We consider the following problem: the drift of the wealth process of two companies, modelled by a two-dimensional Brownian motion, is controllable such that the total drift adds up to a constant. The aim is to maximize the probability that both companies survive. We assume that the volatility of one company is small with respect to the other, and use methods from singular perturbation theory to construct a formal approximation of the value function. Moreover, we validate this formal result by explicitly constructing a strategy that provides a target functional, approximating the value function uniformly on the whole state space.

In this paper we consider the problem of averaging for a class of piecewise deterministic Markov processes (PDMPs) whose dynamic is constrained by the presence of a boundary. On reaching the boundary, the process is forced to jump away from it. We assume that this boundary is attractive for the process in question in the sense that its averaged flow is not tangent to it. Our averaging result relies strongly on the existence of densities for the process, allowing us to study the average number of crossings of a smooth hypersurface by an unconstrained PDMP and to deduce from this study averaging results for constrained PDMPs.

We show that all self-adjoint extensions of semibounded Sturm–Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in \{1,2\}$. This characterization generalizes the well-known analog for semibounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as

$$ \begin{align*} \boldsymbol{A}_{ {}_{\scriptstyle \Theta}}=\boldsymbol{A}_0+\mathbf{B}\Theta\mathbf{B}^*, \end{align*} $$

$\boldsymbol {A}_0$

$\Theta $

$\mathbb {C}^d$

$\boldsymbol {A}_0$

$\mathcal {H}_{-1}(\boldsymbol {A}_0)$

$\Theta $

The merging of boundary triples with perturbation theory provides a more holistic view of the operator’s matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.

As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

The flexibility of the free-floating flexible space manipulator system’s link and joint may affect the control accuracy and cause vibrations. We studied the dynamics modeling, joint trajectory tracking control, and vibration suppressing problem of free-floating flexible-link and flexible-joints space manipulator system with external interference and uncertain parameter. The system’s dynamic equations are established using linear momentum conservation, angular momentum conservation, assumed mode method, and Lagrange equation. Then, the system’s singular perturbation model is established, and a hybrid control is presented. For the slow subsystem, a robust fuzzy sliding mode control is proposed to realize the joint desired trajectory tracking. For the fast subsystem, a speed difference feedback control and a linear-quadratic optimal control are designed to suppress the vibration caused by the flexible joints and the flexible link separately. The simulation comparison experiments under different conditions are taken. The simulate results demonstrate the proposed hybrid control’s validity.

This paper presents a robust adaptive output feedback tracking controller for the flexible-joint robot manipulators to deal with the unknown upper bounds of parameter uncertainties and external disturbances. With applying the singular perturbation theory and integral manifold concept, the complex nonlinear coupled system of the flexible-joint robot manipulators is divided into a slow subsystem and a fast subsystem. A robust adaptive control scheme based on an improved linear parameterization expression is designed for the slow subsystem, and a saturation function is applied in the robust control term to make the torque output smooth. In the meantime, different from the previous approaches, the second-order derivative term of elastic torque is avoided by using the proposed computed torque method, which simplifies the implementation of the fast control law. Moreover, to carry out the whole control system with only position measurements, an approximate differential filter is involved to generate pseudo velocity signals for links and joint motors. In addition, an explicit but strict stability proof of the control system based on the theory of singularly perturbed systems is presented. Finally, simulation results verify the superior dynamic performance of the proposed controller.

This paper deals with a more general class of singularly perturbed boundary value problem for a differential-difference equations with small shifts. In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter ε and the shifts depend on the small parameter ε has been considered. The fitted-mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layer. The cubic B-spline basis functions with fitted-mesh are considered in the procedure which yield a tridiagonal system which can be solved efficiently by using any well-known algorithm. The stability and parameter-uniform convergence analysis of the proposed method have been discussed. The method has been shown to have almost second-order parameter-uniform convergence. The effect of small parameters on the boundary layer has also been discussed. To demonstrate the performance of the proposed scheme, several numerical experiments have been carried out.

We consider the Halfin–Whitt diffusion process Xd(t), which is used, for example, as an approximation to the m-server M/M/m queue. We use recently obtained integral representations for the transient density p(x,t) of this diffusion process, and obtain various asymptotic results for the density. The asymptotic limit assumes that a drift parameter β in the model is large, and the state variable x and the initial condition x0 (with Xd(0) = x0 > 0) are also large. We obtain some alternate representations for the density, which involve sums and/or contour integrals, and expand these using a combination of the saddle point method, Laplace method and singularity analysis. The results give some insight into how steady state is achieved, and how if x0 > 0 the probability mass migrates from Xd(t) > 0 to the range Xd(t) < 0, which is where it concentrates as t → ∞, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L2 norm and first order convergence in the energy norm on graded meshes, which is independent of ɛ. In contrast with the conventional methods, our method is much more accurate and effective.

In this paper we present a first supercloseness analysis for higher-order Galerkin FEM applied to a singularly perturbed convection-diffusion problem. Using a solution decomposition and a special representation of our finite element space, we are able to prove a supercloseness property of p + 1/4 in the energy norm where the polynomial order p ≥ 3 is odd.

We propose two variants of tailored finite point (TFP) methods for discretizing two dimensional singular perturbed eigenvalue (SPE) problems. A continuation method and an iterative method are exploited for solving discretized systems of equations to obtain the eigen-pairs of the SPE. We study the analytical solutions of two special cases of the SPE, and provide an asymptotic analysis for the solutions. The theoretical results are verified in the numerical experiments. The numerical results demonstrate that the proposed schemes effectively resolve the delta function like of the eigenfunctions on relatively coarse grid.

This work is devoted to analyze a numerical scheme for the approximation of the linear heat equation’s controls. It is known that, due to the regularizing effect, the efficient computation of the null controls for parabolic type equations is a difficult problem. A possible cure for the bad numerical behavior of the approximating controls consists of adding a singular perturbation depending on a small parameter ε which transforms the heat equation into a wave equation. A space discretization of step h leads us to a system of ordinary differential equations. The aim of this paper is to show that there exists a sequence of exact controls of the corresponding perturbed semi-discrete systems which converges to a control of the original heat equation when both h (the mesh size) and ε (the perturbation parameter) tend to zero.

In this paper, we study singularly perturbed Filippov systems. More specifically, our main question is to know how the dynamics of Filippov systems is affected by singular perturbations. We extend the Fenichel theory developed in Fenichel (J. Differ. Equ., 1979, Vol. 31, pp. 53–98) to these systems. In addition, the study of non-smooth constrained systems is considered.

We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations \hbox{$\{\mathcal E_\eps\}_\eps,\,\{\widetilde{\mathcalE}_\eps\}_\eps$}$\mathrm{\{}{\mathrm{ℰ}}_{\mathit{\epsilon }}{\mathrm{\}}}_{\mathit{\epsilon }}\mathit{,}\hspace{0.17em}\mathrm{\{}\begin{array}{}{}_{\mathrm{􏽥}}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}{\mathrm{\}}}_{\mathit{\epsilon }}$ of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize the Γ-limit as ε → 0 of ℰε, and show that in general the Γ-limits of ℰε and \hbox{$\widetilde{\mathcal E}_\eps$}$\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ-limit of \hbox{$\widetilde{\mathcal E}_\eps$}$\begin{array}{}{\mathrm{􏽥}}_{}\\ {\mathrm{ℰ}}_{\mathit{\epsilon }}\end{array}$ strictly contains the domain of the Γ-limit of ℰε.

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

We consider quasilinear optimal control problems involving a thick two-level junction Ωε which consists of the junction body Ω0 and a large number of thin cylinders with the cross-section of order 𝒪(ε2). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls given on their lateral surfaces and bases respectively. In addition, the quasilinear boundary conditions depend on parameters ε, α, β and the thin cylinders from each level are ε-periodically alternated. Using the Buttazzo–Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis (as ε → 0) of these problems are made for different values of α and β and different kinds of controls. We have showed that there are three qualitatively different cases. Application for an optimal control problem involving a thick one-level junction with cascade controls is presented as well.

This paper investigates American puts on a dividend-paying underlying whose volatility is a function of both time and underlying asset price. The asymptotic behaviour of the critical price near expiry is deduced by means of singular perturbation methods. It turns out that if the underlying dividend is greater than the risk-free interest rate, the behaviour of the critical price is parabolic, otherwise an extra logarithmic factor appears, which is similar to the constant volatility case. The results of this paper complement numerical approaches used to calculate the option values and the optimal exercise price at times that are not close to expiry.

In this note, we consider a nonlinear diffusion equation with a bistable reaction termarising in population dynamics. Given a rather general initial data, we investigate itsbehavior for small times as the reaction coefficient tends to infinity: we prove ageneration of interface property.

In this article, impedance control of a two link flexible link manipulators is addressed. The concept of impedance control of flexible link robots is rather new and is being addressed for the first time by the authors. Impedance Control provides a universal approach to the control of flexible robots, in both constrained and unconstrained maneuvers. The initial part of the paper concerns the use of Hamilton's principle to derive the mathematical equations governing the dynamics of joint angles, vibration of the flexible links and the constraining forces. The approximate elastic deformations are then derived by means of the Assumed-Mode-Method (AMM). Using the singular perturbation method, the dynamic of the manipulator is decomposed into fast and slow subsystems. The slow dynamic corresponds to the rigid manipulator and the fast dynamic is due to vibrations of flexible links. The sliding mode control (SMC) theory has been used as the means to achieve the 2nd order target impedance for the slow dynamics. A controller based on state feedback is also designed to stabilize the fast dynamics. The composite controller is constructed by using the slow and fast controllers. Simulation results for a 2-DOF robot in which only the 2nd link is flexible confirm that the controller performs remarkably well under various simulation conditions.

A novel composite control scheme for a manipulator with flexible links and joints is presented that uses the singular perturbation technique (SPT) to divide the manipulator dynamics into reduced order slow and fast subsystems. A neural network controller is then applied for the slow subsystem and a state-feedback H∞ controller for the fast subsystem. Results are presented that demonstrate improved performance over an alternative SPT-based controller that uses inverse dynamics and LQR controllers.