Principle of optimality

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. 

Bellman s (1957) principle of optimality An optimal policy has the property that, whatever the initial state and the initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. ... 

A well-known approach to the principle of optimization was first scribbled centuries ago on the walls of an ancient Roman bathhouse in connection with a choice between two aspirants for emperor of Rome. It read— De doubus malis, minus est semper aligendum —of two evils, always choose the lesser. 

J. Nielsen, Principles of optimal metabolic network operation. Mol. Syst. Biol. 3, 126 (2007). 

For readers with no prior knowledge of optimization methods In the textbook of Box et.al. [14] the basic principles of optimization are also explained. The sequential simplex method is presented in Walters et.al. [20]. Multi-criteria optimization is presented in Chapter 4 on an introductory level. For those readers who want to know more about multicriteria optimization, see the references given in Section 1.3.4 and Chapter 4. 

As depicted in Figure 3.11, the principles of optimization (ALARA) and dose limitation embodied in the radiation paradigm may be thought of as defining a top-down approach to management of stochastic risks. Given that radiation exposures have been justified, the radiation paradigm has two basic elements ... 

Chapter 4 introduced the essential principles of optimal control. Here we describe a number of applications to the control of molecular processes. 

Though the estimates that illustrate the principle of optimal motion are based on simplified models and approximation formulas (2.44 - 2.46), they have, nevertheless, made it possible to drew some conclusions which are apparently useful in taking into account the treatment of complex elementary acts of chemical and biochemical processes (Likhtenshtein, 1988a). 

Mayne et al. (1998) provide a quote from Lee and Markus (1967, p. 423) which essentially describes the MFC algorithm. Nour-Eldm (1971, p. 41), among others, explicitly describes the on-line constrained optimization idea, starting from the principle of optimality in dynamic programming ... 

Rawlings and Muske (1993) have shown that this idea can be extended to unstable processes. In addition to guaranteeing stability, their approach provides a computationally efficient method of on-line implementation. Their idea is to start with a finite control (decision) horizon but an infinite prediction (objective function) horizon, i.e., m < < and p = , and then use the principle of optimality and results from optimal control theory to substitute the infinite prediction horizon objective by a finite prediction horizon objective plus a terminal penalty term of the form... 

In this chapter we shall describe the principle of optimality in its general bearing on chemical processes, give a review of cognate work, and sketch its outline. 

The principle of optimality thus brings a vital organization into the search for the optimal policy of a multistage decision process. Bellman (1957) has annunciated in the following terms ... 

This is the feed state of the subsequent R — 1) stages which, according to the principle of optimality, must use an optimal R — l)-stage policy with respect to this state. This will result in a value/R i(pR) of the objective functipn, and when qR is chosen correctly this will give/r(Pr+i), the maximum of the objective function. Thus... 

To extract the optimal P-stage policy with respect to the feed state pR+i, we enter section P of this table at the state pR+i and find immediately from the last column the maximum value of the objective function. In the third column is given the optimal policy for stage P, and in the fourth, the resulting state of the stream when this policy is used. Since by the principle of optimality the remaining stages use an optimal (P — 1)-stage pohcy with respect to pR, we may enter section (P — 1) of the table at this state pR and read off the optimal... 

Then the principle of optimality yields the functional equation... 

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