A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods

The solution of the Riccati differential equation (RDE) is shown to be asymptotically close to the solution of the projection Riccati differential equation (PRDE). The asymptotic behaviour of the latter is analyzed in an explicit formula. The almost-periodic asymptote of the solution of the PRDE is computed by an algorithm based upon the concepts of an aperiodic/almost-periodic generator (APG) decomposition of a linear map and unit row-staircase form of a polynomial matrix. The analysis ultimately provides a convergence criterion. In particular, it is shown that the solution of the PRDE always converges in the aperiodic case.

This paper is concerned with the effects of redundant control inputs on the quadratic optimal performance index in finite-time optimal control. Actually, the effects of redundant control input on quadratic performance index is equal to study the effects of the redundant control input on the solution of Riccati differential equation (RDE). Hence, some sufficient conditions are presented to strictly decrease the solution of RDE with the column of input matrix increased. Especially, a necessary and sufficient condition is derived to strictly decrease the solution of the homogeneous RDE after control input extensions. Moreover, one sufficient condition is obtained, which guarantees that the minimum control energy index strictly decreases with input extension. It is shown that redundant control inputs can improve the system performance based on these results. Some examples are taken to illustrate the effectiveness of related problems.

Convergence of the Time-Invariant Riccati Differential Equation towards Its Strong Solution for Stabilizable Systems

Solutions of symmetric Riccati differential equations (RDEs for short) are in the usual applications positive semidefinite matrices. Moreover, in the class of semidefinite matrices, solutions of different RDEs are also monotone, with respect to properly ordered data. Positivity and monotonicity are essential properties of RDEs. In Dieci and Eirola (1994), we showed that, generally, a direct discretization of the RDE cannot maintain positivity, and be of order greater than one. To get higher order, and to maintain positivity, we are thus forced to look into indirect solution procedures. Here, we consider the problem of how to maintain monotonicity in the numerical solutions of RDEs. Naturally, to obtain order greater than one, we are again forced to look into indirect solution procedures. Still, the restrictions imposed by monotonicity are more stringent that those of positivity, and not all of the successful indirect solution procedures of Dieci and Eirola (1994) maintain monotonicity. We prove that by using symplectic Runge-Kutta (RK) schemes with positive weights (e.g., Gauss schemes) on the underlying Hamiltonian matrix, we eventually maintain monotonicity in the computed solutions of RDEs.

Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method

The traditional reproducing kernel method (TRKM) cannot obtain satisfactory numerical results for solving the partial differential equation (PDE). In this study, for the first time, the abovementioned problems are solved by adaptive piecewise interpolation reproducing kernel method (APIRKM) to obtain the exact and approximate solutions of partial differential equations by means of series expansion using reconstructed kernel function. The highlight of this paper is to obtain more accurate approximate solution and save more time through adaptive discovery. Numerical solutions of the three examples show that the present method is more advantageous than TRKM.

Iterative reproducing kernel Hilbert spaces method for Riccati differential equations

Two data analytic research areas—penalized splines and reproducing kernel methods—have become very vibrant since the mid-1990s. This article shows how the former can be embedded in the latter via theory for reproducing kernel Hilbert spaces. This connection facilitates cross-fertilization between the two bodies of research. In particular, connections between support vector machines and penalized splines are established. These allow for significant reductions in computational complexity, and easier incorporation of special structure such as additivity.

In this study, an iterative approximation is proposed by using the reproducing kernel method (RKM) for the nonlinear advection equation. To apply the iterative RKM, specific reproducing kernel spaces are defined and their kernel functions are presented. The proposed method requires homogenising the initial or boundary conditions of the problem under consideration. After homogenising the initial condition of the advection equation, a linear operator selection is made, and then the approximate solution is constructed using orthonormal basis functions in serial form. Convergence analysis of the approximate solution is demonstrated through the lemma and theorem. Numerical outcomes are provided in the form of graphics and tables to show the efficiency and accuracy of the presented method.

Purpose: In this paper, a new algorithm is presented for solving one-dimensional parabolic partial differential equations subject to integral conditions. Methods: The algorithm is based on the transverse method of lines and the reproducing kernel method. The transverse method of lines can reduce a one-dimensional parabolic partial differential equation subject to integral conditions to a series of ordinary differential equations(ODEs) with integral boundary conditions. The reproducing kernel method is a relative new analytical technique, which can solve successfully ODEs with integral boundary conditions.

Rational solutions of Riccati differential equation with coefficients rational

Chang transformation is a decoupling technique for singularly perturbed linear systems. However, for linear slowly time-varying systems, construction of the transformation requires solutions of Differential Riccati Equation (DRE) and Differential Sylvester Equations (DSE). In this paper, through the construction of a contraction mapping for the remainders R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> and R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> , Theorem 1 provides iterative solutions in an interval of the singular perturbation parameter ε for the DRE and DSE, which are key steps for Chang transformation construction. Based on the iterative solutions, the concepts of pth-order approximated Chang transformation, decoupled system, pth-order approximated slow and fast systems are established, thereby facilitating the analysis in subsequent investigation on estimate of the Singular Perturbation Margin (SPM) for Linear Slowly Time-Varying systems and Nonlinear Slowly Time-Varying systems.

Generalized multiple scale reproducing kernel particle methods

Max-plus representation for the fundamental solution of the time-varying differential Riccati equation

In this study, iterative reproducing kernel method (RKM) will be applied in order to observe the effect of the method on numerical solutions of fractional order Boussinesq equation. Hilbert spaces and their kernel functions, linear operators and base functions which are necessary to obtain the reproducing kernel function are clearly explained. Iterative solution is constituted in a serial form by using reproducing kernel function. Then convergence of RKM solution is shown with lemma and theorem. Two problems, “good” Boussinesq and generalized Boussinesq equations, are examined by using RKM for different fractional values. Results are presented with tables and graphics.