Optimal control theory solution

The above formulation is a well posed problem in optimal control theory and its solution can be obtained by the application of Pontryagin s Minimum Principle (Sage and White (1977)). 

Appendix F. Convergence of the Iterative Solution of the Optimal Control Theory Equations... 

APPENDIX F CONVERGENCE OF THE ITERATIVE SOLUTION OF THE OPTIMAL CONTROL THEORY EQUATIONS... 

Messina et al. consider a system with two electronic states g) and e). The system is partitioned into a subset of degrees of freedom that are to be controlled, labeled Z, and a background subset of degrees of freedom, labeled x the dynamics of the Z subset, which is to be controlled, is treated exactly, whereas the dynamics of the x subset is described with the time-dependent Hartree approximation. The formulation of the calculation is similar to the weak-response optimal control theory analysis of Wilson et al. described in Section IV [28-32], The solution of the time-dependent Schrodinger equation for this system can be represented in the form... 

For the numerical solution of optimal control problems, there are basically two well-established approaches, the indirect approach, e. g., via the solution of multipoint boxmdary-value problems based on the necessary conditions of optimal control theory, and the direct approach via the solution of constrained nonlinear programming problems based on discretizations of the control and/or the state variables. The application of an indirect method is not advisable if the equations are too complicated or a moderate accuracy of the numerical solution is commensurate with the model accuracy. Therefore, the easier-to-handle direct approach has been chosen here. Direct collocation methods, see, e. g., Stryk [6], as well as direct multiple shooting methods, see, e. g., Bock and Plitt [1], belong to this approach. In view of forthcoming large scale problems, we will focus here on the direct multiple shooting method, since only the control variables have to be discretized for this method. This leads to lower dimensional nonlinear programming problems. 

Numerical optimisation approaches have been traditionally used in design of selective excitation and inversion pulses. This usually involved a variation of a small number of parameters that describe a subset of pulse shapes. Kobzar et al proposed to apply the optimal control theory to remove the restriction of the predetermined pulse shape and to obtain a general solution that meets the criteria of maximum rf-amplitude, maximum pulse duration, temporal digitisation of the pulse and compensations for Bi-field inhomogeneity. The application of the method allowed to improve most of the published selective pulses. The method was extended further to optimise coherence transfer steps in coupled spin systems.The reported optimised propagators maximised the... 

Tailored polymer resins are frequently required for a given application. Fontoura et al. used NIR spectroscopy for in-line and in situ monitoring and control of monomer conversion and polymer average molecular weight during styrene solution polymerization. Two process control strategies, one based on the optimal control theory and the other on model predictive control, were implemented both theoretically and experimentally [67]. 

Method of Solution The fundamental numerical problem of optimal control theory is the solution of the two-point boundary-value problem, which invariably arises from the application of the maximum principle to determine optimal control profiles. The state and... 

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

The situation is quite different when inequality constraints are included in the MPC on-line optimization problem. In the sequel, we will refer to inequality constrained MPC simply as constrained MPC. For constrained MPC, no closed-form (explicit) solution can be written. Because different inequahty constraints may be active at each time, a constrained MPC controller is not linear, making the entire closed loop nonlinear. To analyze and design constrained MPC systems requires an approach that is not based on linear control theory. We will present the basic ideas in Section III. We will then present some examples that show the interesting behavior that MPC may demonstrate, and we will subsequently explain how MPC theory can conceptually simplify and practically improve MPC. 

Kupriyanov studied relaxation of spin 1/2 in the scalar coupled spin system AMX with quadrupolar nuclei in the presence of cross correlation effects. Khaneja et a/. presented optimal control of spin dynamics in the presence of relaxation. Eykyn et a/. studied selective cross-polarization in solution state NMR of scalar coupled spin 1/2 and quadrupolar nuclei. Tokatli applied the product operator theory to spin 5/2 nuclei. Mahesh et a/. used strongly coupled spins for quantum information processing. Luy and Glaser " inves-... 

Nonlinear mathematical programming. For a wide variety of problems concerning the optimal control of chemical-engineering processes, the mathematical programming (exploratory methods) is the most routine tool to obtain numerical solutions [13,16-20,24,25]. In contrast to the classical optimization theory, in mathematical programming special... 

Theory and numerical methods for the solution of optimal control problems have reached a high standard. There is a wide range of applications, the most challenging of which are from the field of aerospace engineering and robotics see for example the survey paper [5]. 

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