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Our aim through this paper is to describe a Krylov based projection method in order to reduce high-order dynamical systems. We focus on differential algebraic equations (DAEs) of index-2 that arise from spatial discretization of Stokes equations. An efficient algorithm based on a projection technique onto an extended block Krylov subspace that appropriately allows us to construct a reduced order system is described. Numerical results are provided to confirm the performance of the derived method compared with other known ones. [ABSTRACT FROM AUTHOR]

• A novel online mode-free integral reinforcement learning algorithm is proposed to solve the mutiplayer non-zero sum games. • The online learning is used to compute the corresponding N coupled algebraic Riccati equations. • The policy iterative algorithm is applied to solve the coupled algebraic Riccati equations corresponding to the multiplayer nonzero sum games. In this paper, a novel online mode-free integral reinforcement learning algorithm is proposed to solve the multiplayer non-zero sum games. We first collect and learn the subsystems information of states and inputs; then we use the online learning to compute the corresponding N coupled algebraic Riccati equations. The policy iterative algorithm proposed in this paper can solve the coupled algebraic Riccati equations corresponding to the multiplayer non-zero sum games. Finally, the effectiveness and feasibility of the design method of this paper is proved by simulation example with three players. [ABSTRACT FROM AUTHOR]

This paper is concerned with a M -matrix algebraic Riccati equation (MARE) X D X − A X − X B + C = 0 for which A is block-diagonal and its defining matrix W = B − D − C A is a nonsingular or irreducible singular M -matrix. Such an MARE can be decomposed into many coupled algebraic Riccati equations (AREs) that can be solved by the Jacobi- or Gauss–Seidel-like iteration updating scheme at the outer-loop while by a doubling algorithm in the inner loop for each coupled ARE, as first proposed by Meini (2013). The goals of this paper are two-fold. One is to resolve a critical technical detail in Meini's algorithm that was not addressed. It is about whether each ARE in the inner loop has a minimal nonnegative solution. It is proved that the defining matrix of each coupled ARE during a doubling iteration is indeed a nonsingular or irreducible singular M -matrix and, as a result, they do have minimal nonnegative solutions and a doubling algorithm is an efficient way to compute them. The other goal is to design a highly accurate implementation of the doubling algorithm for the inner loop so that all entries of the minimal nonnegative solution to the original MARE are calculated with high entrywise relative accuracies, regardless of their magnitudes. This is made possible by a novel way of constructing triplet representations for the coupled ARE during doubling iterations. Numerical examples are presented to demonstrate that the resulting algorithm can indeed deliver an entrywise relatively accurate solution. [ABSTRACT FROM AUTHOR]

• For optimal control, the stochastic information with current iterative results is used to solve the stochastic algebraic Riccati equation, which has stochastic characteristics. • Calculate the increment instead of the value itself to improve the calculation accuracy. • Newton's method with line search solve the potentially disastrous problem of the first Newton step. • The convergence and even quadratic convergence of the proposed incremental Newton's iterative algorithm have been discussed. In this paper, a novel incremental Newton's iterative algorithm for investigating the optimal control problem of Itô stochastic systems is presented. Newton's method is employed under the Fréchet derivative framework to iteratively solve a stochastic algebraic Riccati equation. Under moderate conditions, the convergence and even quadratic convergence of the proposed incremental Newton's iterative algorithm are discussed, respectively. In addition, the Newton's method is extended to the one with linear search. In the end, numerical results are given to demonstrate the effectiveness and superiority of the proposed algorithms. [ABSTRACT FROM AUTHOR]

• A novel numerical approach for solving SIVIE has been proposed first time ever. • Operational matrices are used to transform the SIVIE into a system of algebraic equations. • Resultant system of algebraic equations has been solved by the collocation method. • This technique has the advantage of being simple to implement. • Error estimate, convergence, and stability analysis are also established. In this paper, a numerical method is implemented to solve the stochastic Itô-Volterra integral equations. In this approach, operational matrices have been applied to reduce the stochastic Itô-Volterra integral equations to linear algebraic equations. Then collocation method is applied to solve the algebraic equations. The error, convergence, and stability analysis of the proposed method are discussed. Also, the steps of the proposed method have been presented in the form of an algorithm. Numerical examples are introduced to confirm the efficiency and reliability of the proposed scheme. [ABSTRACT FROM AUTHOR]