Stability of optimal filter higher-order derivatives

In many scenarios, a state-space model depends on a parameter which needs to be inferred from data. Using stochastic gradient search and the optimal filter first-order derivatives, the parameter can be estimated online. To analyze the asymptotic behavior of such methods, it is necessary to establish results on the existence and stability of the optimal filter higher-order derivatives. These properties are studied here. Under regularity conditions, we show that the optimal filter higher-order derivatives exist and forget initial conditions exponentially fast. We also show that the same derivatives are geometrically ergodic.     

Adaptive methods for periodic initial value problems of second order differential equations

In this paper numerical methods involving higher order derivatives for the solution of periodic initial value problems of second order differential equations are derived. The methods depend upon a parameter p > 0 and reduce to their classical counter parts as p → 0. The methods are periodically stable when the parameter p is chosen as the square of the frequency of the linear homogeneous equation. The numerical methods involving derivatives of order up to 2q are of polynomial order 2q and trigonometric order one. Numerical results are presented for both the linear and nonlinear problems. The applicability of implicit adaptive methods to linear systems is illustrated.     

关于非线性狄立克雷问题的一致有效渐近解

本文研究最高阶导数项带小参数的二阶拟线性椭圆型方程的狄立克雷问题.在退化方程不存在奇点的情形下,当参数ε是充分小时,证明了解的存在性和唯一性,并在整个区域导出解的一致有效渐近近似式.     

Singular perturbations in a class of problems of optimal control with integral convex criterion

The problem of optimal control is investigated with a linear law of motion and convex quality criterion. A small positive parameter appears in front of the derivatives of some of the unknowns in the law of motion. The behaviour of the optimal solution is studied when the small parameter approaches zero with some assumptions that are different from thos encountered in the literature.     

A linear system of differential equations with small parameter at a part of derivatives,a deviation of the argument,and a turning point

For a system of differential equations with small parameter at a part of derivatives, a linear deviation of the argument, and a turning point, we obtained conditions, under which its solutions are solutions of a system of differential equations with small parameter at a part of derivatives such that its matrices possess the asymptotic expansions at |ε| ≤ ε₀ with the coefficients holomorphic at |x| ≤ x ₀ . The existence and the infinite differentiability of a solution of the system of differential equations with small parameter at a part of derivatives and with a linear deviation of the argument in the presence of a turning point are proved.     

Two-Dimensional Parabolic Inverse Source Problem with Final Overdetermination in Reproducing Kernel Space

A new method of the reproducing kernel Hilbert space is applied to a twodimensional parabolic inverse source problem with the final overdetermination. The exact and approximate solutions are both obtained in a reproducing kernel space. The approximate solution and its partial derivatives are proved to converge to the exact solution and its partial derivatives, respectively. A technique is proposed to improve some existing methods. Numerical results show that the method is of high precision, and confirm the robustness of our method for reconstructing source parameter.     

Solvability for a higher-order Hadamard fractional differential model with a sign-changing nonlinearity dependent on the parameter $\varrho$

Until now most of the results are obtained in the sense of fractional derivatives such as Caputo and Riemann-Liouville, and there are few models using the Hadamard fractional derivatives. In this paper, based on the properties of the Green"s function, the existence of positive solutions are obtained for a Hadamard fractional differential equation with a higher-order sign-changing nonlinearity under some conditions by the fixed point theorem, and the existence of positive solutions is dependent on the parameter $\varrho$ for the Semipositive problem.     

On a system of singularly perturbed second-order quasilinear ordinary differential equations in the critical cases

A singularly perturbed system of second-order quasilinear ordinary differential equations with a small parameter multiplying the second derivatives is examined in the case where the coefficient matrix of the first derivatives is singular and does not depend on the unknown functions.     

Inverse heat problem of determining time-dependent source parameter in reproducing kernel space

A new method by the reproducing kernel Hilbert space is applied to an inverse heat problem of determining a time-dependent source parameter. The problem is reduced to a system of linear equations. The exact and approximate solutions are both obtained in a reproducing kernel space. The approximate solution and its partial derivatives are proved to converge to the exact solution and its partial derivatives, respectively. The proposed method improves the existing method. Our numerical results show that the method is of high precision.     

Numerical analysis of a strongly coupled system of two singularly perturbed convection–diffusion problems

A system of two coupled singularly perturbed convection–diffusion ordinary differential equations is examined. The diffusion term in each equation is multiplied by a small parameter, and the equations are coupled through their convective terms. The problem does not satisfy a conventional maximum principle. Its solution is decomposed into regular and layer components. Bounds on the derivatives of these components are established that show explicitly their dependence on the small parameter. A numerical method consisting of simple upwinding and an appropriate piecewise-uniform Shishkin mesh is shown to generate numerical approximations that are essentially first order convergent, uniformly in the small parameter, to the true solution in the discrete maximum norm.      

Linear system of differential equations with turning point

A system of linear differential equations with small parameter as a coefficient of a part of derivatives is reduced to the canonical form and the properties of the transformation matrix are investigated.     

Stationary waiting time derivatives

We investigate the stability of waiting-time derivatives when inputs to a queueing system-service times and interarrival times-depend on a parameter. We give conditions under which the sequence of waiting-time derivatives admits a stationary distribution, and under which the derivatives converge to the stationary regime from all initial conditions. Further hypotheses ensure that the expectation of a stationary waiting-time derivative is, in fact, the derivative of the expected stationary waiting time. This validates the use of simulation-based infinitesimal perturbation analysis estimates with a variety of queueing processes.We examine waiting-time sequences satisfying recursive equations. Our basic assumption is that the input and its derivatives are stationary and ergodic. Under monotonicity conditions, the method of Loynes establishes the convergence of the derivatives. Even without such conditions, the derivatives obey a linear difference equation with random coefficients, and we exploit this fact to find stability conditions.     

Existence and stability of solutions with boundary layers in multidimensional singularly perturbed reaction-diffusion-advection problems

This paper deals with the boundary-value problem for a nonlinear elliptic equation containing a small parameter multiplying the derivatives and degenerating into a finite equation as the small parameter tends to zero. The existence theorem for the solution with a boundary layer and its Lyapunov stability are proved.     

Differential stability of solutions to convex,control constrained optimal control problems

A family of convex, control constrained optimal control problems that depend on a real parameter is considered. It is shown that under some regularity conditions on data the solutions of these problems, as well as the associated Lagrange multipliers are directionally differentiable with respect to parameter. The respective right-derivatives are given as the solution and the associated Lagrange multipliers for some quadratic optimal control problem. If a condition of strict complementarity type hold, then directional derivatives become continuous ones.     

Generalized variational calculus in terms of multi-parameters fractional derivatives

In this paper, we briefly introduce two generalizations of work presented a few years ago on fractional variational formulations. In the first generalization, we consider the Hilfer’s generalized fractional derivative that in some sense interpolates between Riemann–Liouville and Caputo fractional derivatives. In the second generalization, we develop a fractional variational formulation in terms of a three parameter fractional derivative. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed.     

Differential sensitivity of solutions of convex constrained optimal control problems for discrete systems

A family of optimal control problems for discrete systems that depend on a real parameter is considered. The problems are strongly convex and subject to state and control constraints. Some regularity conditions are imposed on the constraints.The control problems are reformulated as mathematical programming problems. It is shown that both the primal and dual optimal variables for these problems are right-differentiable functions of a parameter. The right-derivatives are characterized as solutions to auxiliary quadratic control problems. Conditions of continuous differentiability are discussed, and some estimates of the rate of convergence of the difference quotients to the respective derivatives are given.     

TRAPEZOIDAL PLATE BENDING ELEMENT WITH DOUBLE SET PARAMETERS

Using double set parameter method, a 12-parameter trapezoidal plate bending element is presented. The first set of degrees of freedom, which make the element convergent, are the values at the four vertices and the middle points of the four sides together with the mean values of the outer normal derivatives along four sides. The second set of degree of freedom, which make the number of unknowns in the resulting discrete system small and computation convenient are values and the first derivatives at the four vertices of the element. The convergence of the element is proved.     

An Unified Computational Framework for Parameter Identification of Material Models in Finite Inelasticity

Parameter identification processes concern the determination of parameters in a material model in order to fit experimental data. We provide a distinct, unified algorithmic setting of a generic class of material models and discuss the associated gradient–based optimization problem. Gradient–based optimization algorithms need derivatives of the objective function with respect to the material parameter vector κ . In order to obtain the necessary derivatives, an analytical sensitivity analysis is pointed out for the unified class of algorithmic material models. The quality of the parameter identification is demonstrated for a representative example.     

On the Shapiro–Lopatinkii condition for elliptic problems

This paper considers elliptic problems with high-order derivatives multiplied by a small parameter. We found the algebraic conditions for an operator and the boundary conditions that guarantee the Fredholm property. An a priori estimate for the solution with a constant independent of the small parameter is proved. These results are known for elliptic boundary-value problems with small parameter in the half-space R _n ⁺. We extend them to the case of bounded domains with smooth boundary. The small-parameter coercive conditions are formulated, and a two-sided estimate is proved.     

On variational principles in which the Lagrangian function involves third order derivatives

Summary Recently variational principles whose Lagrangian functions involve third order derivatives of the position vector have been considered with a view to applying them to certain aspects of elementary particle theory. It is known that definite consistency conditions arise when parameter invariance of the associated action integral is required. By invoking two additional assumptions, which have been adopted in the past, a general parameter invariant Lagrangian is deduced. The structure of the corresponding momentum vector is investigated. It is shown that a large class of the resulting equations of motion are derivable from a different approach due to Rund. Borelowski's equations of motion are also derived. Although no radiation effects are considered the parameter invariant counterpart of the Abraham vector plays an important role in the theory.