liouville integrability of the finite dimensional hamiltonian systems derived from principal chiral field for finite dimensional hamiltonian systems derived from 1 1 dimensional integrable systems if they have lax representations then the lax operator creates a set of conserved integrals when these conserved integrals are in involution it is believed quite popularly that there will be enough functionally independent ones among them to guarantee the liouville integrability of the hamiltonian systems at least for those derived from physical problems in this article we give a counterexample based on the u 2 principal chiral field it is proved that the finite dimensional hamiltonian systems derived from the u 2 principal chiral field are liouville integrable moreover their lax operator gives a set of involutive conserved integrals but they are not enough to guarantee the integrability of the hamiltonian systems

oscillation of nonlinear partial difference equations with delays abstract in this paper we first derive a discrete gaussian formula and then apply the formula to two classes of nonlinear parabolic and hyperbolic partial difference equations with delays to obtain sufficient conditions under which every solution of the two classes of nonlinear partial difference equations with delays is oscillatory

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

numerical chaos symplectic integrators and exponentially small splitting distances two discrete versions of duffing s equation are derived as reductions of discretizations of the nonlinear schrodinger nls equation inheriting the properties of the nsl discretizations one is shown to be integrable and using a mel nikov type analysis the other discretization is shown to be nonintegrable arguments are given why it is possible for the nonintegrable scheme to restore the homoclinic orbit at an exponentially fast rate which is faster than the order of the approximation as h 0 this suggests why there can be fundamental differences between symplectic and nonsymplectic discretizations of certain continuous hamiltonian systems

oscillation of the solutions of parabolic differential equations of neutral type sufficient conditions are obtained for the oscillation of the solutions of linear parabolic differential equations of neutral type of the form t u si 1m l i t u x t t i a t u si 1sa i ft u x t p i p x t u si 1kp i x t u x t s i x t e wx 0 where w is a bounded domain in r n with a piecewise smooth boundary

forced oscillation of solutions of certain hyperbolic equations of neutral type some oscillation properties of solutions of certain hyperbolic equations of neutral type are studied and sufficient conditions for the oscillation of the solutions of a boundary value problem in a cylindrical domain are given

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

oscillation of solutions of neutral partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of neutral partial functional differential equations of the form t p t t u x t i 1 l i t u x t i a t u x t k 1 s a k t u x t k t q x t u x t j 1 m q j x t f j u x t j x t 0 g where is a bounded domain inrnwith a piecewise smooth boundary and is the laplacian in the euclideann spacern

the oscillation of hyperbolic functional differential equations this paper deals with the oscillation properties of a class of hyperbolic functional differential equations and obtains a set of criterions by using some results of functional differential inequality these results reveal that the vary difference between hyperbolic functional differential equations and hyperbolic differential equations

maximum principle for a hybrid system via singular perturbations we prove a maximum principle of pontryagin type for the time optimal control of a hybrid system the system is nonlinear and consists of a controlled coupled ode pde the control input acts via the boundary and the interior of the domain systems of this type frequently arise in modeling population dynamics in a contaminated environment for the regularization we use ekeland s variational principle along with a singular perturbation technique for this purpose we introduce a new trajectory on an additional exterior domain whose size may be considered as a singular perturbation parameter it turns out that the maximum principle is stable with respect to the size of the additional exterior domain the technique allows us to obtain necessary optimality conditions without involving measure boundary data

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

pontryagin maximum principle for semilinear and quasilinearparabolic equations with pointwise state constraints this paper studies the first order necessary conditions for the optimal controls of semilinear and quasilinear parabolic partial differential equations with pointwise state constraints a pontryagin type maximum principle is obtained

time optimal control systhesis for discrete time hybrid automata in this paper the problem of time optimal control for hybrid systems with discrete time dynamics is considered the hybrid controller steers all trajectories starting from a maximal set to a given target set in minimum time we derive an algorithm that computes this maximal winning set also algorithms for the computation of level sets associated with the value function rather than the value function itself are presented we show that by solving the reachability problem for the discrete time hybrid automata we obtain the time optimal solution as well the control synthesis is subject to hard constraints on both control inputs and states for linear discrete time dynamics linear programming and quantifier elimination techniques are employed for the backward reachability analysis emphasis is given on the computation of operators for non convex sets using an extended convex hull approach a two tank example is considered in order to demonstrate the techniques of the paper

the nonlinearized eigenvalue problem of the toda hierarchy in the lie poisson framework abstract a 3 3 discrete eigenvalue problem associated with toda hierarchy is presented after the nonlinearization procedure the 3 3 discrete eigenvalue problem is turned into an integrable poisson map on the poisson manifold r 3 n with a lie poisson structure as a reduction of the lie poisson structure on the co adjoint orbit the standard symplectic structure on the symplectic manifold r 2 n is obtained the poisson map restricted on the leaves of the symplectic foliation is reduced to a usual symplectic map which is exactly the nonlinearized 2 2 eigenvalue problem

oscillation of solutions of systems of neutral type partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of systems of neutral type partial functional differential equations of the form where is a bounded domain in r n with a piecewise smooth boundary and is the laplacian in euclidean n space r n the obtained results are illustrated by some interesting examples

necessary and sufficient conditions for oscillation of neutral delay parabolic differential equations necessary and sufficient conditions are established for the oscillation of neutral delay parabolic differential equations of the form t u x t pu x t t a t du x t k 1 sa k t du x t r k t j 1 mq j t u x t s j t x t wx 0 g where w is a bounded domain in r n with a piecewise smooth boundary w and d is the laplacian in euclidean n space r n

reachability and optimal control for linear hybrid automata a quantifier elimination approach this paper considers an optimal control problem for linear hybrid automata lha first we present a controller synthesis algorithm based on reachability analysis the algorithm computes the maximal initial set from which the controller drives the system to a given target set it is shown that using quantifier elimination qe an under approximation of the maximal reachable set can be derived next a weighted time optimal control problem is solved by transforming it into a constrained optimization problem whose constraints are a set of inequalities with quantifiers quantifier elimination qe techniques are employed in order to derive the quantifier free inequalities that are shown to be linear thus the optimal cost is obtained using linear programming for any state belonging to the maximal initial set the optimal switching times and the optimal continuous control inputs are computed these are used in order to derive a hybrid controller which is optimal with respect to the cost function our results are applied to an air traffic management example which is of practical interest

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

an integrable poisson map generated from the eigenvalue problem of the lotka volterra hierarchy a 3 3 discrete eigenvalue problem associated with the lotka volterra hierarchy is studied and the corresponding nonlinearized one an integrable poisson map with a lie poisson structure is also presented moreover a 2 2 nonlinearized eigenvalue problem which also begets the lotka volterra hierarchy is proved to be a reduction of the poisson map on the leaves of the symplectic foliation

integrable mechanical systems related to the harry dym hierarchy it is shown that all stationary and restricted flows of the harry dym hierarchy have a parametrization as a set of newton equations they are new classes of integrable mechanical systems properties of the harry dym hierarchy are used for deriving systematically spectral parameter dependent lax representations and bi hamiltonian formulation for some of these systems they yield complete sets of integrals in involution which render these systems to be completely integrable

limits of singularly perturbed control problems with statistical dynamics of fast motions we describe the limit behavior of admissible trajectories in a singularly perturbed control system as the small parameter tends to zero a general case is considered where in the limit the fast motion may infinitely rapidly oscillate in time invariant measures of the parameterized fast flow are employed to describe the limit behavior and construct the limit control problem the notion of relatively slow controls is introduced approximating properties of the limit problem within the families of relatively slow controls are verified the results are illustrated by examples

optimal problems for nonlinear parabolic boundary control systems this paper considers optimal problems for boundary control systems with the control acting through a general nonlinear boundary condition the problems include constraints on the control and target conditions the final result is a version of pontryagin s maximum principle this setup covers such physical situations as stefan boltzmannboundary conditions and is based on an abstract theory of nonlinear programming problems

suboptimization of singularly perturbed control systems a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems the technique is applicable in particular in the case when the optima

integrable maps for the garnier and for the neumann system we construct integrable maps for the garnier and for the neumann system they are related to the toda hierarchy exactly in the same way as the garnier and the neumann systems are related to the kdv hierarchy as restricted flows here we give lax representations for these maps and prove that they are completely integrable