Partial differential equations 22 INDEX

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. 

It is useful to regard (4.19) as a wave equation in which the term S = —p,od2PNL/dt2 acts as a source radiating in a linear medium of refractive index n. Because Pnl (and therefore S) is a nonlinear function of E, Equation (4.19) is a nonlinear partial differential equation in E. This is the basic equation that underlies the theory of nonlinear optics. 

In order to finite-difference the model partial differential equations, we need values of the state variables at discrete distances, yi,V2, --yN, and zi,Z2,...Zn, and at discrete times, 0, Ai, 2At,..., (n/ — l)At. Here N is the number of grid blocks along the quadrant boundar>% At is the time step, and n/ = t/fAt. The reservoir quadrant is therefore replaced by a system of grid blocks shown in Figure 8.37. The integer i is used as the index in the y direction, and the integer j as the index in the z direction. In addition, the index n is used to denote time. Hence pe use the following notation to identify a process variable... 

A simplified model of ignition and burning of polymers which docs not require one to work with partial differential equations has been proposed by Rychly [49]. It was applied to the combustion carried out in a mass loss cone calorimeter system [50]. A series of simulations of heat release rate curves was performed for polymers with intumescent additives [51]. The model was also used for the prediction of limiting oxygen index (LOI) [52],... 

The parameter n represents the index of the power law. Ui and W] are the averaged velocities due to the Poiseuille flow. For n=l, the Newtonian eases are obtained. The integrations are performed and substituted in the equation of motion and the equation of continuity. Hence, three non-linear partial differential equations are found. By... 

By a reactor model, we mean a system of equations (algebraic, ordinary, or partial differential, functional or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single, concentration-like variables, Uo and u. The relation of input and output will also depend on a set of parameters (such as Damkohler number. Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when w0 = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(jc)) and the lumped output is... 

We now return to Equation A2.3 and substitute into it Equation A2.4. We then obtain A2.7, where N is the total number of basis orbitals being used. The variation principle now has to be applied to A2.7 to find the values of the c s which will give the best iji s possible with the chosen basis. The energy is minimized simultaneously with respect to all the c s by carrying out a partial differentiation with respect to each c and making the derivatives of the energy satisfy A2.8. The result, after some manipulations, is a set of N equations of the form A2.9, where the index i takes a different value for each equation. 

On the framework of the detectability motion given in [5], the Jacobian matrix of ( ) is the observability matrix, and its non-singularity provides the robust partial observability of the reactor motion x(t), with observability index K = 1 and the differential equation in Eq. 6 is the unobservable dynamics whose unique solution is the rmobservable motion (Eq. 3a). 

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