A guidance problem for a differential inclusion with parameter

A differential inclusion with parameter on a finite time interval is studied. Questions related to determining the solvability set in the problem of guiding a differential inclusion to a compact target set in the phase space are discussed.     

Singular solutions for a class of Grusin type operators

We construct singular solutions for a one-parameter family of partial differential equations with double characteristics and with complex lower order terms. The parameter belongs to a discrete set which is described in terms of the spectrum of a related differential operator.

     

Almost periodic and almost automorphic dynamics for scalar convex differential equations

We study the set of bounded trajectories for the flow defined by a class of scalar convex differential equations depending on a parameter. It is found that there exists precisely one value of the parameter for which almost automorphic but not almost periodic dynamics may appear. Even for this parameter value, the occurrence of almost periodic dynamics is shown to be residual in some cases. The dependence of this parameter on the functions defining the differential equations is also studied. The authors were partly supported by Junta de Castilla y León under project VA024/03, and C.I.C.Y.T. under project BFM2002-03815.     

A level set reduced basis approach to parameter estimation

We introduce an efficient level set framework to parameter estimation problems governed by parametrized partial differential equations. The main ingredients are: (i) an “admissible region” approach to parameter estimation; (ii) the certified reduced basis method for efficient and reliable solution of parametrized partial differential equations; and (iii) a parameter-space level set method for construction of the admissible region. The method can handle nonconvex and multiply connected regions. Numerical results for two examples in design and inverse problems illustrate the versatility of the approach.     

Some properties of viability problems depending on a parameter

This paper is devoted to proving the existence of solutions to differential inclusions, viable in a compact set and continuously depending on a parameter. It contains an inequality of the Filippov-Gronwall type for such solutions and a version of the relaxation theorem, uniform on the parameter.The work was performed when the author was visiting SISSA, Trieste.     

流过半无限竖直可渗透壁面时的磁流体动力学自由对流

研究不可压缩粘性导电流体,流过半无限竖直可渗透平板时,将其偏微分形式的流动和传热的基本控制方程,应用适当的相似变换,简化为非线性的常微分方程组．对两种抽吸参数:大的和小的抽吸参数,采用摄动法得到变换后方程的近似解．数值结果表明,随着磁场参数和抽吸参数的增大,任意点的速度场在减小;磁场参数的影响,引起热边界层厚度的增大;速度和温度场随着热汇参数的增大而减小．     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Structure of stability and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: II

We consider families of linear differential systems depending on a real parameter that occurs only as a factor multiplying the matrix of the system. The asymptotic stability set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are asymptotically stable. We prove that a set on the real axis is the asymptotic stability set of such a family if and only if it is an F _σδ -set lying entirely on an open ray with origin at zero. In addition, for any set of this kind, the coefficient matrix of a family whose asymptotic stability set coincides with this set can be chosen to be infinitely differentiable and uniformly bounded on the time half-line.     

A penalty function-based differential evolution algorithm for constrained global optimization

We propose a differential evolution-based algorithm for constrained global optimization. Although differential evolution has been used as the underlying global solver, central to our approach is the penalty function that we introduce. The adaptive nature of the penalty function makes the results of the algorithm mostly insensitive to low values of the penalty parameter. We have also demonstrated both empirically and theoretically that the high value of the penalty parameter is detrimental to convergence, specially for functions with multiple local minimizers. Hence, the penalty function can dispense with the penalty parameter. We have extensively tested our penalty function-based DE algorithm on a set of 24 benchmark test problems. Results obtained are compared with those of some recent algorithms.     

Optimal estimation of parameters and states in stochastic time-varying systems with time delay

In this study estimation of parameters and states in stochastic linear and nonlinear delay differential systems with time-varying coefficients and constant delay is explored. The approach consists of first employing a continuous time approximation to approximate the stochastic delay differential equation with a set of stochastic ordinary differential equations. Then the problem of parameter estimation in the resulting stochastic differential system is represented as an optimal filtering problem using a state augmentation technique. By adapting the extended Kalman–Bucy filter to the resulting system, the unknown parameters of the time-delayed system are estimated from noise-corrupted, possibly incomplete measurements of the states.     

Fractional-order attractors synthesis via parameter switchings

In this paper we provide numerical evidence, via graphics generated with the help of computer simulations, that switching the control parameter of a dynamical system belonging to a class of fractional-order systems in a deterministic way, one obtains an attractor which belongs to the class of all admissible attractors of the considered system. For this purpose, while a multistep numerical method for fractional-order differential equations approximates the solution to the mathematical model, the control parameter is switched periodically every few integration steps. The switch is made inside of a considered set of admissible parameter values. Moreover, the synthesized attractor matches the attractor obtained with the control parameter replaced with the averaged switched parameter values. The results are verified in this paper on a representative system, the fractional-order Lü system. In this way we were able to extend the applicability of the algorithm presented in earlier papers using a numerical method for fractional differential equations.     

Chaos in vibroimpact systems with one degree of freedom in a neighborhood of chatter generation: II

We study the rectilinear motion of a mass point with impacts against a stopper. The motion between the impacts is described by a second-order ordinary differential equation with a parameter. The impact recovery coefficient also depends on the parameter of the vibroimpact system. We describe a bifurcation that leads to the generation of a Smale horseshoe in a parametric neighborhood of the chatter phenomenon. We prove the existence of an invariant set described by symbolic dynamics.     

Chaos in vibroimpact systems with one degree of freedom in a neighborhood of chatter generation: I

We study the rectilinear motion of a mass point with impacts against a delimiter. The motion between the impacts is described by a second-order ordinary differential equation with a parameter. The impact recovery coefficient also depends on the parameter of the vibroimpact system. We describe a bifurcation that leads to the generation of a Smale horseshoe in a parametric neighborhood of the chatter phenomenon. We prove the existence of an invariant set described by symbolic dynamics.     

Reciprocal Diffusions and Symmetries of Parabolic PDE: The Nonflat Case

We introduce a new set of Reciprocal Characteristics for the class of reciprocal diffusions naturally associated to a general parabolic second-order linear differential operator. All the coefficients of this operator, including the diffusion matrix, depend on time. This set of reciprocal characteristics is provided by the study of the symmetries of the differential operator. The Riemannian metric defined by the diffusion matrix is of central importance. Our reciprocal characteristics are the natural extension of Ovsiannikov's differential invariants to the time dependent parabolic case. We also show that the symmetries of the PDE coincide with the one parameter families of transformations which leave the usual stochastic Lagrangian as well as a modified Onsager–Machlup Lagrangian invariant.     

The Power Matrix, coadjoint action and quadratic differentials

The coefficients of a quadratic differential which is changing under the Loewner flow satisfy a well-known differential system studied by Schiffer, Schaeffer and Spencer, and others. By work of Roth, this differential system can be interpreted as Hamilton's equations. We apply the power matrix to interpret this differential system in terms of the coadjoint action of the matrix group on the dual of its Lie algebra. As an application, we derive a set of integral invariants of Hamilton's equations which is in a certain sense complete. In function theoretic terms, these are expressions in the coefficients of the quadratic differential and Loewner map which are independent of the parameter in the Loewner flow.     

Stability and bifurcation analysis in a system of four coupled neurons with multiple delays

A system of delay differential equations is studied which represent a model for four neurons with time delayed connections between the neurons and time delayed feedback from each neuron to itself. The linear stability and bifurcation of the system are studied in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements. Meanwhile, the bifurcation set are drawn in the parameter space.     

Unification of a differential inclusion with parameter in the guidance problem

A differential inclusion (d.i.) with a parameter α ∈ ?, where ? is a compact set in a finite-dimensional Euclidean space is considered. The problem under study is one of guiding a d.i. to a given compact set M in the phase space of the d.i. at a fixed moment of time. Issues concerning the unification of the d.i. with a parameter are discussed. A specific feature of the present unification is that it is performed for a parameterdependent d.i. that cannot be initially structured by a vector function similar to those determining the dynamics of systems in control and conflict control problems.     

Differential evolution with preferential crossover

We study the mutation operation of the differential evolution algorithm. In particular, we study the effect of the scaling parameter of the differential vector in mutation. We derive the probability density function of points generated by mutation and thereby identify some drawbacks of the scaling parameter. We also visualize the drawbacks using simulation. We then propose a crossover rule, called the preferential crossover rule, to reduce the drawbacks. The preferential crossover rule uses points from an auxiliary population set. We also introduce a variable scaling parameter in mutation. Motivations for these changes are provided. A numerical study is carried out using 50 test problems, many of which are inspired by practical applications. Numerical results suggest that the proposed modification reduces the number of function evaluations and cpu time considerably.     

Existence of natural and projectively equivariant quantizations

We study the existence of natural and projectively equivariant quantizations for differential operators acting between order 1 vector bundles over a smooth manifold M. To that aim, we make use of the Thomas-Whitehead approach of projective structures and construct a Casimir operator depending on a projective Cartan connection. We attach a scalar parameter to every space of differential operators, and prove the existence of a quantization except when this parameter belongs to a discrete set of resonant values.     

A family of degenerate differential operators

Certain second-order partial differential operators, expressed as sums of squares of complex vector fields, are shown not to beC ^∞ hypoelliptic even at a point, rather than merely in an open set. The proof is based on an asymptotic analysis of a family of ordinary differential operators depending on a complex parameter. Research supported by the National Science Foundation.