Controlled Nonlinear Stochastic Delay Equations: Part II: Approximations and Pipe-Flow Representations

This is the second part of a work dealing with key issues that have not been addressed in the modeling and numerical optimization of nonlinear stochastic delay systems. We consider new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. Part I was concerned with issues concerning the class of admissible controls and their approximations, since the classical definitions are inadequate for our models. This part is concerned with transportation equation representations and their approximations. Such representations of nonlinear stochastic delay models have been crucial in the development of numerical algorithms with much reduced memory and computational requirements. The representations for the new models are not obvious and are developed. They also provide a template for the adaptation of the Markov chain approximation numerical methods.     

Numerical methods for controls for nonlinear stochastic systems with delays and jumps: applications to admission control

We consider numerical methods of the Markov chain approximation type for computing optimal controls and value functions for systems governed by nonlinear stochastic delay equations. Earlier work did not allow Poisson random measure driving processes or delays that are concentrated on points with positive probability. In addition, the Poisson measures can be controlled. Previous proofs are not adequate for the present case. The algorithms are developed and convergence proved as the approximating parameters go to their limits. One motivating example concerns admissions control to a network, where the file arrival process is governed by a Poisson process, and arrivals might be admitted or not, according to the control, which leads to a controlled Poisson process. Numerical data for such an example are presented. The original problem is recast in terms of a transportation equation, which allows the development of practical algorithms. For the problems of interest, alternative methods can entail prohibitive memory and computational requirements.     

Topological methods for the global controllability of nonlinear systems

Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.     

Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems—Strong convergence of optimal controls

A class of optimal control problems for a parabolic equation with nonlinear boundary condition and constraints on the control and the state is considered. Associated approximate problems are established, where the equation of state is defined by a semidiscrete Ritz-Galerkin method. Moreover, we are able to allow for the discretization of admissible controls. We show the convergence of the approximate controls to the solution of the exact control problem, as the discretization parameter tends toward zero. This result holds true under the assumption of a certain sufficient second-order optimality condition.Dedicated to the 60th birthday of Lothar von Wolfersdorf     

Optimal control problem with phase constraints for a linear system of equations with delay

The optimal control problem for a linear system of differential equations with delay for which the initial function on the initial set is the control function is considered. Phase constraints with variable times are imposed on the trajectory of the object. Necessary optimality conditions of controls in admissible classes are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1647–1652, December, 1991.     

Robust guaranteed cost control for uncertain linear differential systems of neutral type

In this paper, the robust guaranteed cost control problem for a class of uncertain linear differential systems of neutral type with a given quadratic cost functions is investigated. The uncertainty is assumed to be norm-bounded and time-varying nonlinear. The problem is to design a state feedback control laws such that the closed-loop system is robustly stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainty and time delay. A criterion for the existence of such controllers is derived based on the matrix inequality approach combined with the Lyapunov method. A parameterized characterization of the robust guaranteed cost controllers is given in terms of the feasible solutions to the certain matrix inequalities. A numerical example is given to illustrate the proposed method.     

On the Control of a Nonlinear Dynamic System in a Time-Optimal Problem with State Constraints

An optimal control problem for a nonlinear dynamic system is studied. The required control must satisfy given constraints and provide the fulfilment of a number of conditions on the current state of the system. For the construction of admissible controls in this problem, we propose an approach based on the ideas of solution of control problems with a guide. The results of numerical simulation are presented.     

Constrained LQR problems in elliptic distributed control systems with point observations—convergence results

In this paper constrained LQR problems in distributed control systems governed by the elliptic equation with point observations are studied. A variational inequality approach coupled with potential theory in a Banach space setting is adopted. First the admissible control set is extended to be bounded by two functions, and feedback characterization of the optimal control in terms of the optimal state is derived; then two numerical algorithms proposed in [5] are modified, and the strong convergence and uniform convergence in Banach space are proved. This verifies that the numerical algorithm is insensitive to the partition number of the boundary. Since our control variables are truncated below and above by two functions inL ^p and in our numerical computation only the layer density not the control variable is assumed to be piecewise smooth, uniform convergence guarantees a better convergence. Finally numerical computation for an example is carried out and confirms the analysis. This research was supported in part by NSF Grant DMS-9404380 and by an IRI Award of Texas A&M University. The current address of the first author is the Department of Mathematical Science, University of Nevada at Las Vegas, Las Vegas, NV 89154, USA.     

Adaptive estimation of linear functionals in functional linear models

We consider the estimation of the value of a linear functional of the slope parameter in functional linear regression, where scalar responses are modeled in dependence of randomfunctions. In Johannes and Schenk [2010] it has been shown that a plug-in estimator based on dimension reduction and additional thresholding can attain minimax optimal rates of convergence up to a constant. However, this estimation procedure requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter based on a combination of model selection and Lepski??s method, which is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning parameter is selected as theminimizer of a stochastic penalized contrast function imitating Lepski??smethod among a random collection of admissible values. We show that this adaptive procedure attains the lower bound for the minimax risk up to a logarithmic factor over a wide range of classes of slope functions and covariance operators. In particular, our theory covers pointwise estimation as well as the estimation of local averages of the slope parameter.     

Hopf bifurcation analysis in a fluid flow model of Internet congestion control algorithm

This paper focuses on the Hopf bifurcation analysis of some classes of nonlinear time-delay models, namely fluid flow models, for the Internet congestion control algorithm of TCP/AQM networks. Using tools from control and bifurcation theory, it is proved that there exists a critical value of communication delay for the stability of the network. When the delay passes through the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the stability of the bifurcation and direction of the bifurcating periodic solutions are determined by applying the normal form theory and the center manifold theorem. Finally, some numerical examples are given to verify the theoretical analysis.     

Numerical Approximation Of The Phase-Field Transition System With Non-Homogeneous Cauchy-Neumann Boundary Conditions In Both Unknown Functions Via Fractional Steps Method

The paper concerns with the proof of the convergence for an iterative scheme of fractional steps type associated to the phase-field transition system endowed with non-homogeneous Cauchy-Neumann boundary conditions, in both unknown functions. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic system. On the basis of this approach, a numerical algorithm in 2D case is introduced and an industrial implementation is made.     

On the computation of linearly constrained stationary points

We study a numerical method for the computation of linearly constrained stationary points. The proposed method can be interpreted as a projected gradient method with constant stepsize in which one allows perturbations in the admissible set and controls these perturbations in each iteration. The method is applicable to some classes of overdetermined problems to which the projected gradient method may not be directly applicable. Illustrative numerical examples are given.     

Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

The porous medium equation (PME)is a typical nonlinear degenerate parabolic equation. We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al., J. Comput. Phys., 385 (2019), pp. 13–32], where the trajectory equation can be obtained and two numerical schemes have been developed based on different dissipative energy laws. It is also proved that the nonlinear scheme, based on $f$log $f$ as the total energy form of the dissipative law, is uniquely solvable on an admissible convex set and preserves the corresponding discrete dissipation law. Moreover, under certain smoothness assumption, we have also obtained the second order convergence in space and the first order convergence in time for the scheme. In this paper, we provide a rigorous proof of the error estimate by a careful higher order asymptotic expansion and two step error estimates. The latter technique contains a rough estimate to control the highly nonlinear term in a discrete $W$^1,∞norm and a refined estimate is applied to derive the optimal error order.     

Extrapolation spaces and controllability of impulsive semilinear functional differential inclusions with infinite delay in Fréchet spaces

In this article, we investigate some classes of semilinear impulsive functional differential inclusions with infinite delay. It is assumed that the linear part is possibly neither densely defined nor it satisfies the Hille–Yosida condition on a Banach space, namely the extrapolated space. Our approach is based on the theory of extrapolation spaces combined with a recent Frigon nonlinear alternative for multivalued admissible contractions in Fréchet spaces.     

Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem

We study the numerical approximation of distributed nonlinear optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Our main result are error estimates for optimal controls in the maximum norm. Characterization results are stated for optimal and discretized optimal control. Moreover, the uniform convergence of discretized controls to optimal controls is proven under natural assumptions.     

LDG method for reaction-diffusion dynamical systems with time delay

In this paper we introduce a local discontinuous Galerkin method to solve nonlinear reaction-diffusion dynamical systems with time delay. Stability and convergence of the schemes are obtained. Finally, numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method.     

Trajectory sensitivity analysis of hybrid systems with sliding motion

This paper concerns hybrid control systems exhibiting the sliding motion. It is assumed that the system’s motion on the switching surface is described by index-2 differential–algebraic equations (DAEs), which guarantee the accurate tracking of the sliding motion surface. For those systems the sensitivity analysis is performed with the help of solutions to system’s linearized equations. The paper states conditions under which the solutions to the linearized equations for original DAEs and the solutions to linearized equations for underlying ordinary differential equations (ODEs) exhibit similar properties. Due to the presence of sliding motion, we restrict the class of admissible control functions to piecewise differentiable functions. The presented sensitivity analysis might be useful in deriving the weak maximum principle for optimal control problems with hybrid systems exhibiting sliding motion and in establishing the global convergence of algorithms for solving those problems.     

Operator splitting for dissipative delay equations

We investigate Lie–Trotter product formulae for abstract nonlinear evolution equations with delay. Using results from the theory of nonlinear contraction semigroups in Hilbert spaces, we explain the convergence of the splitting procedure. The order of convergence is also investigated in detail, and some numerical illustrations are presented.     

What Monte Carlo models can do and cannot do efficiently?

The question “what Monte Carlo models can do and cannot do efficiently” is discussed for some functional spaces that define the regularity of the input data. Data classes important for practical computations are considered: classes of functions with bounded derivatives and Hölder type conditions, as well as Korobov-like spaces.

Theoretical performance analysis of some algorithms with unimprovable rate of convergence is given. Estimates of computational complexity of two classes of algorithms – deterministic and randomized for both problems – numerical multidimensional integration and calculation of linear functionals of the solution of a class of integral equations are presented.     

A preconditioned descent algorithm for variational inequalities of the second kind involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian operator

This paper is concerned with the numerical solution of a class of variational inequalities of the second kind, involving the p-Laplacian operator. This kind of problems arise, for instance, in the mathematical modelling of non-Newtonian fluids. We study these problems by using a regularization approach, based on a Huber smoothing process. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to the solution of the original problem is verified. We propose a preconditioned descent method for the numerical solution of these problems and analyze the convergence of this method in function spaces. The existence of admissible descent directions is established by variational methods and admissible steps are obtained by a backtracking algorithm which approximates the objective functional by polynomial models. Finally, several numerical experiments are carried out to show the efficiency of the methodology here introduced.