asymptotic behavior of solutions of a k hessian evolution equation abstract we study the long time behavior of solutions of the k hessian evolution equation u t s k d 2 u posed on a bounded domain of the n dimensional space with homogeneous boundary conditions to this end we construct a separable solution and we show that the long time behavior of u is precisely described by this special solution further we initiate the study of that equation on the entire space providing a new class of explicit and radially symmetric self similar solutions that we call k barenblatt solutions these solutions present some common properties as those of well known barenblatt solutions for the porous media equation and the p laplacian equation

the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

a new decomposition method for stochastic dynamic stabilization abstract the aim of this paper is to prove that the stabilizability of a controlled nonlinear stochastic system can be deduced from the stabilizability of a reduced system for which the existence of a stabilizing feedback law can be decided more easily this result extends the well known deterministic chain of integrator lemma by using a different scheme of reduction for the original stochastic system

stability of an adaptive regulator for partially known nonlinear stochastic systems we investigate the properties of a fast identification style of control algorithm applied to a class of stochastic dynamical systems in continuous time which are sampled at a constant rate the algorithm does not assume that the system dynamics are known and estimates them using a simple filter under a mild smoothness condition on the system dynamics we show that when the sampling rate is sufficiently fast the control algorithm stabilizes the system in the sense that the sampled closed loop system becomes an ergodic markov chain moreover an explicit bound is given for the expected deviation of the system state from the origin the result is also adapted for the case where state measurement is subject to random noise

exact similarity solutions to some nonlinear diffusion equations the author gives new closed form similarity solutions to n dimensional radially symmetric nonlinear diffusion equations with power law and exponential diffusivities these solutions are generalisations of known similarity solutions

brief paper stochastic minimax control for stabilizing uncertain quasi integrable hamiltonian systems a procedure for designing feedback control to asymptotically stabilize with probability one quasi integrable hamiltonian systems with bounded uncertain parametric disturbances is proposed first the partially averaged ito stochastic differential equations are derived from given system by using the stochastic averaging method for quasi integrable hamiltonian systems second the hamilton jacobi issacs hji equation for the ergodic control problem of the averaged system and a performance index with undetermined cost function is established based on the principle of optimality this equation is then solved to yield the worst disturbances and the associated optimal controls third the asymptotic lyapunov stability with probability one of the optimally controlled system with worst disturbances is analyzed by evaluating the maximal lyapunov exponent of the fully averaged ito equations finally the cost function and feedback control are determined by the requirement of stabilizing the worst disturbed system a simple example is worked out to illustrate the application of the proposed procedure and the effects of optimal control on stabilizing the uncertain system

stochastic adaptive backstepping controller design by introducing dynamic signal and changing supply function a more general class of stochastic non linear systems with unmodelled dynamics and uncertain non linear functions are considered in this paper with the concept of isps being extended to stochastic case by combining changing supply function technique with backstepping design technique an adaptive output feedback controller is proposed it is shown that all the solutions of the closed loop system are uniformly bounded in probability and the output can be regulated to an arbitrarily small neighbourhood of the origin in probability a simulation example demonstrates the control scheme

lyapunov like techniques for stochastic stability the purpose of this paper is to study the stabilizability problem for control stochastic nonlinear systems driven by a wiener process sufficient conditions for the existence of stabilizing feedback laws that are smooth except possibly at the equilibrium point of the system are provided by means of stochastic lyapunov like techniques the notion of dynamic asymptotic stability in probability of control stochastic differential systems is introduced and the stabilization by means of dynamic controllers is studied

adaptive output feedback tracking of stochastic nonlinear systems we address the adaptive stabilization and tracking problems for a class of output feedback canonical systems driven by wiener noises of unknown covariance filtered transformation and backstepping techniques are employed in the stochastic control design we obtain two adaptive controllers that guarantee the global stability in probability for vanishing perturbations or the input to state stability in probability for nonvanishing perturbations respectively the tracking error can converge to a small residual set around the origin in the sense of mean quartic value

classification of type i and type ii behaviors for a supercritical nonlinear heat equation we study blow up of radially symmetric solutions of the nonlinear heat equation ut u u p 1u either on rn or on a finite ball under the dirichlet boundary conditions we assume p ps n 2n 2 and that the initial data is bounded possibly sign changing our first goal is to establish various characterizations of type i and type ii blow ups among many other things we show that the following conditions are equivalent a the blow up is of type ii b the rescaled solution w y s converges to either y or y as s where denotes the singular stationary solution c u x t x tends to 1 as x 0 where t is the blow up time our second goal is to study continuation beyond blow up among other things we show that if a blow up is of type i and incomplete then its limit l1 continuation becomes smooth immediately after blow up and that type i blow up implies type i regularization that is t t 1 p 1 u t l is bounded as t t we also give various criteria for complete and incomplete blow ups

some properties of finite time stable stochastic nonlinear systems in this paper we study some properties of finite time stable stochastic nonlinear systems we begin by showing several continuous dependence theorems of solutions on initial values under some conditions on the coefficients of stochastic systems we then derive some regular properties of its stochastic settling time for a finite time stable stochastic nonlinear system we show continuity positive definiteness and boundedness of the expected stochastic settling time under appropriate conditions finally a lyapunov function is constructed by making use of the expectation of the stochastic settling time and the infinitesimal generator of the stochastic system defined on the lyapunov function is also given and hence resulting in a converse lyapunov theorem of finite time stochastic stability

bifurcation and admissible solutions for the hessian equation abstract we study the following eigenvalue problem of k hessian equation s k d 2 u k f u in b u 0 on b global bifurcation result is established for this problem as applications of the bifurcation result we determine the intervals of for the existence nonexistence uniqueness and multiplicity of radially symmetric k admissible solutions for this problem

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

the liouville bratu gelfand problem for radial operators abstract we determine precise existence and multiplicity results for radial solutions of the liouville bratu gelfand problem associated with a class of quasilinear radial operators which includes perturbations of k hessian and p laplace operators

nonlinearity and constrained quantum motion the dynamical equation satisfied by the density matrix when a quantum system is subjected to one or more constraints arising from conserved quantities is derived the resulting nonlinear evolution of the density matrix has the property that it is independent of the specific composition of the pure state mixture generating the initial state of the system

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

quantum mechanics as an infinite dimensional hamiltonian system with uncertainty structure part ii making reference to the formalism developed in part i to formulate schrodinger quantum mechanics the properties of kahlerian functions in general almost kahlerian manifolds are studied

symplectic approach to quantum constraints a general prescription for the treatment of constrained quantum motion is outlined we consider in particular constraints defined by algebraic submanifolds of the quantum state space the resulting formalism is applied to obtain solutions to the constrained dynamics of systems of multiple spin particles when the motion is constrained to a certain product space containing all of the energy eigenstates the dynamics thus obtained are quasi unitary in the sense that the equations of motion take a form identical to that of unitary motion but with different boundary conditions when the constrained subspace is a product space of disentangled states the associated motion is more intricate nevertheless the equations of motion satisfied by the dynamical variables are obtained in closed form

output feedback control design for strict feedback stochastic nonlinear systems under a risk sensitive cost in this note we study the problem of output feedback control design for a class of strict feedback stochastic nonlinear systems under an infinite horizon risk sensitive cost criterion the controller designed can guarantee an arbitrary small long term average cost for arbitrary risk sensitivity parameter and achieve boundedness in probability for the closed loop system using the integrator backstepping methodology furthermore the controller preserves the equilibrium at the origin of the nonlinear system