence of at least two nontrivial solutions.

Multiple solutions for strongly resonant periodic systems

We considered a semilinear, second order periodic system. We assumed that the differential operator x → − x ″ − A x has zero as an eigenvalue and has no negative eigenvalues. Also we imposed a strong resonance condition (with respect to the zero eigenvalue) on the potential function F ( t , x ) . Using the second deformation theorem, we established the existence of at least two nontrivial solutions. To do this we needed to conduct a detailed analysis of the Cerami compactness condition, which is actually of independent interest.

Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential

We study a semilinear nonautonomous second order periodic system with a nonsmooth potential function which exhibits a quadratic or superquadratic growth. We establish the existence of a solution, using minimax methods of the nonsmooth critical point theory.