Riccati equation

About: Riccati equation is a research topic. Over the lifetime, 10428 publications have been published within this topic receiving 210015 citations. The topic is also known as: Riccati's differential equation.

Application of Variational Iteration Method to a General Riccati Equation

Abstract: In this paper, the variational iteration method (VIM) is applied to the solution of general Riccati differential equations. The equations under consideration includes one with variable coefficient and one in matrix form. In VIM, a correction functional is constructed by a general Lagrange multiplier which can be identified via a variational theory. The VIM yields an approximate solution in the form of a quickly convergent series. Comparisons with exact solution and the fourth-order RungeKutta method show that the VIM is a powerful method for the solution of nonlinear equations. The present paper may be a suitable and fruitful exercise for teaching and better understanding techniques in advanced undergraduate courses on classical mechanics.

Some Chandrasekhar-type algorithms for quadratic regulators

Abstract: The by-now classical method for the quadratic regulator problem is based on the solution of an n × n matrix nonlinear Riccati differential equation, where n is the dimension of the state-vector. Care has to be exercised in numerical solution of the Riccati equation to ensure nonnegative-definiteness of its solution, from which the optimum m × n feedback gain matrix K(?) is calculated by a further matrix multiplication. For constant-parameter systems, we present a new algorithm that requires only the solution of n(m + p) simultaneous equations: the nm elements of the feed-back gain matrix K(?) and the np elements of a rank-p square-root of the derivative of P(?), where p is the rank of the nonnegative-definite weighting matrix Q that measures the contribution of the state trajectory to the cost functional. If n is large compared with p and m, our algorithm can provide considerable computational savings over direct solution of the Riccati equation, where n(n + 1)/2 simultaneous equations have to be solved. Also the square-root feature means that with reasonable care the automatic nonnegative-definiteness of the derivative matrix-P(?) can be carried over to P(?) itself. Similar results can be obtained for indefinite Q matrices, but with n(m + 2p) equations rather than n(m + p). The equations of our algorithm have the same form as certain famous equations introduced into astrophysics by S. Chandrasekhar, which explains our terminology. The method used in the paper can also be applied to Lyapunov differential equations, as discussed in an Appendix, and to the linear least-squares estimation of stationary processes, as discussed elsewhere.

Properties of the solutions of rational matrix difference equations

Abstract: We prove a comparison theorem for the solutions of a rational matrix difference equation, generalizing the Riccati difference equation, and existence and convergence results for the solutions of this equation. Moreover, we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing or almost stabilizing solution.

Theory of chiral multilayers

Abstract: We investigate reflection from and transmission through chiral multilayers with discrete and continuous variations in material characteristics. Both boundary-value and initial-value approaches are used. The S-parameter matrix and associated copolarized and cross-polarized reflection and transmission coefficients are derived from the chiral constitutive relations, Maxwell’s equations, and boundary conditions. A generalized matrix Riccati equation is found for the reflection and transmission coefficients of arbitrary chiral multilayers by using an initial-value approach and Ambarzumian’s principle of invariant embedding. All results are exact and applicable to both normal and oblique incidence. Special emphasis is given to the physical principles involved, to special cases, and to salient features.

Stochastic Control for Linear Systems Driven by Fractional Noises

Abstract: This paper is concerned with optimal control of stochastic linear systems involving fractional Brownian motion (FBM). First, as a prerequisite for studying the underlying control problems, some new results on stochastic integrals and stochastic differential equations associated with FBM are established. Then, three control models are formulated and studied. In the first two models, the state is scalar-valued and the control is taken as Markovian. Either the problems are completely solved based on a Riccati equation (for model 1, where the cost is a quadratic functional on state and control variables) or optimality is characterized (for model 2, where the cost is a power functional). The last control model under investigation is a general one, where the system involves the Stratonovich integral with respect to FBM, the state is multidimensional, and the admissible controls are not limited to being Markovian. A new Riccati-type equation, which is a backward stochastic differential equation involving both FBM and normal Brownian motion, is introduced. Optimal control and optimal value of the model are explicitly obtained based on the solution to this Riccati-type equation.