Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

     

Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors

This paper considers the bilinear minimax control problem of an important class of parabolic systems with Robin boundary conditions. Such systems are linear on state variables when the control and disturbance are fixed, and linear on the control or disturbance when the state variables are fixed. The objective is to maintain target state variables by taking account the influence of noises in data, while a desired power level and adjustment costs are taken into consideration. Firstly we introduce some classes of bilinear systems and obtain the existence and the uniqueness of the solution, as well as stability under mild assumptions. Afterwards the minimax control problem is formulated. We show the existence of an optimal solution, and we also find necessary optimality conditions. Finally, to illustrate the abstract results, we present two examples of neutron fission systems.     

Stabilization of stationary affine control systems with discrete time

We obtain new sufficient conditions for the local and global asymptotic stabilization of the zero solution of a nonlinear affine control system with discrete time and with constant coefficients by a continuous state feedback. We assume that the zero solution of the free system is Lyapunov stable. For systems with linear drift, we construct a bounded control in the problem of global asymptotic state and output stabilization. Corollaries for bilinear systems are obtained.     

Controller design for stabilizing bilinear systems with state-based switching

Some nonlinear systems can be approximated by switching bilinear systems. In this paper, we proposed a method to design state-based stabilizing controller for switching bilinear systems. Based on the similarity between switching bilinear systems and switching linear systems, corresponding switching linear systems are obtained for switching bilinear systems by applying state-based feedback control laws. Instead, we consider asymptotically stabilizing the corresponding switching linear system through solving a number of relaxed LMI conditions. Stabilizing controllers for switching bilinear systems can be derived based on the results of the corresponding switching linear systems. The stability of the controller is proved step by step through the decreasing of the multiple Lyapunov functions along the state trajectory. The effectiveness of the method is demonstrated by both a theoretical example and an example of urban traffic network with traffic signals.     

On some solutions of Chapman-Kolmogorov equation for discrete-state Markov processes with continuous time

The nonlinear equation mentioned in the title is the basic one in the theory of Markov processes. In the case of a discrete-state process, its solution is given by the transition probability function. Usually, solving this equation amounts to solving a linear equation. In 1932, S. N. Bernstein posed the problem of direct determination of the solution. In 1961, such solutions were given in terms of bilinear series by O. V. Sarmanov for stationary continuous-state Markov processes. In 2007, several solutions were obtained by the author in terms of generalized bilinear series without placing Sarmanov’s restrictions. In this paper, our results are extended to discrete-state processes. Two solutions of the Chapman-Kolmogorov equation are derived by means of reducing it to some functional equation. The solutions are represented in the form of a bilinear sum and its generalizations, each term of the sum being proportional to the product of two orthogonal functions. The results obtained are illustrated by two-state processes, which exemplify the assertions derived in this paper. Another example is used to show that the Chapman-Kolmogorov equation has a solution which is devoid of probabilistic sense.     

Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation

A set of sufficient conditions consisting of systems of linear partial differential equations is obtained which guarantees that the Wronskian determinant solves the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form. Upon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield rational solutions, solitons, negatons, positons and interaction solutions.     

Bilinear Control and Application to Flexible a.c. Transmission Systems

The purpose of this paper is an integrated overview of bilinear systems (BLS) research which has evolved over the past few decades, and a new result on control of flexible a.c. transmission systems (FACTS) is presented. BLS may be derived in many cases from principles of physics, chemistry, biology, socioeconomics, and engineering. In other cases, BLS are more accurate approximations to nonlinear systems than are traditional linear systems, as shown for example by the added bilinear terms (in state and control) for the Taylor series.While an appropriately designed linear control system may be optimum relative to some quadratic performance index without added constraints, bilinear or parametric control can be designed to improve more global performance and indeed to increase the region of attainable states. Such controllability and stabilization of BLS and of a series line-capacitor controlled FACTS is presented.     

Pfaffian solutions to a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation and its modified counterpart

A class of exact Pfaffian solutions to a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation is obtained. A set of sufficient conditions consisting of systems of linear partial differential equations involving free parameters is generated to guarantee that the Pfaffian solves the equation. A Bäcklund transformation of the equation is presented. The equation is transformed into a set of bilinear equations, and a few classes of traveling wave solutions, rational solutions and Pfaffian solutions to the extended bilinear equations are furnished. Examples of the Pfaffian solutions are explicitly computed, and a few solutions are plotted.     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Necessary Optimality Conditions for Some Control Problems of Elliptic Equations with Venttsel Boundary Conditions

In this paper we derive a necessary optimality condition for a local optimal solution of some control problems. These optimal control problems are governed by a semi-linear Vettsel boundary value problem of a linear elliptic equation. The control is applied to the state equation via the boundary and a functional of the control together with the solution of the state equation under such a control will be minimized. A constraint on the solution of the state equation is also considered.