Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations

A Chebyshev collocation method, an expansion method, has been proposed in order to solve the systems of higher-order linear integro-differential equations. This method transforms the IDE system and the given conditions into the matrix equations via Chebyshev collocation points. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Chebyshev coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the systems of differential and integral equations.     

Integraph solution of differential equations

In a previous article¹ a continuously recording integraph was described, by means of which differential equations, involving only one integration, could be solved. The present article describes a revision of this machine such that an equation involving two successive integrations, corresponding to practically any second-order total differential equation, with all terminal conditions included, can be solved. The need for a workable means of solving the differential equations involving empirical and discontinuous coefficients which occur repeatedly in electrical engineering and physics is recalled. In the machine described such solutions are effected by means of suitable interlinked integrating devices, the result being plotted continuously as a function of the independent variable. Tests and simple solutions show the over-all error to be approximately 1 or 2 per cent. The various sources of this error are discussed.     

Mechanical solution of algebraic equations

The paper describes a machine for determining the real and complex roots of higher-degree algebraic equations. The principle of operation is found in the correspondence between sine wave quantities and complex numbers. The particular machine is designed for equations of the eighth degree, and finds all the roots with engineering accuracy in the space of a few minutes.Though designed primarily with a view to determining the indicial admittance of electric networks, the machine should find utility in other fields of applied mathematics as well.     

Existence of solutions for nonlinear fractional q-difference integral equations with two fractional orders and nonlocal four-point boundary conditions

We investigate the existence of solutions for a nonlinear fractional q-difference integral equation (q-variant of the Langevin equation) with two different fractional orders and nonlocal four-point boundary conditions. Our results are based on some classical fixed point theorems. An illustrative example is also presented.     

Output feedback stabilization of high-order nonlinear systems with polynomial nonlinearity

In this paper, the problem of output feedback stabilization for high-order nonlinear systems with more general low-order and high-order nonlinearities multiplied by a polynomial-type output-dependent growth rate is studied. By constructing the novel Lyapunov function and observer, based on the homogeneous domination and adding a power integrator methods, an output feedback controller is developed to guarantee that the equilibrium of the closed-loop system is globally uniformly asymptotically stable.     

Sliding mode filtering for stochastic systems with polynomial state and observation equations

In this paper, the mean-square and mean-module filtering problems for polynomial system states over polynomial observations are studied proceeding from the general expression for the stochastic Ito differentials of the estimate and the error variance. The paper deals with the general case of nonlinear polynomial states and observations. As a result, the Ito differentials for the estimates and error variances corresponding to the stated filtering problems are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the sliding mode filtering equations for any polynomial state over observations with any polynomial drift is then established. In the examples, the obtained sliding mode filters are applied to solve the third-order sensor filtering problems for a quadratic state, assuming a conditionally Gaussian initial condition for the extended second-order state vector. The simulation results show that the designed sliding mode filters yield reliable and rapidly converging estimates.     

Local stabilization of coupled nonlinear parabolic equations by boundary control

This paper considers local stabilization of a boundary control system coupled by nonlinear parabolic equations. Based on backstepping approach, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Local exponential stabilization of the closed loop is established. A system with three coupled nonlinear parabolic equations is simulated, which show that the obtained feedback control law is feasible.     

Analytical and numerical stability of nonlinear neutral delay integro-differential equations

In this paper, we are concerned with the analytical and numerical stability of nonlinear neutral delay integro-differential equations (NDIDEs). First, sufficient conditions for the analytical stability of nonlinear NDIDEs with a variable delay are derived. Then, we show that any A-stable linear multistep method can preserve the asymptotic stability of the analytical solution for nonlinear NDIDEs with a constant delay. At last, we validate our conclusions by numerical experiments.     

Null controllability of some nonlinear degenerate 1D parabolic equations

The main goal of the present paper is twofold: (i) to establish the well-posedness of a class of nonlinear degenerate parabolic equations and (ii) to investigate the related null controllability and decay rate properties. In a previous step, we consider an appropriate regularized system, where a small parameter α is involved. More precisely, the usual nonlinear term b(x)uu_x is replaced by b(x)zu_x, where $z={(\text{Id}.?{\alpha }^{2}A)}^{?1}u$ and A is a Poisson–Dirichlet operator. We investigate the behavior of the null controls and their associated states as α → 0.     

Solution of linear state-space equations and parameter estimation in feedback systems using laguerre polynomial expansion

This paper analyzes the application of Laguerre polynomial expansion to linear systems. It can be applied to the solution of linear state equations by using an algebraic matrix to determine the coefficients of the Laguerre expansion. It also can be applied to system identification by using the expansion to determine the coefficients in the transfer function. Examples are given to demonstrate the accuracy of finite order expansion by Laguerre polynomials.     

Output feedback tracking control for polynomial nonlinear systems with measurement noises and mismatched disturbances

In this paper, the output feedback tracking control problem is investigated for polynomial nonlinear systems (PNSs) with measurement noises and mismatched disturbances. First, in order to suppress measurement noises, a polynomial observer is introduced to simultaneously estimate states and mismatched disturbances. Next, based on the idea of backstepping control, a novel output feedback controller is designed for PNSs to compensate mismatched disturbances. Command filters are employed to avoid the repeated derivatives of virtual control and measurement noises in the recursive controller design. Then, a sufficient condition in terms of the parameter-dependent linear matrix inequality (PDLMI) is derived to guarantee the boundedness of tracking errors and estimation errors. By utilizing the sum of squares (SOS) decomposition technique, the PDLMI is solved to obtain desired controller parameters. Finally, an example of dynamic point-the-bit rotary steerable drilling tool system is performed to demonstrate the effectiveness and feasibility of the proposed strategy.     

Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems

This paper presents the optimal control problem for a non-linear polynomial system with respect to a Bolza-Meyer criterion with a non-quadratic non-integral term. The optimal solution is obtained as a sliding mode control, whereas the conventional polynomial-quadratic regulator does not lead to a causal solution and, therefore, fails. Performance of the obtained optimal controller is verified in the illustrative example against the conventional polynomial-quadratic regulator that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode control.