Riccati differential equation

Equations (102)-(104) are the Riccati differential equations that have no solutions in quadratures for arbitrary jq and k2. They can, however, be easily solved numerically to obtain the desired second moment of the distribution (and hence the weight average degree of polymerization) as a function of time, according to Eq. (101). 

This equation has the form of the Riccati differential equation with electrode potential dependent coefficients (see Appendix) ... 

This equation needs to be solved by reducing the quadratic expression by means of the Riccati differential equation (see Chapter 2) and then making it exact through an appropriate integrating factor, x(y). The problem is the exponential term that multiplies the first derivate, which is a function affecting the solution of (15.54). 

Consider the reflection of a normally incident time-harmonic electromagnetic wave from an inhomogeneous layered medium of unknown refractive index n(x). The complex reflection coefficient r(k,x) satisfies the Riccati nonlinear differential equation [2] ... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

It is straightforward to see that the superpotential nonlinear first-order differential equation, a Riccati equation. 

Chapter 5 considers optimal regulator control problems. The Kalman linear quadratic regulator (LQR) problem is developed, and this optimal multivariable proportional controller is shown to be easily computable using the Riccati matrix differential equation. The regulator problem with unmeasurable... 

Solving the above differential equations, P(t) and m,(t) are computed, and then from Eq. 26, the feedback matrix K(t) and the control force F(t) are calculated. Equation 24 is called Riccati equation ... 

---