Lyapunov inequalities and stability for linear Hamiltonian systems

In this paper, we will establish several Lyapunov inequalities for linear Hamiltonian systems, which unite and generalize the most known ones. For planar linear Hamiltonian systems, the connection between Lyapunov inequalities and estimates of eigenvalues of stationary Dirac operators will be given, and some optimal stability criterion will be proved.     

Properties of Hamiltonian systems in the Pontryagin maximum principle for economic growth problems

We consider an optimal control problem with a functional defined by an improper integral. We study the concavity properties of the maximized Hamiltonian and analyze the Hamiltonian systems in the Pontryagin maximum principle. On the basis of this analysis, we propose an algorithm for constructing an optimal trajectory by gluing the dynamics of the Hamiltonian systems. The algorithm is illustrated by calculating an optimal economic growth trajectory for macroeconomic data.     

Method of separation of variables and Hamiltonian system

In the theory of mechanics and/or mathematical physics problems in a prismatic domain, the method of separation of variables ususally leads to the Sturm–Liouville-type eigenproblems of self-adjoint operators, and then the eigenfunction expansion method can be used in equation solving. However, a number of important application problems cannot lead to self-adjoint operator for the transverse coordinate. From the minimum potential energy variational principle, by selection of the state and its dual variables, the generalized variational principle is deduced. Then, based on the analogy between the theory of structural mechanics and optimal control, the present article leads the problem to the Hamiltonian system. The finite-dimensional theory for the Hamiltonian system is extended to the corresponding theory of the Hamiltonian operator matrix and adjoint symplectic spaces. The adjoint symplectic orthonormality relation is proved for the whole state eigenfunction vectors, and then the expansion of an arbitrary whole state function vector by the eigenfunction vectors is established. Thus the range of classical method of separation of variables is considerably extended. The eigenproblem derived from a plate bending problem in a strip domain is used for illustration. © 1993 John Wiley & Sons, Inc.     

Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems

A nonlinear stochastic optimal time-delay control strategy for quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal control problem of quasi-integrable Hamiltonian system with time-delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed feedback control forces are approximated by the control forces without time-delay and the original problem is converted into a stochastic optimal control problem without time-delay. After that, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. As an example, the stochastic time-delay optimal control of two coupled van der Pol oscillators under stochastic excitation is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.     

Hamiltonian paths in Cayley graphs

The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log₂|G|, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction of 3-regular Hamiltonian expanders.     

The Hamiltonian connectivity of rectangular supergrid graphs

A Hamiltonian path of a graph is a simple path which visits each vertex of the graph exactly once. The Hamiltonian path problem is to determine whether a graph contains a Hamiltonian path. A graph is called Hamiltonian connected if there exists a Hamiltonian path between any two distinct vertices. In this paper, we will study the Hamiltonian connectivity of rectangular supergrid graphs. Supergrid graphs were first introduced by us and include grid graphs and triangular grid graphs as subgraphs. The Hamiltonian path problem for grid graphs and triangular grid graphs was known to be NP-complete. Recently, we have proved that the Hamiltonian path problem for supergrid graphs is also NP-complete. The Hamiltonian paths on supergrid graphs can be applied to compute the stitching traces of computer sewing machines. Rectangular supergrid graphs form a popular subclass of supergrid graphs, and they have strong structure. In this paper, we provide a constructive proof to show that rectangular supergrid graphs are Hamiltonian connected except one trivial forbidden condition. Based on the constructive proof, we present a linear-time algorithm to construct a longest path between any two given vertices in a rectangular supergrid graph.     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

On dispersive stability of Hamiltonian systems on lattices

We study the long-time dynamics of oscillations in lattices of infinitely many particles interacting via certain non-linear potentials. The aim is to proof dispersive stability of such Hamiltonian systems analogously to results known for PDEs. To do so we first recapitulate the dynamics of linear Hamiltonian systems on an infinite chain and give optimal decay rates based on the dispersion relation. Based on this we proof that if the non-linearity is weak enough, the non-linear system shows a similar behaviour like its linearization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symplectic integration of Hamiltonian wave equations

Summary The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poisson structure separately, then form the the resulting ODE's. The stability, accuracy, and dispersion of different explicit splitting methods are analyzed, and we give the circumstances under which the best results can be obtained; in particular, when the Hamiltonian can be split into linear and nonlinear terms. Many different treatments and examples are compared.     

Hamiltonian decomposition of complete bipartiter-hypergraphs

In [1] the concepts of paths and cycles of a hypergraph were introduced. In this paper, we give the concepts for bipartite hypergraph and Hamiltonian paths and cycles of a hypergraph, and prove that the complete bipartite 3-hypergraph withq vertices in each part is Hamiltonian decomposable whereq is a prime. This research is supported by the National Natural Science Foundation of China (No.19831080).     

Hamiltonian path graphs

The Hamiltonian path graph H(G) of a graph G is that graph having the same vertex set as G and in which two vertices u and v are adjacent if and only if G contains a Hamiltonian u-v path. A characterization of Hamiltonian graphs isomorphic to their Hamiltonian path graphs is presented.     

Singular Sturmian separation theorems on unbounded intervals for linear Hamiltonian systems

In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case.     

Hamiltonian Approach to Secondary Quantization

Structures and objects used in Hamiltonian secondary quantization are discussed. By the secondary quantization of a Hamiltonian system ?, we mean the Schrödinger quantization of another Hamiltonian system ?₁ for which the Hamiltonian equation is the Schrödinger one obtained by the quantization of the original Hamiltonian system ?. The phase space of ?₁ is the realification ?_R of the complex Hilbert space ? of the quantum analogue of ? equipped with the natural symplectic structure. The role of a configuration space is played by the maximal real subspace of ?.     

Conditions implying the vanishing of the Hamiltonian at infinity in optimal control problems

In an infinite horizon optimal control problem the Hamiltonian vanishes at infinity when the differential equation is autonomous and the integrand in the criterion satisfies some weak integrability conditions. A generalization of Michel’s result (in Econometrica 50:975–985, 1982) is obtained.     

Hamiltonian Systems on Time Scales

Linear and nonlinear Hamiltonian systems are studied on time scales . We unify symplectic flow properties of discrete and continuous Hamiltonian systems. A chain rule which unifies discrete and continuous settings is presented for our so-called alpha derivatives on generalized time scales. This chain rule allows transformation of linear Hamiltonian systems on time scales under simultaneous change of independent and dependent variables, thus extending the change of dependent variables recently obtained by Do lý and Hilscher. We also give the Legendre transformation for nonlinear Euler–Lagrange equations on time scales to Hamiltonian systems on time scales.     

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov-type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.