Trajectory optimization

To exemplify both aspects of the formalism and for illustration purposes, we divide the present manuscript into two major parts. We start with calculations of trajectories using approximate solution of atomically detailed equations (approach B). We then proceed to derive the equations for the conditional probability from which a rate constant can be extracted. We end with a simple numerical example of trajectory optimization. More complex problems are (and will be) discussed elsewhere [7]. 

Here we examine the control of migration in a periodically driven nonlinear oscillator. Our aim is to demonstrate that application of the approximate solution found from the statistical analysis of fluctuational trajectories optimizes (minimizes) the energy of the control function. We compare the performance of some known adaptive control algorithms to that of the control function found through our analysis. 

For simplicity, we restrict ourselves to having a feed flowrate that starts at some value F0 and ramps down with a constant slope S. This practical approach to trajectory optimization is recommended by Smith and Choong6 for batch processes. We want to find the values of F0 and S that achieve the desired conversion and selectivity. There will be many pairs of values that will satisfy the two criteria. Each will have a different batch time and a different amount of C produced. 

It is easy to perceive from the above example that optimal control involves optimization of an objective functional subject to the equations of change in a system and additional constraints, if any. Because of this fact, optimal control is also known as dynamic or trajectory optimization. 

Semibatch copolymerization is most often done in an attempt to maintain a reasonably constant copolymer composition when the comonomers are of widely varying reactivities [1]. Semibatching of initiator is often done to maintain temperature control in a heat transfer-limited kettle, and semibatch addition of initiator or chain transfer agent may be used to maintain a desired MWD. Quantitative strategies for semibatching may be developed through empirical experimentation at the bench or pilot scale, or, if accurate mathematical models are available, classical trajectory optimization techniques may be used [2,3]. 

Table 4.1 Summary of single-trajectory optimization algorithm performance.

In more complex reaction schemes, the terrain of the objective function will not point so directly to the optimal conditions. Thus, an expanded multi-trajectory optimization system is designed, allowing for the optimization of such a complex reaction system. This analysis takes into account the changing behavior of the objective function by re-analyzing the objective terrain and changing the search direction during the optimization [39,40]. 

The ideabehindthis remains acentral objective that such improvements may be reflected in terms of obstacle avoidance, response speed, trajectory optimization, and achievement of objectives. 

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