State vectors Jacobi

The Jacobi state vectors without diagonalization of the Jacobi matrix... 

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. 

This is an ordinary eigenvalue problem in which the tridiagonal Jacobi matrix Jxj is given in Eq. (60) with M = oo. The residues dk) are defined by Eq. (15), where IT ) is the exact complete state vector normalized to Co 0. The same type of definition for dk is valid for an approximation such as Eq. (67), provided that normalization is properly included according to Eq. (69) ... 

This recurrence relation and the available triple set an, fin uk] are sufficient to completely determine the state vector Q without any diagonalization of the associated Jacobi matrix U = c0J, which is given in Eq. (60). Of course, diagonalization of J might be used to obtain the eigenvalues uk, but this is not the only approach at our disposal. Alternatively, the same set uk is also obtainable by rooting the characteristic polynomial or eigenpolynomial from... 

Let now the model be nonlinear. Then the Jacobi matrix depends on the unknown vector z and the reconcilation consists of a number of steps, say of a sequence of approximations z " if the sequence converges then the limit value, say z, represents a point on the solution manifold M, thus an estimate of the actual value of the state vector. So as to have an a priori idea of what can be expected, one can proceed as follows. 

We suppose that the state vector z can take its values in some N-dimensional interval Vet/ where ll is the admissible region (8.5.8). The interval can be assessed as some neighbourhood of a vector Zq e fSf. A first information can be obtained in the same manner as above, in the linear case. Taking different Zq e we can examine the behaviour of the Jacobi matrix Dg(Zo) on fW (restricted to t thus on r U). [We can also, in the case of balance models, start from different values of the independent parameters representing the degrees of freedom and determining Zq e fW see Sections 8.2 and 8.3. But such procedure may be rather tedious.] In the reconciliation, however, also the behaviour of Dg(z) in a neighbourhood of the solution manifold is relevant. 

Details of the method outlined below can be found i n [5], A stochastic system of several extensive variables X. is supposed tc be described by a master equation which can be explicitly written when the transition probabilities per unit time W( Xj Xj ) are known. In a reaction diffusion system, X. may be the number of chemical species a in a cell located by the vector r and is denoted by X. = X. Introducing the toaka tic, pot ntiai U defined by P = exp(-S - N U), where P is the probability, N is proportional to the total volume of the system and S stands for the normalization factor, we switch to the quasicontinu-ous intensive variables x = X /N, where N may be the mean number of particles in one cell of a reaction-diffusion system. If we assume that for all states for which liJ ( X j -> X1 ) are nonnegligible, x - xj is much smaller than 1, the equation for U can be expressed, at the zeroth order in 1/N, in terms of xj and 3U/9xj. liie thus obtain a Hamilton Jacobi type of equation ... 

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