We consider multi-grid, or more appropriately, multi-level techniques for the numerical solution of operator Lyapunov and algebraic Riccati equations. The Riccati equation, which is quadratic, plays an essential role in the solution of linear-quadratic optimal control problems. The linear Lyapunov equation is important in the stability theory for linear systems and its solution is the primary s...

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and Riccati mapping method, we can acquire solutions of SBLMPE. Since utilized to explain incompressible liquid in fluid mechanics, acquired may be applied lot fascinating physical phenomena. To address how motion effects...

In the study of nonlinear physics, the search of new exact solutions of nonlinear evolution equations (NEEs) is one of the most important problems. Various methods for obtaining exact solutions to NEEs have been proposed, such as the Lie group method of infinitesimal transformations [1], the nonclassical Lie group method [2], the Clarkson and Kruskal direct method (CK) [3 – 5], the conditional ...

We consider multi-grid, or more appropriately, multi-level techniques for the numerical solution of operator Lyapunov and algebraic Riccati equations. The Riccati equation, which is quadratic, plays an essential role in the solution of linear-quadratic optimal control problems. The linear Lyapunov equation is important in the stability theory for linear systems and its solution is the primary s...

This paper presents a simple and efficient method for determining the solution of Riccati differential equation with coefficients rational. In case the differential Galois group of the differential equation (E l) : y = ry, r ∈ C(x) is reducible, we look for the rational solutions of Riccati differential equation θ + θ 2 = r, by reducing the number of check to be made and by accelerating the sea...

In this paper, we study the linear quadratic Nash games for infinite horizon singularly perturbed systems. In order to solve the problem, we must solve a pair of cross–coupled algebraic Riccati equations with a small positive parameter ε. As a matter of fact, we propose a new algorithm, which combines Lyapunov iterations and the generalized Lyapunov equation direct method, to solve the cross–co...

In this paper based on a system of Riccati equations with variable coefficients, we presented a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccat equation with constant coefficient expansion method to constr...

In this paper, asymmetric Algebraic Riccati Equations are analyzed. We derive a new parametrization of the set of solutions, which may be viewed as alternative to the classical one based on Lagrangian subspaces of a “pseudo-Hamiltonian” matrix. Generalizing the symmetric case, the proposed parametrization is given in terms of pairs of invariant subspaces of two related “feedback” matrices. Exte...