Free Final Time

We first consider optimal control problems in which the final time is free, i.e., not specified or fixed. Thus, in addition to finding optimal control functions, we need to determine the optimal final time as well in these problems. 

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Integral constraints could be equality or inequality constraints. We first consider integral equality constraints in an optimal control problem with free state and free final time. 

Free Final Time and Free Final State... 

The optimal / is —7.51, corresponding to the final product concentration of 7.51 g/cm. When the final time was free and available for optimization in the last example, we got better results — more production in less time. The reason is that free final time affords more freedom (than when it is fixed) in the optimal control of a process. 

Free Final Time and Final State on Hypersurfaces... 

Because the Stokes pulse precedes but overlaps the pump pulse, initially Up and all population initially in field-free state 11) coincides with flo(0)- At the final time, ilp Q5 so all of the population in flo(0) projects onto the target state 6). Note that flo(0) has no projeetion on the intermediate field-free state 5 ). The Rabi frequencies of the Stokes and pump pulses that are required for efficient STIRAP-generated population transfer satisfy the condition [66]... 

Table 2.2 shows timescales required to increase the mass by given mass increments and average mass-infall rates during these periods for three different final masses of low-mass stars. The formation of the initial stellar embryo requires about one free-fall time and is very short. The subsequent addition of 80% of the final mass requires about four free-fall times, i.e. is rather rapid. The mass accretion rate... 

This is the optimal control problem of Section 5.2 (p. 126) in which both the final time and the final state are unspecified or free. Using vectors, the objective is to minimize the functional... 

Table 7.1 lists the parameters used to solve the optimal control problem with the above algorithm. In this problem, the final time and final states are free. It is desired to find two controls, u t) and U2 t), and the final time tf that maximize the final product concentration 2/4(tf). 

The term in Nf dNf vanishes, for it is just the transversality condition imposed as a final boundary condition on the adjoint function such that the Lagrangian should be stationary. For an optimum system, SL is nonnegative. If, as is true with a free end time, St is arbitrary in sign, then Hf must be zero. Being zero at the final time and constant during the control period, the Hamilton density of an autonomous optimum free end time system is identically zero. 

Finally I should like to thank, for their continuous cooperation and help, the members of the Regional Computing Center of the University, and especially its Director, Dr. F. Wolf for his understanding and for providing as much free computer time as possible for this work and also for helping to obtain time on the computer of the Leibnitz Computing Center of the Technical University of Munich. 

In an irreversible process the temperature and pressure of the system (and other properties such as the chemical potentials to be defined later) are not necessarily definable at some intemiediate time between the equilibrium initial state and the equilibrium final state they may vary greatly from one point to another. One can usually define T and p for each small volume element. (These volume elements must not be too small e.g. for gases, it is impossible to define T, p, S, etc for volume elements smaller than the cube of the mean free... 

More complex ions are created lower in the atmosphere. Almost all ions below 70-80 km are cluster ions. Below this altitude range free electrons disappear and negative ions fonn. Tln-ee-body reactions become important. Even though the complexity of the ions increases, the detemiination of the final species follows a rather simple scheme. For positive ions, fomiation of H (H20) is rapid, occurring in times of the order of milliseconds or shorter in the stratosphere and troposphere. After fomiation of H (H20), the chemistry involves reaction with species that have a higher proton affinity than that of H2O. The resulting species can be... 

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